A Physically Constrained KPP–Rate-and-State Reaction–Diffusion Framework for Stable Large-Scale Modeling of Stress Evolution

Abstract

The emergence of large-scale models and machine learning has transformed the modeling of complex nonlinear systems, such as postseismic stress evolution. However, purely data-driven approaches often lack interpretability and numerical stability, leading to physically inconsistent long-term predictions. This study addresses these limitations by introducing a coupled Kolmogorov–Petrovsky–Piskunov–Rate-and-State (KPP–RS) reaction–diffusion system as a rigorous physical prior for large-scale modeling of stress-driven dynamics. Using analytic semigroup theory and Banach’s fixed-point theorem, we establish the global existence and uniqueness of solutions, ensuring that the governing dynamics are mathematically well posed—a necessary prerequisite for stable learning-based frameworks. We further prove the global dissipativity of the system and identify a bounded absorbing set in the H¹ phase space, which imposes intrinsic physical constraints and limits unphysical parameter exploration in large-scale optimization or black-box modeling. In addition, a Courant–Friedrichs–Lewy (CFL) stability condition is derived, providing a theoretical benchmark for time-step selection in numerical implementations, including physics-informed or hybrid neural architectures. This analytical framework supplies a mathematically controlled foundation for developing robust, interpretable, and stable pattern-recognition or time-series representations in complex geophysical systems.

1. Introduction

The rapid advancement of large-scale models and deep learning algorithms has significantly transformed the landscape of complex nonlinear system modeling. In geophysics, these data-driven approaches are increasingly employed to recognize spatiotemporal patterns and classify seismic activities. However, a major challenge persists: purely black-box models often struggle with interpretability and numerical stability. When applied to postseismic stress evolution, these models may produce predictions that violate fundamental physical principles, especially during long-term simulations where error accumulation can lead to unphysical results. This highlights the urgent need for physics-informed frameworks that can provide rigorous mathematical constraints for large-scale learning-based representations.

This study addresses these limitations by introducing a coupled Kolmogorov–Petrovsky–Piskunov–Rate-and-State (KPP–RS) reaction–diffusion system. Unlike traditional empirical models, the KPP–RS system offers a deterministic backbone that ensures the physical consistency of stress-driven dynamics. By employing analytic semigroup theory, we demonstrate that the system is mathematically well-posed—a critical property that guarantees the existence of stable features for machine learning classifiers. Furthermore, our analysis of global dissipativity and the identification of a bounded absorbing set in H¹ space provide intrinsic boundaries for large-scale optimization algorithms. These mathematical bounds effectively constrain the parameter search space, preventing the model from exploring unphysical states during the training process.

Reaction–diffusion equations have long provided a fundamental framework for modeling nonlinear propagation and saturating growth. Classical works by Fisher [1] and subsequent developments of the Fisher–KPP equation [2] revealed how logistic reaction terms coupled with diffusion generate bounded, dissipative dynamics, inspiring broad applications in nonlinear PDE theory and complex systems [3,4]. Such models offer a natural mathematical structure for systems in which local amplification competes with spatial redistribution, leading to controlled long-time behavior.

Earthquake processes exhibit closely analogous characteristics. Fault-zone degradation, microcrack accumulation, and stress amplification often follow a logistic-type growth pattern, as demonstrated by distributed damage mechanics [5] and nonlinear viscoelastic rheology [6]. In this context, logistic reaction terms represent the saturation of stress or damage as faults approach their strength limits. Laboratory and numerical studies further reveal accelerating deformation near failure [7], crackling noise consistent with critical dynamics [8], and hierarchical rupture precursors [9]. Foundational studies in geodynamics [10] and comprehensive reviews of earthquake mechanics [11] consistently emphasize the roles of saturation and dissipation in brittle deformation and in the evolution of fault strength.

Stress redistribution introduces additional layers of complexity. Aftershock migration has been associated with diffusion-like or diffusion-controlled processes [12], postseismic relaxation [13], cascading triggering [14], and dynamic stress perturbations [15]. Fluid-induced stress transfer can also generate diffusion-controlled clustering [16], and numerical models combining diffusion with seismicity production successfully reproduce observed aftershock patterns [17]. Reaction–diffusion formulations have likewise been applied to volcanic and tectonic systems [18], reinforcing the relevance of diffusion operators in geophysical stress modeling.

Fault friction provides a third essential mechanism. The rate-and-state friction (RSF) theory developed by Dieterich [19] and Ruina [20] explains how friction evolves with slip and internal state variables. RSF-based PDE models reproduce nucleation, slip acceleration, and rupture complexity across multiple scales [21,22]. Thermal weakening and shear localization further influence fault instability and energy dissipation during rupture [23]. Recent advances in physics-informed machine learning have demonstrated the capability to retrieve RSF parameters directly from experimental data [24], and the studies demonstrate that RSF state evolution governs aftershock productivity and temporal decay [25]. Statistical analyses of earthquake clustering reveal fractal migration, scale-free behavior, and triggering cascades [26], suggesting that aftershock sequences cannot be adequately described by static Coulomb stress transfer alone.

Despite extensive progress in nonlinear reaction, stress diffusion, and frictional state modeling, to the best of our knowledge, no prior study has unified all three mechanisms into a mathematically rigorous PDE framework with proven existence, boundedness, and numerical stability. Many aftershock models rely on simulation-based approaches that do not address fundamental questions of well-posedness or long-time control, thereby limiting their physical interpretability and predictive reliability.

In this work, we construct and analyze a coupled KPP–RS reaction–diffusion system that integrates

• Nonlinear stress amplification with logistic saturation.
• Spatial redistribution through diffusion.
• Rate-and-state-controlled frictional memory.

Using analytic semigroup theory and a Banach fixed-point argument, we establish local well-posedness. Global existence and boundedness are then proven through comparison principles and energy estimates. In addition, a Courant–Friedrichs–Lewy (CFL) condition is derived to ensure numerical stability for explicit discretizations, with convergence established within the Lax–Richtmyer framework. From a computational perspective, the lack of well-posedness and boundedness in governing equations poses a fundamental limitation for large-scale or learning-based modeling, where stability and reproducibility are essential.

By proving the existence of a bounded absorbing set, this study provides a rigorous foundation for future investigations into the ergodic behavior and statistical predictability of stress-driven cascades. Within this framework, long-term stress relaxation may be interpreted as a deterministic constraint on cumulative energy release, offering a physically meaningful perspective on the concept of seismic “energy debt.” Beyond seismology, the proposed KPP–RS system contributes to the broader theory of nonlinear reaction–diffusion systems with memory, clarifying how bounded dissipative dynamics can generate complex, seemingly chaotic spatiotemporal patterns in geophysical fields.

The present study focuses on the rigorous functional-analytic analysis of the coupled KPP–RS reaction–diffusion system. In particular, we establish global existence, uniqueness, and dissipativity of solutions, which are fundamental requirements for stable large-scale modeling of stress-driven dynamics. The identification of a bounded absorbing set in the H¹ phase space further ensures long-term boundedness of the stress–state evolution and prevents unphysical divergence in numerical simulations. Empirical applications, parameter calibration, and forecasting performance of the Dynamic Stress Gradient Framework (DSGF) have been investigated in a companion study [27] recently accepted for publication in Nonlinear Processes in Geophysics. The results developed here provide the theoretical framework ensuring well-posedness and long-term boundedness of the applied model.

2. Materials and Methods

2.1. Materials

This section provides a complete mathematical foundation for the coupled KPP–RS reaction–diffusion system, which models postseismic stress evolution and fault-state interaction. We establish local well-posedness, prove global existence under dissipative assumptions, and derive a discrete stability condition consistent with physical constraints and numerical implementation. All proofs are presented in a unified and rigorous framework.

2.1.1. Functional Setting and Preliminaries

Let $\Omega \subset {\mathbb{R}}^2$ be a bounded, smooth domain with unit outward normal $\mathit{n}$. We study the coupled PDE system:

$$\left\{\begin{array}{c}{\displaystyle \raisebox{1ex}{$\partial \sigma $}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.} = D_{\sigma }∆\sigma + F\left(\sigma , V\right), \mathrm{i}\mathrm{n} \Omega \times \left(0, T\right)\\ {\displaystyle \raisebox{1ex}{$\partial V$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.} = \mathrm{G}\left(\sigma , V\right),\mathrm{i}\mathrm{n} \Omega \times \left(0, T\right)\end{array}\right.$$

(1)

Initial conditions:

$$\sigma (0, \mathit{x}) = {\sigma }_{0, } V\left(0, \mathit{x}\right) ={ V}_0\mathit{x} \mathrm{in} \Omega$$

Boundary condition:

The stress field satisfies homogeneous Neumann boundary conditions:

$${\partial }_n\sigma =0 \mathrm{on} \partial \Omega \times \left(0,T\right)$$

(2)

Physically, this condition represents a traction-free or no-flux boundary, indicating the absence of artificial stress injection across the domain boundary. In geophysical applications, such an assumption corresponds to an isolated fault patch or a computational subdomain in which boundary stress flux is negligible—an approach commonly adopted in regional-scale postseismic modeling. From a mathematical standpoint, the specification of homogeneous Neumann conditions is essential for defining the diffusion operator with a proper domain and for guaranteeing the generation of an analytic semigroup. This, in turn, ensures the well-posedness and stability properties established in the subsequent analysis.

No boundary condition is required for V, because Equation (1) contains no spatial derivatives.

In this system,

• σ denotes the stress field;
• V is the slip-rate or state variable;
• F and G describe nonlinear couplings of KPP-type (reaction–diffusion) and Rate-and-State frictional feedback.
• D_σ > 0 is a diffusivity coefficient;

We define

$$\mathbf{X}={ H}^1\left(\Omega \right) \times L^2\left(\Omega \right)$$

$$\parallel \left(\sigma (t), V(t)\right){\parallel }_{\mathrm{x}}=\parallel \sigma \left(t\right){\parallel }_{H^1\left(\Omega \right)}+\parallel V\left(t\right){\parallel }_{L^2\left(\Omega \right)}$$

$${\mathbf{X}}_T=\mathrm{C}\left(\left[0, T\right]; \mathbf{X}\right)$$

$${\parallel \left(\sigma ,V\right)\parallel }_{{\mathbf{X}}_T}=\underset{t\in \left\{0, T\right]}{\sup }\left(\parallel \left(\sigma ,V\right)\left(t\right){\parallel }_{\mathbf{X}}\right)$$

where σ∈ C([0, T]; H¹(Ω)) and V∈ C([0, T]; L²(Ω)). We clarified that ${\parallel \sigma \parallel }_{H^1\left(\Omega \right)}$ denotes the standard $H^1$-norm, i.e.,

${\parallel \sigma \parallel }_{H^1\left(\Omega \right)}={\left(\parallel \sigma {\parallel }_{L^2\left(\Omega \right)}^2+ \parallel \mathsf{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2\right)}^{1/2}$ which includes both the zero-order and first-order terms.

The functional space $\mathbf{X}=H^1\left(\Omega \right)\times L^2\left(\Omega \right)$ is chosen not arbitrarily but to reflect the physical structure of postseismic stress evolution. The stress field σ(x, t) necessarily admits spatial gradients, which are required to model stress diffusion and redistribution. Therefore, σ belongs to H¹(Ω) by definition. In contrast, the rate-and-state variable V(x, t) has no spatial derivative in the governing equations, and thus L²(Ω) is the most general and physically meaningful space. This functional setting is consistent with the standard theory of reaction–diffusion systems and accurately reflects the physical nature of dynamic stress evolution in the crust.

The coupled KPP–RS system used in this work is defined by two nonlinear reaction terms:

• KPP-type stress reaction term

In the present KPP–RS system, the coupling to the state variable is introduced through the rate-and-state law $G\left(\sigma ,V\right)$. The stress reaction term is chosen as a pure KPP-type logistic nonlinearity and is independent of $V$; hence we write $F\left(\sigma \right)$ instead of $F\left(\sigma ,V\right)$.:

$$F\left(\sigma \right)=a\sigma -b{\sigma }^2$$

(3)

where $a>0$ represents stress amplification and $b>0$ enforces saturation, consistent with distributed-damage mechanics and KPP-type dissipation.

This choice ensures:

(i)

A global logistic upper bound $\sigma \le a/b$

(ii)

Polynomial-type nonlinearities required for Sobolev embedding.

(iii)

Satisfaction of the dissipativity condition in Assumption 3.

2.

RSF evolution term

For the rate-and-state variable, we employ a standard form consistent with the Dieterich–Ruina friction laws.

$$G\left(\sigma ,V\right)=\alpha -\beta V+\gamma \sigma$$

(4)

where $\alpha , \beta , \gamma >0$ are material constants. A concise interpretation of these parameters is as follows:

(i)

α (Baseline Forcing): Represents a baseline forcing term or tectonic loading that drives the continuous evolution of the state variable.

(ii)

β (Relaxation/Healing Rate): Characterizes the linear relaxation or healing of the state variable. The reciprocal 1/β defines the intrinsic relaxation timescale, governing how fast the system dissipates perturbations

(iii)

γ (Stress Sensitivity): Measures the sensitivity of the state variable to stress variations, introducing a linear stress–state interaction that accounts for stress-dependent weakening or strengthening mechanisms.

In addition, the term $-\beta V$ introduces linear damping; $\gamma \sigma$ incorporates stress–dependent weakening or strengthening.

This form satisfies:

(i)

Local Lipschitz continuity;

(ii)

Bounded invariant interval for $V$ (Assumption 4);

(iii)

Linear growth structure required for Assumption 1.

Although the general framework allows fully coupled nonlinearities $F\left(\sigma ,V\right)$, in the present study the stress reaction term is chosen to be independent of $V$, i.e., $F\left(\sigma ,V\right)\equiv F\left(\sigma \right)$. The coupling between stress and frictional state is introduced solely through the rate-and-state evolution $G\left(\sigma ,V\right)$. This triangular coupling structure preserves the KPP dissipative mechanism and simplifies the energy analysis without loss of physical relevance.

The mathematical analysis in this section assumes that all coefficients—such as the diffusion coefficient $D_{\mathit{\sigma }}$, the KPP parameters $a,b$ and the rate-and-state parameters $\alpha ,\beta$—are strictly positive constants; this assumption is not in contradiction with the physical model presented in our companion study [27], where these coefficients may vary slowly in time. While the companion study [27] focuses on empirical forecasting performance and practical implementation of the DSGF framework, the present manuscript is devoted to its rigorous mathematical analysis. In particular, we establish existence, uniqueness, dissipativity, and stability properties of the coupled nonlinear KPP–RS system. The results presented here provide the theoretical foundation that guarantees well-posedness and long-term boundedness of the framework underlying those applied investigations. The constant-coefficient system should be viewed as the core mathematical backbone of the full model. In practice, immediately after a major earthquake, the physical parameters governing stress diffusion and frictional state evolution change only gradually; thus, over short time intervals, the system is well approximated by a constant-coefficient formulation. Analytically, the constant-coefficient case provides the key semigroup bounds and an a priori estimate that persists under bounded, slowly varying perturbations. Consequently, the well-posedness and stability properties established here carry over to the time-dependent coefficient model, ensuring that its short-time or quasi-steady dynamics remain mathematically controlled.

2.1.2. Standing Assumptions

For the parabolic diffusion equation governing the stress field,

$${\partial }_t\sigma =D_{\sigma }\Delta \sigma +F\left(\sigma \right),F\left(\sigma \right)=a\sigma -b{\sigma }^2$$

is essential to specify boundary conditions on $\partial \Omega$. In the present formulation, the stress reaction term depends only on the stress variable, i.e., $F\left(\sigma ,V\right)\equiv F\left(\sigma \right)$. The coupling to the state variable is introduced through $G\left(\sigma ,V\right)$. In this work, the stress field $\sigma$ is assumed to satisfy homogeneous Neumann boundary conditions, representing a traction-free or no-flux constraint on the fault-plane boundary:

$$\begin{array}{cccc}& {\partial }_{\mathit{n}}\sigma =0 \mathrm{on} \partial \Omega \times \left(0,T\right),& & \left(\mathrm{BC}\right)\end{array}$$

where $\mathit{n}$ denotes the outward unit normal vector along $\partial \Omega$. This boundary condition directly determines the domain of the diffusion operator

Definition of the diffusion operator:

Let A = D_σΔ with homogeneous Neumann boundary conditions, considered as an unbounded operator on $L^2\left(\Omega \right)$ with domain

$$D\left(A\right)=u\in H^2\left(\Omega \right):{\partial }_{\mathit{n}}u=0 \mathrm{on} \partial \Omega$$

With this domain, the Laplacian $\mathsf{\Delta }$ is self-adjoint and non-positive on $L^2\left(\Omega \right)$, and $\mathit{A}$ generates a strongly continuous analytic semigroup $\left\{S_{\sigma }\right(t){\}}_{t\ge 0}$, which is used in the mild-solution formulation established later. The explicit specification of boundary conditions is therefore mathematically indispensable, as it ensures:

• The well-definedness of the diffusion operator $A$;
• The validity of energy estimates and integration-by-parts identities;
• The applicability of analytic semigroup theory for establishing local existence and uniqueness.

To guarantee well-posedness and the physical admissibility of the coupled KPP–RS system, we impose the following structural conditions on the nonlinear reaction terms F(σ) and G(σ, V).

Assumption 1.

Local Lipschitz continuity and polynomial growth.

Let $\mathbf{X} = H^1\left(\Omega \right) \times L^2\left(\Omega \right)$. The nonlinear maps $F:H^1(\Omega ) \to L^2(\Omega )$ and $\mathrm{G}:\mathbf{X} \to L^2(\Omega )$ are locally Lipschitz on bounded sets: for every $R > 0$, there exist constants $L_F\left(R),{ L}_{\mathrm{G}}\right(R) > 0$ such that for all pairs $({\sigma }_1, V_1)$ and $({\sigma }_2,{ V}_2)$ in X with $\parallel {\sigma }_{\mathrm{i}}{\parallel }_{H^1} + \parallel V_{\mathrm{i}}{\parallel }_{L^{2 }}\le R$ (for i = 1, 2),

$$\parallel F\left({\sigma }_1\right)-F\left({\sigma }_2\right){\parallel }_{L^2} \le { L}_F\left(R\right)\hspace{0.17em}\parallel {\sigma }_1-{ \sigma }_2{\parallel }_{H^1}$$

(5)

$$\parallel \mathrm{G}({\sigma }_1,{ V}_1)-\mathrm{G}({\sigma }_2,V_2){\parallel }_{L^2} \le { L}_{\mathrm{G}}(R\left)\right(\parallel {\sigma }_1-{\sigma }_2{\parallel }_{H^1}+\parallel V_1-V_2{\parallel }_{L^2})$$

(6)

Moreover, $F$ and $\mathrm{G}$ satisfy the following growth bounds: there exist constants ${\mathrm{C}}_F,{\mathrm{C}}_{\mathrm{G}} > 0$ such that for all $(\sigma , V) \in \mathbf{X}$,

$$\parallel F\left(\sigma \right){\parallel }_{L^2}\le C_F\left(1+\parallel \sigma {\parallel }_{H^1}\right)$$

(7)

$$\parallel G(\sigma ,V){\parallel }_{L^2}\le C_G\left(1+\parallel \sigma {\parallel }_{H^1}+\parallel {V\parallel }_{L^2}\right)$$

(8)

Remark 1.

Two-dimensional setting and Sobolev embeddings.

Throughout this work, we assume that the spatial domain $\Omega \subset {\mathbb{R}}^2$ is a bounded, connected domain with sufficiently smooth boundary. The analysis is, therefore, carried out in a two-dimensional setting. The spatial domain is adopted as a planar approximation, consistent with standard two-dimensional fault-plane modeling commonly employed in regional stress analyses. This representation captures the dominant along-strike and down-dip stress variations while maintaining analytical tractability. In two dimensions, the Sobolev embedding theorem implies that

$$H^1\left(\Omega \right)\hookrightarrow L^p\left(\Omega \right), \mathrm{for} \mathrm{all} 2\le p<\mathrm{\infty }.$$

which provides the necessary control over the KPP-type quadratic nonlinearities. This embedding ensures that nonlinear reaction terms remain bounded within the energy framework, thereby enabling the derivation of global dissipativity and absorbing set estimates.

In particular, there exists a constant $C_{\Omega }>0$ such that

$${\parallel u\parallel }_{L^4(\Omega )}\le C_{\Omega }\parallel u{\parallel }_{H^1(\Omega )},\forall \hspace{0.17em}u\in H^1(\Omega ).$$

This embedding plays a crucial role in controlling the nonlinear reaction terms appearing in the KPP-type stress evolution equation. For example, quadratic nonlinearities of the form ${\sigma }^2$ satisfy

$${\parallel {\sigma }^2\parallel }_{L^2\left(\Omega \right)}=\parallel \sigma {\parallel }_{L^4\left(\Omega \right)}^2 \le C_{\Omega }\parallel \sigma {\parallel }_{H^1(\Omega )}^2,$$

which allows the nonlinear terms to be estimated consistently within the energy framework. We emphasize that the two-dimensional assumption is essential for the above embedding properties and is consistent with the planar approximation commonly adopted in regional-scale postseismic stress modeling.

Assumption 2.

Analytic Semigroup and Smoothing Estimates.

The diffusion operator $A=D_{\sigma }∆$ with homogeneous Neumann boundary conditions generates a strongly continuous analytic semigroup $\left\{S_{\sigma }\right(t){\}}_{t\ge 0}$ on $L^2\left(\Omega \right)$. Moreover, for all $tt>0$, the operator

$$S_{\sigma }\left(t\right):L^2\left(\Omega \right)\to H^1\left(\Omega \right)$$

is bounded and satisfies the smoothing estimates:

$${‖S_{\sigma }\left(t\right)f‖}_{H^1}\le C_{\Omega }{‖f‖}_{H^1}$$

$${‖S_{\sigma }\left(t\right)f‖}_{L^2}\le C_{\Omega }{t}^{{\displaystyle \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{‖f‖}_{L^2}$$

(9)

for all t > 0. Here, $C_{\Omega }$ > 0 is the semigroup bound, defined as

$$C_{\Omega }=\underset{t\ge 0}{\mathrm{s}\mathrm{u}\mathrm{p}}\left(\parallel S_{\sigma }\left(t\right){\parallel }_{{\mathcal{L}(H}^1)}\right)$$

which is intrinsically determined by the geometry and Poincaré characteristics of the domain $\Omega$ and is consistent with Remark 1. The constant $C_{\Omega }$ also expresses the fact that the diffusion part of the stress evolution does not amplify the H¹ norm and, thus, never amplifies the stress gradients. In physical terms, diffusion always smooths stress rather than creating new peaks. Thus, the evolution operator is uniformly bounded.

Assumption 3.

KPP-Type Dissipativity (Logistic Upper Bound).

There are two parameters a, b > 0 such that:

$$F\left(\sigma \right)\le a\sigma -b{\sigma }^2,\mathrm{for}\mathrm{all}\sigma \in \mathrm{R}$$

This implies, by the comparison principle, σ(x, t) ≤ a/b for all t ≥ 0, providing a uniform L^∞ bound for the stress field.

Assumption 4.

Invariance and Boundedness of the State Variable.

There exists a bounded interval $𝒱\subset \mathrm{R}$ such that if $V_0\left(\mathit{x}\right)\in 𝒱$ for almost every $\mathit{x}\in \Omega$, then the corresponding solution satisfies $V\left(\mathit{x},t\right)\in 𝒱$ for all $t\ge 0$. Moreover, the function $G\left(\sigma ,V\right)$ is locally Lipschitz continuous on $H^1\left(\Omega \right)\times 𝒱$.

In addition, since the stress field $\sigma$ is uniformly bounded in time (Lemma 1), the evolution equation

$${\partial }_{t}V\left(\mathit{x},t\right)=G\left(\sigma \left(\mathit{x},t\right),V\left(\mathit{x},t\right)\right)$$

can be regarded, for each fixed $\mathit{x}\in \Omega$, as a non-autonomous ordinary differential equation with uniformly bounded coefficients. Classical ODE comparison arguments then imply the existence of a compact absorbing interval $\left[0,M_V\right]\subset 𝒱$, with $M_V>0$, such that

$$0\le V\left(\mathit{x},t\right)\le M_V \mathrm{for} \mathrm{all} t\ge 0 \mathrm{and} \mathrm{a}.\mathrm{e}. \mathit{x}\in \Omega .$$

This prevents blow-up of the frictional state variable and ensures physical admissibility. In practice, a bounded invariant interval $𝒱$ is physically justified because the frictional state variable V represents an internal variable with a finite admissible range (e.g., contact age, state of asperity population, or slip memory). Many widely used rate-and-state laws, including Dieterich–Ruina aging and slip laws, naturally restrict V to a finite domain. Logistic-type dissipation is one possible mechanism ensuring such invariance but is not needed for the present analysis. These parameters can be interpreted in

Table 1. Mathematical–Physical Interpretation of Assumptions 1–4.

Label	Mathematical Meaning	Physical Interpretation
1	Nonlinear terms F and G are locally Lipschitz and satisfy linear growth conditions.	Stress interaction and frictional response evolve continuously with respect to perturbations. Excludes instantaneous or singular amplification mechanisms in the modeled system.
2	The diffusion operator DσΔ with homogeneous Neumann boundary conditions generates an analytic semigroup on L2(Ω), ensuring smoothing and bounded evolution.	Represents spatial redistribution of stress through diffusive mechanisms (e.g., viscoelastic relaxation or fluid-assisted diffusion). Imposes spatial regularization of stress variations.
3	KPP logistic-type upper bound: σ ≦ a/b	Encodes a saturation mechanism limiting stress growth. Reflects finite fault strength or bounded stress accumulation in the continuum approximation.
4	The state variable V remains in a bounded invariant set $𝒱$	Prevents unbounded evolution of the rate-and-state variable. Represents physically admissible ranges for frictional state evolution.

.

2.2. Local Well-Posedness (Banach Fixed-Point Framework)

2.2.1. Definition of the Fixed-Point Operator

We employ the mild formulation under the Neumann semigroup. A pair (σ, V) ∈ X_T is called a mild solution if it satisfies for all t ∈ [0, T]

$$\sigma \left(t\right)=S_{\sigma }\left(t\right){\sigma }_0+{\int }_0^t{S}_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right), V\left(s\right)\right)ds$$

(10)

$${V\left(t\right)=V}_0+{\int }_0^{t}G\left(\sigma \left(s\right), V\left(s\right)\right)ds$$

(11)

The mild formulation under the analytic semigroup framework follows the classical theory of [28,29].

Define the operator $\mathit{\Phi }:{\mathbf{X}}_T \to {\mathbf{X}}_T$ by

$$\mathit{\Phi }(\sigma , V)=\left({\Phi }_1\right(\sigma , V), {\Phi }_2(\sigma , V\left)\right)$$

with $\left({\Phi }_1, {\Phi }_2\right)$ given as the right-hand sides of Equations (10) and (11). By definition, a mild solution on [0, T] is precisely a fixed point of Φ, i.e.,

$$\mathit{\Phi }(\sigma , V)=(\sigma , V)$$

2.2.2. Mapping Φ into a Closed Ball in X_T

Let R > 0 and define the closed ball ${\mathbf{B}}_R=\{\left(\sigma ,V\right)\in {\mathbf{X}}_T:‖({\sigma , V)‖}_{{\mathbf{X}}_T}\le R\}$. Since X_T is a Banach space, the closed ball B_R is complete. Once we show that for T > 0 sufficiently small, Φ(B_R)⊂B_R and Φ is a strict contraction on B_R. Therefore, Banach’s Fixed-Point Theorem ensures the existence of a unique fixed point. First, the estimate for Φ₁ is provided. Take any (σ, V) ∈ B_R and t ∈ [0,T], based on Equations (10) and (11), we get:

$$\left\{\begin{array}{c}{\Phi }_1\left(\sigma ,V\right)\left(t\right)=S_{\sigma }\left(t\right){\sigma }_0+{\int }_0^t{S}_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right), V\left(s\right)\right)ds\\ {\Phi }_2\left(\sigma ,V\right)\left(t\right)=V_0+{\int }_0^{t}G(\left(\sigma \left(s\right), V\left(s\right)\right)ds\end{array}\right.$$

From the property of H¹-norm,

$$\parallel {\Phi }_1\left(\sigma ,V\right)\left(t\right){\parallel }_{H^1}\le \parallel S_{\sigma }\left(t\right){\sigma }_0{\parallel }_{H^1}+\parallel {\int }_0^t{S}_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right),V\left(s\right)\right)ds{\parallel }_{H^1}$$

The following expression can be obtained by applying the constant $C_{\Omega }$, so the superior upper bound of $\parallel S_{\sigma }\left(t\right){\parallel }_{{\mathcal{L}(H}^1)}$ in Assumption 2

$$\parallel S_{\sigma }\left(t\right){\sigma }_0{\parallel }_{H^1}\le C_{\Omega }\parallel {\sigma }_0{\parallel }_{H^1}$$

For the integration term of right-hand side in ${\mathsf{\Phi }}_1\left(\sigma ,V\right)\left(t\right)$, the smooth effect of $L^2\to H^1$ is utilized to derive as:

$$\parallel {\int }_0^t{S}_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right),V\left(s\right)\right)ds{\parallel }_{H^1}\le {\int }_0^t\parallel S_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{H^1}ds$$

(12)

Applying the estimation of semigroup

$$\parallel S_{\sigma }\left(t\right)f{\parallel }_{L^2}\le C_{\Omega }\parallel f{\parallel }_{L^2}, \parallel \mathsf{\nabla }{S}_{\sigma }\left(t\right)f{\parallel }_{L^2}\le {C_{\Omega }t}^{-1/2}\parallel f{\parallel }_{L^2}$$

Hence, $\parallel S_{\sigma }\left(t\right)f{\parallel }_{H^1}\le \parallel S_{\sigma }\left(t\right)f{\parallel }_{L^2}+\parallel \mathsf{\nabla }{S}_{\sigma }\left(t\right)f{\parallel }_{L^2}\le {C_{\Omega }(1+t}^{-1/2})\parallel f{\parallel }_{L^2}$

We obtain

$$\parallel S_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{H^1}\le C_{\Omega }(1+(t-s{)}^{-1/2})\parallel F\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{L^2}$$

(13)

Therefore, Substituting Equations (13) into (12)

$$\parallel {\int }_0^t{S}_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right),V\left(s\right)\right)ds{\parallel }_{H^1}$$

$$\le C_{\Omega }{\int }_0^t(1+(t-s{)}^{-{\displaystyle \frac{1}{2}}})\parallel F\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{L^2}ds$$

$$\le C_{\Omega }\underset{0\le s\le T}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel F\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{L^2}{\int }_0^t(1+(t-s{)}^{-{\displaystyle \frac{1}{2}}})ds$$

$$=C_{\Omega }\left(t+2t^{1/2}\right)\underset{0\le s\le T}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel F\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{L^2}$$

(14)

Taking the supremum over $t\in \left[0,T\right]$,

$$\underset{0\le t\le T}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel {\int }_0^t{S}_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right),V\left(s\right)\right)ds{\parallel }_{H^1}\le C_{\Omega }\underset{0\le s\le T}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel F\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{L^2}\left(T+2T^{1/2}\right)$$

Using the linear growth property of Assumption 1 and the inequality $\parallel \sigma {\parallel }_{L^2}\le \parallel \sigma {\parallel }_{H^1}$, the last term of Equation (14) has the following relationship for all $\left(\sigma ,V\right)\in {\mathbf{B}}_R$ and $t\in \left[0,T\right]$:

$$\underset{t\in \left\{0,T\right]}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel F\left(\sigma \left(t\right),V\left(t\right)\right){\parallel }_{L^2}\le C_F\left(1+\underset{t\in \left\{0,T\right]}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel \sigma \left(t\right){\parallel }_{H^1}+\underset{t\in \left\{0,T\right]}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel V\left(t\right){\parallel }_{L^2}\right)\le C_F\left(1+R\right)$$

We can further obtain,

$$\underset{t\in \left\{0,T\right]}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel {\int }_0^t{S}_{\sigma }\left(t-s\right)F\left(\sigma \left(s\right),V\left(s\right)\right)ds{\parallel }_{H^1}\le C_F\left(1+R\right)\left(C_{\Omega }T+2C_{\Omega }{T}^{1/2}\right)$$

Combining the upper bound of $\parallel S_{\sigma }\left(t\right){\sigma }_0{\parallel }_{H^1}$ with above equation, the superior upper bound of ${\Phi }_1\left(\sigma ,V\right)\left(t\right)$ in H¹-norm has the following relation:

$$\underset{t\in \left\{0,T\right]}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel {\Phi }_1\left(\sigma ,V\right)\left(t\right){\parallel }_{H^1}\le C_{\Omega }\parallel {\sigma }_0{\parallel }_{H^1}+C_F\left(1+R\right)\left(C_{\Omega }T+2C_{\Omega }{T}^{1/2}\right)$$

(15)

where $C$ and $C_F$ are positive constants. Here $R$ > 0 is the radius of the closed ball ${\mathbf{B}}_R=\left\{\left(\sigma ,V\right)\in {\mathbf{X}}_T:‖({\sigma ,V)‖}_{{\mathbf{X}}_T}\le R\right\}.$ It makes the estimate consistent with the full decomposition in Equation (14).

Similarly, for (σ,V) ∈ B_R,

$$‖{\Phi }_2\left(\sigma ,V\right)\left(t\right){\parallel }_{L^2}\le \parallel V_0{\parallel }_{L^2}+{\int }_0^t‖G\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{L^2}ds$$

Using the linear growth of G in Assumption 1, we can obtain

$$\parallel G\left(\sigma \left(s\right),V\left(s\right)\right){\parallel }_{L^2}\le C_G\left(1+\parallel \sigma {\parallel }_{H^1}+\parallel V{\parallel }_{L^2}\right)\le C_G\left(1+\parallel \left(\sigma ,V\right){\parallel }_{X_T}\right)\le C_G\left(1+R\right)$$

It implies

$$\underset{t\in [0,T]}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel {\Phi }_2\left(\sigma ,V\right)\left(t\right){\parallel }_{L^2}\le \parallel V_0{\parallel }_{L^2}+C_G\left(1+R\right)T$$

(16)

where $C_G$ denotes the growth constant of G in (Assumption 1).

2.2.3. Self-Mapping Condition

According to Equations (15) and (16), it indicates that for all (σ, V) ∈ B_R the inequality equation describes the upper bound of $\parallel \mathit{\Phi }\left(\sigma ,V\right){\parallel }_{{\mathbf{X}}_T}$ can be expressed as:

$$\parallel \mathit{\Phi }\left(\sigma ,V\right){\parallel }_{{\mathbf{X}}_T}\le C_{\Omega }\parallel {\sigma }_0{\parallel }_{H^1}+\parallel V_0{\parallel }_{L^2}+(1+R)\left[\right(2C_F{C}_{\Omega }{T}^{1/2})+(C_{\Omega }{C}_F+C_G\left)T\right)]$$

where $C_{\Omega },C_F,C_G$ are positive constants. This corrected expression is then used to choose $R$ and $T$ such that $\mathsf{\Phi }$ maps $B_R$ into itself.

If R is chosen by the following condition

$$R=2\left(C_{\Omega }\parallel {\sigma }_0{\parallel }_{H^1}+\parallel V_0{\parallel }_{L^2}\right)$$

and a T > 0, which is quite small can be taken to satisfy the following equation:

$$M\left(T\right)=\left[\left(2C_F{C}_{\Omega }{T}^{1/2}\right)+\left(C_{\Omega }{C}_F+C_G\right)T\right]\le {\displaystyle \frac{R}{2\left(1+R\right)}}$$

then it can correspond to

$$\parallel \mathit{\Phi }\left(\sigma ,V\right){\parallel }_{{\mathbf{X}}_T}<R$$

that is Φ(B_R)⊂B_R.

2.2.4. Contraction Mapping

To apply the Banach Fixed-Point Theorem, we now show that Φ: B_R⊂X_T→X_T is a strict contraction for sufficiently small T. Let U_i = (σ_i,V_i)∈B_R (i = 1, 2) and δσ = σ₁ − σ₂, δV = V₁ − V₂, for any t ∈ [0, T], it has

$${\Phi }_1\left({\sigma }_1,V_1\right)\left(t\right)-{\Phi }_2\left({\sigma }_2,V_2\right)\left(t\right)={\int }_0^t{S}_{\sigma }\left(t-s\right)\left(F\left({\sigma }_1\left(s\right),V_1\left(s\right)\right)-F\left({\sigma }_2\left(s\right),V_2\left(s\right)\right)\right)ds$$

Using the same smoothing estimate of F in Equation (13), we obtain

$$\parallel {\Phi }_1\left({\sigma }_1,V_1\right)\left(t\right)-{\Phi }_2\left({\sigma }_2,V_2\right)\left(t\right){\parallel }_{{C(\left[0,T\right];H}^1)}$$

$$\le 2C_{\Omega }{T}^{{\displaystyle \frac{1}{2}}}\underset{0\le s\le T}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel F\left({\sigma }_1\left(s\right),V_1\left(s\right)\right)-F\left({\sigma }_2\left(s\right),V_2\left(s\right)\right){\parallel }_{L^2}$$

(17)

The right term of Equation (17) can be modified by using the Lipschitz condition in Assumption 1 as:

$$\parallel F\left({\sigma }_1\left(s\right),V_1\left(s\right)\right)-F\left({\sigma }_2\left(s\right),V_2\left(s\right)\right){\parallel }_{L^2}$$

$$\le L_F\left(\parallel \mathsf{\delta }\sigma {\parallel }_{L^2}+\parallel \mathsf{\delta }V{\parallel }_{L^2}\right)$$

$$\le L_F\left(\parallel \left(\mathsf{\delta }\sigma , \mathsf{\delta }V\right){\parallel }_{{\mathbf{X}}_T}\right)$$

Therefore,

$$\parallel {\Phi }_1\left({\sigma }_1,V_1\right)\left(t\right)-{\Phi }_1\left({\sigma }_2,V_2\right)\left(t\right){\parallel }_{C\left(\left[0,T\right];H^1\right)} \le 2C_{\Omega }{L}_F{T}^{{\displaystyle \frac{1}{2}}}\parallel \left(\delta \sigma ,\delta V\right){\parallel }_{{\mathbf{X}}_T}$$

(18)

Similarly, for the second component ${\Phi }_2$, we have

$$\parallel {\Phi }_2\left({\sigma }_1,V_1\right)\left(t\right)-{\Phi }_2\left({\sigma }_2,V_2\right)\left(t\right){\parallel }_{C\left(\left[0,T\right];L_2\right)}$$

$$\le {\int }_0^t\left(G\left({\sigma }_1\left(s\right),V_1\left(s\right)\right)-G\left({\sigma }_2\left(s\right),V_2\left(s\right)\right)\right)ds$$

$$\le T{L}_G\parallel \left(\delta \sigma ,\delta V\right){\parallel }_{{\mathbf{X}}_T}$$

(19)

Combining Equations (18) and (19), we can find the contracting relationship:

$$\parallel \mathit{\Phi }\left({\sigma }_1,V_1\right)\left(t\right)-\mathit{\Phi }\left({\sigma }_2,V_2\right)\left(t\right){\parallel }_{{\mathbf{X}}_T}\le k\left(T\right)\parallel \left(\delta \sigma ,\delta V\right){\parallel }_{{\mathbf{X}}_T}$$

(20)

where k(T) = L_F(${\mathit{C}}_{\mathsf{\Omega }}$T + 2${\mathit{C}}_{\mathsf{\Omega }}$T^1/2)+L_GT. Since k(T) → 0 as T → 0, we may choose T₀ > 0 such that k(T₀) < 1. Thus, Φ is a strict contraction on ${\mathbf{B}}_{\mathit{R}}\subset {\mathbf{X}}_{{\mathit{T}}_0}$. This argument follows the standard Banach fixed-point framework [30].

2.2.5. Conclusion: Local Well-Posedness

By Banach’s Fixed-Point Theorem, Φ admits a unique fixed point in ${\mathbf{X}}_{{\mathit{T}}_0}$. This fixed point is precisely the unique mild solution (σ,V) of Equations (10) and (11). Continuous dependence on initial data follows directly from the contraction estimate Equations (20). Therefore, the coupled KPP-RS system is locally well-posed in H¹(Ω) × L²(Ω). Here, it exists a theorem:

Theorem 1.

Local Well-Posedness.

Under Assumptions 1 and 2, for any initial data (σ₀,V₀) ∈ H¹(Ω) × L²(Ω), there exists a time T₀ > 0 and a unique mild solution (σ,V) ∈ C([0, T₀];H¹(Ω) × L²(Ω)) of the coupled KPP–RS system in the sense of Equations (10) and(11). Moreover, the solution depends continuously on the initial data in the H¹(Ω) × L²(Ω) topology.

2.3. Global a Priori Estimates and Extension to a Global Solution

In this section, we derive uniform-in-time estimates for the solution (σ, V) to ensure that the local solution established in Section 2.2 can be extended to a global one. These estimates are based on the dissipative structure of the KPP-RS framework and the geometric properties of the domain Ω. For clarity, the various constants involved in these global estimates—including those previously introduced and those specifically derived in the following lemmas—are summarized in

Table 2. Summary of constants and parameters used in the global stability analysis.

Symbol	Definition/Role	Mathematical Context
$C_{\Omega }$	Domain-dependent embedding bound	Controls LP and semigroup estimates in 2D.
$L_{\mathrm{G}}, L_F$	Lipschitz constants	Define the sensitivity of the nonlinear source terms
${\mathrm{C}}_{\mathrm{G}}, {\mathrm{C}}_F$	Growth bounds	Ensure the linear growth of the mappings F and G.
${\mathrm{C}}_{\mathrm{p}}(\Omega )$	Neumann–Poincaré constant	Relates the ${\mathrm{L}}^2-\mathrm{norm} \mathrm{to} \mathrm{the} \mathrm{gradient} \mathrm{in} H^1 (\Omega$)/R.
$M_{\sigma } , M_V$	Uniform $L^{\mathrm{\infty }}$-bound	Maximum allowable stress and state variable magnitudes

.

We also establish uniform a priori estimates on every finite interval [0, T], which, in turn, guarantees the global extendability of the local mild solution. Throughout, we worked in the Banach space X = H¹(Ω) × L²(Ω). We first recall Assumption 3: There exist constants $a,b>0$ such that

$$F\left(\sigma \right)\le a\sigma -b{\sigma }^2$$

(21)

2.3.1. Lemma 1: Global $L^{\mathbf{\infty }}$-Bound for $\sigma$

Let $\sigma$ satisfy

$${\partial }_t\sigma -D_{\sigma }\Delta \sigma =F\left(\sigma \right),\sigma \left(x, 0\right)={\sigma }_0\left(x\right)$$

Under Assumption 3, the solution obeys the uniform bound

$$\parallel \sigma \left(t\right){\parallel }_{L^{\mathrm{\infty }}}\le M_{\sigma }=\mathrm{m}\mathrm{a}\mathrm{x}\parallel {\sigma }_0{\parallel }_{L^{\mathrm{\infty }}},\hspace{0.17em}a/b,\forall t\ge 0$$

(22)

Proof of Lemma 1.

Consider the logistic ODE

$${\displaystyle \frac{\mathrm{d}w}{\mathrm{d}t}}=aw-b{w}^2,w\left(0\right)=\parallel {\sigma }_0{\parallel }_{L^{\mathrm{\infty }}}$$

The explicit solution satisfies $0<w\left(t\right)\le \mathrm{m}\mathrm{a}\mathrm{x}\parallel {\sigma }_0{\parallel }_{L^{\mathrm{\infty }}},a/b$.

By Assumption 3

$$F\left(\sigma \right)\le a\sigma -b{\sigma }^2$$

Since

$$0\le \sigma \left(t\right)\le {\displaystyle \frac{a}{b}}=M_{\sigma },\forall \left(\mathit{x},t\right)\in \Omega \times \left[0,T\right]$$

The reaction term in Equation (21) is uniformly bounded on [0,T]. This bound is independent of the solution and depends only on the parameters $a,b$.

$${\partial }_t\sigma -D_{\sigma }\Delta \sigma \le a\sigma -b{\sigma }^2=w^{\prime }\left(t\right)$$

and $\sigma \left(\cdot ,0\right)\le w\left(0\right)$. By the parabolic comparison principle (classical analyses in semilinear reaction–diffusion systems [31]), we obtain $\sigma \left(\mathit{x},t\right)\le w\left(t\right)$ for all $t\ge 0$.

Taking the essential supremum over $\mathit{x}\in \Omega$ completes the proof. □

2.3.2. Lemma 2: Dissipative $L^2$-Bound for the Rate-and-State Variable $V$

Let $V$ satisfy

$${\partial }_{t}V=G\left(\sigma ,V\right),V\left(\cdot ,0\right)=V_0.$$

(23)

where

$$G\left(\sigma ,V\right)=\alpha -\beta V+\gamma \sigma , \alpha >0, \beta >0$$

Assume Assumption 4, and suppose that the stress field $\sigma$ is uniformly bounded in $L^{\mathrm{\infty }}\left(\Omega \right)$ for all $t\ge 0$ (Lemma 1). Then there exist constants ${\lambda }_V>0$ and $C_V\ge 0$, depending only on $\alpha ,\beta ,\gamma$, the bound of $\sigma$, and $\mid \Omega \mid$, such that

$${\displaystyle \frac{1}{2}}\parallel V\left(t\right){\parallel }_{L^2}^2 \le -{\lambda }_V\parallel V\left(t\right){\parallel }_{L^2}^2+C_V,t\ge 0$$

Consequently, for all $t\ge 0$,

$$\parallel V\left(t\right){\parallel }_{L^2}^2 \le \parallel V_0{\parallel }_{L^2}^2\hspace{0.17em}{e}^{-2{\lambda }_{V}t}+{\displaystyle \frac{C_V}{{\lambda }_V}}$$

and, in particular, $V$ is uniformly bounded in $L^2\left(\Omega \right)$ for all time.

Proof of Lemma 2.

Multiplying the equation by $V$ to Equation (23) and integrating over Ω gives.

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel V{\parallel }_{L^2}^2={\int }_{\Omega }V\hspace{0.17em}G\left(\sigma ,V\right)\hspace{0.17em}dx$$

(24)

Using the explicit Rate-and-State evolution (Section 2.1),

$$G\left(\sigma ,V\right)=\alpha -\beta V+\gamma \sigma , \alpha >0, \beta >0$$

we obtain

$${\int }_{\Omega }V\hspace{0.17em}G\left(\sigma ,V\right)\hspace{0.17em}dx=\alpha {\int }_{\Omega }V\hspace{0.17em}dx-\beta {\int }_{\Omega }{V}^2\hspace{0.17em}dx+\gamma {\int }_{\Omega }\sigma V\hspace{0.17em}dx$$

(25)

We estimate each term on the right-hand side. First, by Hölder’s inequality and Young’s inequality,

$$\alpha {\int }_{\Omega }V\hspace{0.17em}dx\le \alpha \hspace{0.17em}\mid \Omega {\mid }^{1/2}\parallel V{\parallel }_{L^2}\le {\displaystyle \frac{\beta }{4}}\parallel V{\parallel }_{L^2}^2+{\displaystyle \frac{{\alpha }^2\mid \Omega \mid }{\beta }}$$

(26)

Next, using the uniform $L^{\mathrm{\infty }}$-bound for $\sigma$ from Lemma 1, i.e., $\parallel \sigma \left(t\right){\parallel }_{L^{\mathrm{\infty }}} \le M_{\sigma }$ for all $t\ge 0$, we have

$$\gamma {\int }_{\Omega }\sigma V\hspace{0.17em}dx\le \mid \gamma \mid \hspace{0.17em}\parallel \sigma {\parallel }_{L^{\mathrm{\infty }}}\hspace{0.17em}\parallel V{\parallel }_{L^1}\le \mid \gamma \mid \hspace{0.17em}{M}_{\sigma }\hspace{0.17em}\mid \Omega {\mid }^{1/2}\parallel V{\parallel }_{L^2}\le {\displaystyle \frac{\beta }{4}}\parallel V{\parallel }_{L^2}^2+{\displaystyle \frac{{\gamma }^2{M}_{\sigma }^2\mid \Omega \mid }{\beta }}$$

(27)

Substituting Equations (26) and (27) into (25), and using $-\beta {\int }_{\Omega }{V}^{2}dx=-\beta \parallel V{\parallel }_{L^2}^2$, we obtain

$${\int }_{\Omega }V\hspace{0.17em}G\left(\sigma ,V\right)\hspace{0.17em}dx\le -{\displaystyle \frac{\beta }{2}}\parallel V{\parallel }_{L^2}^2+C_V$$

(28)

where the $C_V$ is defined as ${\displaystyle \frac{\mid \Omega \mid }{\beta }}\left({\alpha }^2+{\gamma }^2{M}_{\sigma }^2\right)$. Combining (24) and (28), we arrive at the dissipative differential inequality

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel V\left(t\right){\parallel }_{L^2}^2\le -{\lambda }_V\parallel V\left(t\right){\parallel }_{L^2}^2+C_V,{\lambda }_V={\displaystyle \frac{\beta }{2}}$$

(29)

Applying Grönwall’s inequality to (29) yields, for all $t\ge 0$,

$$\parallel V\left(t\right){\parallel }_{L^2}^2\le \parallel V_0{\parallel }_{L^2}^2{e}^{-2{\lambda }_{V}t}+{\displaystyle \frac{C_V}{{\lambda }_V}}$$

(30)

In particular,

$$\underset{t\ge 0}{\mathrm{s}\mathrm{u}\mathrm{p}}\parallel V\left(t\right){\parallel }_{L^2}^2\le \parallel V_0{\parallel }_{L^2}^2+{\displaystyle \frac{C_V}{{\lambda }_V}},$$

which proves the global (time-uniform) $L^2$-boundedness of $V$. □

2.3.3. Lemma 3: Global $H^1$-Bound for $\sigma$

The stress field $\sigma$ satisfies

$$\parallel \sigma \left(t\right){\parallel }_{H^1}^2\le \parallel \mathsf{\nabla }{\sigma }_0{\parallel }_{H^1}^2{e}^{-C_{1}t}+{\displaystyle \frac{C_2}{C_1}},0\le t\le T$$

(31)

for some constants $C_1 \mathrm{and} C_2>0$.

Proof of Lemma 3.

Recall the H¹ norm and inner product:

$$\parallel \sigma {\parallel }_{H^1\left(\Omega \right)}={\left(\parallel \sigma {\parallel }_{L^2\left(\Omega \right)}^2+\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2\right)}^{1/2}$$

We derive an $H^1$-estimate by combining an $L^2$-energy estimate and a gradient estimate.

Step 1: $L^2$-estimate

Taking the $L^2\left(\Omega \right)$ inner product of the stress evolution Equation (1) with $\sigma$, and using integration by parts together with the homogeneous Neumann boundary condition ${\partial }_{\mathit{n}}\sigma =0$, we obtain

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \sigma {\parallel }_{L^2\left(\Omega \right)}^2+D_{\sigma }\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2=(F(\sigma ,V),\sigma {)}_{L^2\left(\Omega \right)}$$

Step 2: Gradient estimate

To control the gradient, we test (1) with $-\Delta \sigma$ in $L^2\left(\Omega \right)$:

$$({\partial }_t\sigma ,-\Delta \sigma {)}_{L^2\left(\Omega \right)}=(D_{\sigma }\Delta \sigma ,-\Delta \sigma {)}_{L^2\left(\Omega \right)}+\left(F\right(\sigma ,V),-\Delta \sigma {)}_{L^2\left(\Omega \right)}.$$

Using integration by parts and the Neumann boundary condition, we have

$$({\partial }_t\sigma ,-\Delta \sigma {)}_{L^2\left(\Omega \right)}={\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2$$

and

$${(D}_{\sigma }\Delta \sigma ,-\Delta \sigma {)}_{L^2\left(\Omega \right)}=-D_{\sigma }\parallel \Delta \sigma {\parallel }_{L^2\left(\Omega \right)}^2$$

Hence,

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2+D_{\sigma }\parallel \Delta \sigma {\parallel }_{L^2\left(\Omega \right)}^2=-(F(\sigma ,V),\Delta \sigma {)}_{L^2\left(\Omega \right)}$$

Step 3: ${\mathrm{H}}^1$-energy inequality

Adding above results, we obtain

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \sigma {\parallel }_{H^1\left(\Omega \right)}^2+D_{\sigma }(\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2+\parallel \Delta \sigma {\parallel }_{L^2\left(\Omega \right)}^2)=(F(\sigma ,V),\sigma {)}_{L^2\left(\Omega \right)}-(F(\sigma ,V),\Delta \sigma {)}_{L^2\left(\Omega \right)}$$

(32)

There exists a constant $\mu$ > 0 by using Cauchy–Schwarz and Young’s inequalities, we can derive

$$\left|\left(F\right(\sigma {),\sigma )}_{L^2}\right|\le {\displaystyle \frac{\mu }{2}}\parallel \Delta \sigma {\parallel }_{L^2\left(\Omega \right)}^2+{\displaystyle \frac{1}{2\mu }}\parallel F\left(\sigma ,v\right){\parallel }_{L^2\left(\Omega \right)}^2$$

$$\left|{\left(F\right(\sigma ),\mathsf{\nabla }\sigma )}_{L^2}\right|\le {\displaystyle \frac{D_{\sigma }}{2}}\parallel \Delta \sigma {\parallel }_{L^2\left(\Omega \right)}^2+{\displaystyle \frac{1}{2D_{\sigma }}}\parallel F\left(\sigma ,v\right){\parallel }_{L^2\left(\Omega \right)}^2$$

Since σ and V are uniformly bounded on [0,T] (Lemmas 1 and 2), and F satisfies the structural Assumptions 3 and 4, it has

$$\parallel F\left(\sigma \right){\parallel }_{L^2\left(\Omega \right)}^2\le C_{T }\mathrm{for}\mathrm{constant}{C}_{T }>0$$

Because homogeneous Neumann boundary conditions are imposed, the standard Poincaré inequality cannot be applied directly because constant functions belong to the kernel of the Laplacian. We therefore employ the Neumann–Poincaré inequality in the quotient space. Writing

$$\sigma =(\sigma -\overline{\sigma }\left(t\right))+\overline{\sigma }\left(t\right)$$

where

$$\overline{\sigma }\left(t\right)={\displaystyle \frac{1}{\mid \Omega \mid }}{\int }_{\Omega }\sigma \left(x,t\right)dx$$

For Neumann boundary conditions, the Poincaré inequality holds in the quotient space H¹(Ω)/R:

$$\parallel {\sigma -\overline{\sigma }\left(t\right)\parallel }_{L^2\left(\Omega \right)}^2\le C_P\left(\Omega \right)\parallel {\mathsf{\nabla }\sigma \parallel }_{L^2\left(\Omega \right)}^2$$

where $\overline{\sigma }\left(t\right)$ is the mean stress and $C_P\left(\Omega \right)>0$ is the Poincaré constant depending on the domain. Moreover, the mean value $\overline{\sigma }\left(t\right)$ is uniformly bounded in time due to the logistic-type structure of the reaction term. Consequently, the L²-norm of σ can be controlled by the gradient term up to a bounded mean contribution. Choosing $\mu >0$ sufficiently small so that the term ${\displaystyle \frac{\mu C_P(\Omega )}{2}}\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2$ can be absorbed by the dissipation term $D_{\sigma }\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2$, and using the gradient estimate from Step 2 together with Young’s inequality for the (F(σ,V), $\mathsf{\nabla }$σ)-term, we deduce from (32) that

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \sigma {\parallel }_{H^1\left(\Omega \right)}^2+C_1\parallel \sigma {\parallel }_{H^1\left(\Omega \right)}^2\le C_2$$

for some constants $C_1>0$ and $C_2>0$. To explicitly reflect the dependence of the dissipative rate on the physical parameters, one may take $C_1=\mathrm{m}\mathrm{i}\mathrm{n}\{{\displaystyle \frac{D_{\sigma }}{C_p\left(\Omega \right)}},{\mu }_{\mathbf{0}}\}$ with ${\mu }_{\mathbf{0}}$ > 0 arising from the lower-order dissipation terms in Equation (32). The constant $C_2$ depends on the uniform bounds established previously. This ensures that the global H¹ stability is governed by both the stress redistribution capacity and the domain characteristics.

Applying Gronwall’s inequality yields

$$\parallel \sigma \left(t\right){\parallel }_{H^1\left(\Omega \right)}^2\le \parallel {\mathsf{\nabla }\sigma }_0{\parallel }_{H^1\left(\Omega \right)}^2{e}^{-C_{1}t}+{\displaystyle \frac{C_2}{C_1}},0\le t\le T$$

which proves the desired bound. □

Revised Remark 2.

Bounds and Physical Dependence of the Constants.

The estimates derived in Lemmas 2 and 3 play complementary roles in the analysis. Lemma 2 provides an a priori $L^2$-bound for the state variable $V$, which is sufficient to exclude finite-time blow-up and ensure continuation of local solutions. When the rate-and-state evolution law $G\left(\sigma ,V\right)$ includes an intrinsic damping mechanism (e.g., an aging or slip law with $\beta >0$), the bound in Lemma 2 can be strengthened to a time-uniform estimate, reflecting the dissipative nature of the state-variable dynamics. In this case, $V$ admits a compact absorbing set in $L^2\left(\Omega \right)$. From a modeling standpoint, the absorbing set defines an intrinsic admissible domain for stress–friction evolution, beyond which physically meaningful trajectories cannot exist. The existence of an absorbing set implies that, after a transient period, all trajectories enter and remain in a bounded region of the phase space; this bounded region can be interpreted as the admissible range of physically meaningful stress–state evolution, effectively delimiting the long-term behavior of the KPP-RS framework and preventing unphysical energy growth.

Lemma 3 establishes a uniform-in-time $H^1$-bound for the stress field $\sigma$, which originates from the parabolic diffusion and the KPP-type saturation structure. This stronger estimate is crucial for controlling the long-time dynamics of the coupled system. The constants appearing in Lemmas 2 and 3 depend explicitly on physically meaningful parameters of the model. In particular:

• C₁ arises from the linear and dissipative components of the rate-and-state law and depends on frictional parameters such as β
• C₂ reflects bounded forcing contributions controlled by the KPP saturation parameters $a,b$ and the loading term $\alpha$
• The coercivity constant in the $H^1$-estimate depends on the diffusion coefficient $D_{\sigma }$ and the Poincaré constant of the domain $\Omega$
• The rate ${\lambda }_V$ is determined by the intrinsic damping mechanism of the rate-and-state law (e.g., the parameter $\beta >0$ in the aging or slip law), while the decay rate for the stress field originates from the diffusion coefficient $D_{\sigma }$ combined with the Poincaré constant of the domain.

Consequently, both the decay rates and the residual bounds are fully determined by physically meaningful parameters of the model. Thus, all bounds depend solely on physical parameters and initial data, ensuring transparent and physically interpretable estimates.

2.3.4. Lemma 4: Local Lipschitz Continuity of the Nonlinear Mapping F

Let Ω ⊂ R² be a bounded domain and define

$$\mathbf{X} =H^1(\Omega ) \times L^2(\Omega ), \parallel (\sigma \left(t\right), V\left(t\right){\parallel }_{\mathrm{x}}=\parallel \sigma (t){\parallel }_{H^1\left(\Omega \right)}+\parallel V(t){\parallel }_{L^2\left(\Omega \right)}$$

Suppose that the nonlinear reaction term F(σ,V) is given by a smooth function

$$f:{\mathbb{R}}^2\to \mathrm{R},F\left(\sigma ,V\right)=f\left(\sigma \left(\mathit{x}\right),V\left(\mathit{x}\right)\right)$$

and that Assumptions 3 and 4 hold, so that σ and V remain uniformly bounded in time. Then, for every R > 0 there exists a constant L_F(R) > 0 such that for all (σ_i, V_i)∈X with

$$\parallel ({{\sigma }_i,V_i)\parallel }_x \le R(i=1,2)$$

$$‖F\left({\sigma }_1,V_1\right)-{F\left({\sigma }_2,V_2\right)⌉}_{L^2\left(\Omega \right)}\le L_F\left(R\right)\left(‖{\sigma }_1-{{\sigma }_2‖}_{H^1\left(\Omega \right)}+‖V_1-{V_2‖}_{L^2\left(\Omega \right)}\right)$$

In particular, F: X → L²(Ω) is locally Lipschitz continuous on closed balls

$${\mathbf{B}}_R\left(\mathbf{X}\right)=\left\{\right(\sigma ,V)\in \mathbf{X}:\parallel (\sigma \left(t\right), V\left(t\right){\parallel }_{\mathbf{x}}\le R\}$$

Proof of Lemma 4.

Since Ω⊂R², the Sobolev embedding theorem gives

$$H^1\left(\Omega \right)\hookrightarrow L^p\left(\Omega \right), \forall 2\le p<\infty ,$$

In particular, any polynomial-type expression in σ (e.g., σ², σ^k) belongs to L²(Ω) whenever σ∈H¹(Ω). Although $H^1\left(\Omega \right)$ is not continuously embedded into $L^{\infty }\left(\Omega \right)$ in two dimensions, the maximum principle (refer to Lemma 1) ensures that any solution $\sigma$ originating from bounded initial data remains uniformly bounded in $L^{\infty }\left(\Omega \right)$. Similarly, the evolution equation for V preserves $L^{\infty }\left(\Omega \right)$-bounds (Lemma 2). Therefore, there exist constants $M_{\sigma }, M_V>0$ such that:

$$\parallel {\sigma }_i{\parallel }_{L^{\infty }\left(\Omega \right)}\le M_{\sigma } , \parallel V_i{\parallel }_{L^{\infty }\left(\Omega \right)}\le M_V \mathrm{f}\mathrm{o}\mathrm{r} i=\mathrm{1,2}$$

Consequently, the pointwise values $\left({\sigma }_{\mathrm{i}}\left(\mathrm{x}\right){,V}_{\mathrm{i}}\left(\mathrm{x}\right)\right)$ are confined to the compact rectangle

$$\mathbf{K}=\left[-M_{\sigma },M_{\sigma }\right]\times \left[-M_V,M_V\right]\subset {\mathrm{R}}^2,\mathrm{for}\mathrm{a}.\mathrm{e}.\mathrm{x}\in \Omega$$

Since f ∈ C¹(R²), its partial derivatives ${\partial }_{\sigma }f$ are continuous on K and therefore bounded. There exists a constant C_K > 0 such that

$$\left|{\partial }_{\sigma }f\left(\xi ,\eta \right)\right|+\left|{\partial }_{V}f\left(\xi ,\eta \right)\right|\le C_K,\forall \hspace{0.17em}\left(\xi ,\eta \right)\in K$$

For almost every x ∈ Ω. By the mean value theorem in R² implies

$$F\left({\sigma }_1,V_1\right)\left(\mathit{x}\right)-F\left({\sigma }_2,V_2\right)\left(\mathit{x}\right)=f\left({\sigma }_1\left(\mathit{x}\right),V_1\left(\mathit{x}\right)\right)-f\left({\sigma }_2\left(\mathit{x}\right),V_2\left(\mathit{x}\right)\right)$$

can be written as

$$F\left({\sigma }_1,V_1\right)\left(\mathit{x}\right)-F\left({\sigma }_2,V_2\right)\left(\mathit{x}\right)=a\left(\mathit{x}\right)\left({\sigma }_1\left(\mathit{x}\right)-{\sigma }_2\left(\mathit{x}\right)\right)+b\left(\mathit{x}\right)\left(V_1\left(\mathit{x}\right)-V_2\left(\mathit{x}\right)\right)$$

(33)

for some coefficients a(x), b(x) with

$$\left|a\left(\mathit{x}\right)\right|+\left|b\left(\mathit{x}\right)\right|\le C_K \mathrm{for}\mathrm{a}.\mathrm{e}.\mathit{x}\in \Omega$$

Thus, for almost every x ∈Ω,

$$\left|F\left({\sigma }_1,V_1\right)\left(\mathit{x}\right)-F\left({\sigma }_2,V_2\right)\left(\mathit{x}\right)\right|\le C_K\left(|{\sigma }_1\left(\mathit{x}\right)-{\sigma }_2\left(\mathit{x}\right)|+\left|\right(V_1\left(\mathit{x}\right)-V_2\left(\mathit{x}\right)|\right)$$

(34)

Taking the L²(Ω)-norm and using the triangle inequality, we obtain

$$‖F\left({\sigma }_1,V_1\right)-{F\left({\sigma }_2,V_2\right)⌉}_{L^2\left(\Omega \right)}\le C_K\left(‖{\sigma }_1-{{\sigma }_2‖}_{L^2\left(\Omega \right)}+‖V_1-{V_2‖}_{L^2\left(\Omega \right)}\right)$$

(35)

Finally, since

$$‖{\sigma }_1-{{\sigma }_2‖}_{L^2\left(\Omega \right)}\le ‖{\sigma }_1-{{\sigma }_2‖}_{H^1\left(\Omega \right)}$$

we conclude that

$$‖F\left({\sigma }_1,V_1\right)-{F\left({\sigma }_2,V_2\right)‖}_{L^2\left(\Omega \right)}\le L_F\left(R\right)\left(‖{\sigma }_1-{{\sigma }_2‖}_{H^1\left(\Omega \right)}+‖V_1-{V_2‖}_{L^2\left(\Omega \right)}\right)$$

for some constant L_F(R) depending only on the radius R of the ball B_R(X) through the bounds $M_{\sigma } \mathrm{and} M_V$. In the general setting we allow F(σ, V) = f(σ, V) to cover coupled nonlinearities.

However, in the specific KPP form used in this work,

$$F\left(\sigma ,V\right)=a\sigma -b{\sigma }^2$$

the mapping is independent of V, hence ${\partial }_{V}F\equiv 0$. Therefore, in the Lipschitz estimate (e.g., Equations (33)–(35), the term involving $‖V_1-{V_2‖}_{L^2\left(\Omega \right)}$ vanishes, and the bound reduces to

$$‖F\left({\sigma }_1\right)-{F\left({\sigma }_2\right)‖}_{L^2\left(\Omega \right)}\le L_F\left(R\right)\left(‖{\sigma }_1-{{\sigma }_2‖}_{L^2\left(\Omega \right)}\right)\le L_F\left(R\right)\left(‖{\sigma }_1-{{\sigma }_2‖}_{H^1\left(\Omega \right)}\right)$$

This proves the local Lipschitz continuity of F on B_R(X). □

2.3.5. Lemma 5: Total Energy Inequality

Define the total energy functional

$$E\left(t\right)={\displaystyle \frac{1}{2}}\left(\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2\left(\Omega \right)}^2+\parallel V\left(t\right){\parallel }_{L^2\left(\Omega \right)}^2\right)$$

(36)

Assume that $\sigma$ and $V$ solve

$${\partial }_t\sigma -D_{\sigma }\Delta \sigma =F\left(\sigma \right),{\partial }_{t}V=G\left(\sigma ,V\right)$$

with homogeneous Neumann boundary conditions for $\sigma$, and that the nonlinearities satisfy the KPP and rate-and-state structures described in Section 2.1, namely

$$F\left(\sigma \right)=a\sigma -b{\sigma }^2,a,b>0,$$

$$G\left(\sigma ,V\right)=\alpha -\beta V+\gamma \sigma ,\alpha ,\beta >0.$$

Then there exist constants $\lambda >0$ and $C\ge 0$, depending only on the model parameters $\left(D_{\sigma },a,b,\alpha ,\beta \right)$ and on the geometry of $\Omega$, such that for all $t\ge 0$

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}E\left(t\right)\le -\lambda E\left(t\right)+C$$

(37)

Proof of Lemma 5.

We proceed in two steps, corresponding to the $\sigma$– and $V$–components.

Step 1. $H^1$–estimate for the stress field $\sigma$.

Recall that

$${\partial }_t\sigma =D_{\sigma }\Delta \sigma +F\left(\sigma \right),F\left(\sigma \right)=a\sigma -b{\sigma }^2.$$

Taking the $L^2$ inner product of this equation with $-\Delta \sigma$ and using the homogeneous Neumann boundary condition, we obtain

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel \mathsf{\nabla }\sigma (t){\parallel }_{L^2}^2=({\partial }_t\sigma ,-\Delta \sigma {)}_{L^2}=D_{\sigma }(\Delta \sigma ,-\Delta \sigma {)}_{L^2}+(F(\sigma ,V),-\Delta \sigma {)}_{L^2}.$$

Hence

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2}^2=-D_{\sigma }\parallel \Delta \sigma {\parallel }_{L^2}^2-(F(\sigma ),\Delta \sigma {)}_{L^2}.$$

Using Cauchy–Schwarz and Young’s inequalities, we estimate

$$\mid (F(\sigma ),\Delta \sigma {)}_{L^2}\mid \le \parallel F(\sigma ){\parallel }_{L^2},\hspace{0.17em}\parallel \Delta \sigma {\parallel }_{L^2}\le {\displaystyle \frac{D_{\sigma }}{2}}\parallel \Delta \sigma {\parallel }_{L^2}^2+{\displaystyle \frac{1}{2D_{\sigma }}}\parallel F(\sigma ){\parallel }_{L^2}^2.$$

Therefore,

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2}^2\le -{\displaystyle \frac{D_{\sigma }}{2}}\parallel \Delta \sigma {\parallel }_{L^2}^2+{\displaystyle \frac{1}{2D_{\sigma }}}\parallel F\left(\sigma \right){\parallel }_{L^2}^2$$

By Lemma 1 (logistic invariant region), we have a uniform pointwise bound

$$0\le \sigma \left(\mathit{x},t\right)\le {\displaystyle \frac{a}{b}},$$

so that $F\left(\sigma \right)=a\sigma -b{\sigma }^2$ is uniformly bounded in $L^{\mathrm{\infty }}$, and hence

$$\parallel F\left(\sigma \right){\parallel }_{L^2}^2\le C_F,$$

for constant $C_F>0$ depending only on $a,b$ and $\mid \Omega \mid$. On the other hand, under homogeneous Neumann boundary conditions, a Poincaré-type inequality for the Neumann Laplacian implies that there exists a constant $C_P\left(\Omega \right)$> 0, depending on Ω and the first non-zero Neumann eigenvalue, such that

$${\displaystyle \frac{1}{C_P(\Omega )}}\parallel {\mathsf{\nabla }\sigma \parallel }_{L^2\left(\Omega \right)}^2\le \parallel {\Delta \sigma \parallel }_{L^2\left(\Omega \right)}^2$$

Substituting these bounds into the differential inequality derived from testing the σ-equation with −Δσ, we find

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2}^2\le \left(-{\displaystyle \frac{D_{\sigma }}{C_P(\Omega )}}\right)\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2}^2+{\displaystyle \frac{C_F}{D_{\sigma }}}$$

Thus, there exist constants ${\gamma }_{\sigma }={\displaystyle \frac{D_{\sigma }}{C_P(\Omega )}}>0$ and $C_{\sigma }={\displaystyle \frac{C_F}{D_{\sigma }}}\ge 0$,

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2}^2\le -{\gamma }_{\sigma }\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2}^2+C_{\sigma }$$

(38)

Step 2. $L^2$–estimate for the rate-and-state variable $V$.

For the aging law,

$${\partial }_{t}V=\alpha -\beta V+\gamma \sigma ,\alpha ,\beta >0.$$

Multiplying by $V$ and integrating over $\Omega$, we get

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel V\left(t\right){\parallel }_{L^2}^2={\int }_{\Omega }V\left(\alpha -\beta V+\gamma \sigma \right)\hspace{0.17em}dx=\alpha {\int }_{\Omega }V\hspace{0.17em}dx-\beta \parallel V{\parallel }_{L^2}^2++\gamma {\int }_{\Omega }V\sigma dx$$

Using Hölder and Young’s inequalities, we estimate

$$\alpha {\int }_{\Omega }V\hspace{0.17em}dx\le \alpha \mid \Omega {\mid }^{1/2}\parallel V{\parallel }_{L^2}\le {\displaystyle \frac{\beta }{4}}\parallel V{\parallel }_{L^2}^2+{\displaystyle \frac{{\alpha }^2\mid \Omega \mid }{\beta }}$$

and, by Lemma 1 ($\parallel \sigma {\parallel }_{L^{\mathrm{\infty }}}\le M_{\sigma }$),

$$\gamma {\int }_{\Omega }\sigma V\hspace{0.17em}dx\le \mid \gamma \mid \parallel \sigma {\parallel }_{L^{\mathrm{\infty }}}\parallel V{\parallel }_{L^1}\le \mid \gamma \mid M_{\sigma }\mid \Omega {\mid }^{1/2}\parallel V{\parallel }_{L^2}\le {\displaystyle \frac{\beta }{4}}\parallel V{\parallel }_{L^2}^2+{\displaystyle \frac{{\gamma }^2{M}_{\sigma }^2\mid \Omega \mid }{\beta }}$$

Combining above two equations, we obtain:

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\parallel V{\parallel }_{L^2}^2\le -{\displaystyle \frac{\beta }{2}}\parallel V{\parallel }_{L^2}^2+C_V$$

(39)

where

$$C_V={\displaystyle \frac{\mid \Omega \mid }{\beta }}\left({\alpha }^2+{\gamma }^2{M}_{\sigma }^2\right)$$

Step 3. Total energy inequality.

Recall the definition

$$E\left(t\right)={\displaystyle \frac{1}{2}}\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2}^2+{\displaystyle \frac{1}{2}}\parallel V\left(t\right){\parallel }_{L^2}^2$$

Differentiating and using (38) and (39), we obtain

$$\begin{array}{cc}& {\displaystyle \frac{d}{dt}}E\left(t\right)={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2}^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel V\left(t\right){\parallel }_{L^2}^2\\ & \le {\displaystyle \frac{1}{2}}\left(-{\gamma }_{\sigma }\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2}^2+C_{\sigma }\right)+{\displaystyle \frac{1}{2}}\left(-\beta \parallel V{\parallel }_{L^2}^2+C_V\right)\\ & =-{\displaystyle \frac{{\gamma }_{\sigma }}{2}}\parallel \mathsf{\nabla }\sigma {\parallel }_{L^2}^2-{\displaystyle \frac{\beta }{2}}\parallel V{\parallel }_{L^2}^2+{\displaystyle \frac{C_{\sigma }+C_V}{2}}\end{array}$$

Since

$$\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2}^2\le 2E\left(t\right), \parallel V\left(t\right){\parallel }_{L^2}^2\le 2E\left(t\right),$$

it follows that

$${\displaystyle \frac{d}{dt}}E\left(t\right)\le -\lambda E\left(t\right)+CR,t\ge 0$$

where

$$\lambda =\mathrm{m}\mathrm{i}\mathrm{n}\{{\gamma }_{\sigma },\beta \}>0,CR={\displaystyle \frac{C_{\sigma }+C_V}{2}}$$

This proves the claimed energy estimate. Here $\lambda >0$ depends on the diffusion coefficient $D_{\sigma }$, the domain $\Omega$, and the KPP parameters $a,b$ through the elliptic estimate in Equation (38), whereas $C$ depends additionally on the rate-and-state parameters $\alpha ,\beta$ and the domain size $\mid \Omega \mid$ via Equation (39). Thus, the total energy decay is controlled entirely by physically interpretable parameters of the DSGF model. We emphasize that the following result establishes the dissipativity and long-time boundedness of the system rather than convergence toward a single equilibrium state. The existence of an absorbing set is sufficient to guarantee global-in-time control of solutions and forms the basis for further investigations of asymptotic behaviour. □

Theorem 2.

Global Well-Posedness and Ultimate Boundedness (Dissipativity).

Let Assumptions 1–4 hold. Then the coupled KPP–RS system

$$\left\{\begin{array}{c}{\partial }_t\sigma -D_{\sigma }\Delta \sigma =F\left(\sigma \right),\\ {\partial }_{t}V=G\left(\sigma ,V\right),\end{array}\right.$$

admits a unique global mild (and strong) solution

$$\mathbf{U}\left(t\right)=\left(\sigma \left(t\right),V\left(t\right)\right)\in C\left(\left[0,\mathrm{\infty }\right);\mathbf{X}\right),\mathbf{X}=H^1\left(\Omega \right)\times L^2\left(\Omega \right).$$

Moreover, the solution is asymptotically stable in the sense that there exists a bounded set B ⊂ X (an absorbing set) such that $\mathbf{U}\left(t\right)$∈ B for all sufficiently large t.

Proof of Theorem 2.

Step 1 (Local well-posedness).

The local existence and uniqueness of the mild solution U(t) on [0, T₀] follow directly from the Banach fixed-point argument established in Section 2.2 and Theorem 1.

Step 2 (Total energy inequality).

We utilize the total energy functional

$$E\left(t\right)={\displaystyle \frac{1}{2}}\left(\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2\left(\Omega \right)}^2+\parallel V\left(t\right){\parallel }_{L^2\left(\Omega \right)}^2\right)$$

By Lemma 5, due to the parabolic dissipation of $\sigma$ and the intrinsic damping (−βV) of the rate-and-state variable, the total system energy satisfies the net decay differential inequality:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}E\left(t\right)\le -\lambda E\left(t\right)+C,\mathrm{for}\mathrm{some}\lambda \mathrm{and} C\ge 0.$$

where λ and C depend only on the model parameters, the domain, and the uniform bound on σ from Lemma 1.

Applying the Grönwall inequality to this dissipative form yields the time-uniform bound:

$$E\left(t\right)\le E\left(0\right)e^{-\lambda t}+{\displaystyle \frac{C}{\lambda }}$$

(40)

This energy bound implies that the X-norm of the solution,

$$\parallel \mathbf{U}\left(t\right){\parallel }_{\mathbf{X}}=\parallel \sigma \left(t\right){\parallel }_{H^1\left(\Omega \right)}+\parallel V\left(t\right){\parallel }_{L^2\left(\Omega \right)}$$

remains finite for all t ∈ [0, $\mathrm{\infty }$).

Step 3 (Global extension).

Estimate Equation (40) provides a time-uniform bound on $\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2\left(\Omega \right)}$ and $\parallel V\left(t\right){\parallel }_{L^2\left(\Omega \right)}$. In addition, Lemma 1 yields $\parallel \sigma \left(t\right){\parallel }_{{\mathit{L}}^{\mathrm{\infty }}\left(\mathsf{\Omega }\right)}$ for all t ≥ 0, hence

$$\parallel \sigma (t){\parallel }_{L^2\left(\Omega \right)}\le |\Omega {|}^{1/2}{\mathrm{M}}_{\sigma }$$

From Lemmas 1–3, for every finite T > 0, the corresponding local mild solution satisfies

$$\underset{t\in \ge 0}{\sup }\parallel \mathbf{U}\left(t\right){\parallel }_{\mathbf{X}}=\underset{t\in \left[0,T\right]}{\sup }\left(\parallel \sigma \left(t\right){\parallel }_{H^1}+\parallel V\left(t\right){\parallel }_{L^2}\right)<\mathbf{\infty }$$

Therefore, no blow-up can occur in finite time. By the standard continuation principle for semilinear parabolic systems [28,29], any local solution can be extended as long as its $\mathbf{X}$-norm remains finite.

Step 4 (Ultimate boundedness).

The derived bound Equation (40) shows that

$$\underset{t\to \mathrm{\infty }}{\lim }\sup E\left(t\right)\le {\displaystyle \frac{C}{\lambda }}$$

Hence there exists T_D > 0 such that for all t ≥ T_D

$$E\left(t\right)\le {\displaystyle \frac{2C}{\lambda }}$$

(41)

Define the absorbing set

$$\mathbf{B}=\{\left(\sigma ,V\right)\in \mathbf{X}:{\displaystyle \frac{1}{2}}\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2\left(\Omega \right)}^2+{\displaystyle \frac{1}{2}}\parallel V\left(t\right){\parallel }_{L^2\left(\Omega \right)}^2\le {\displaystyle \frac{2C}{\lambda }},\parallel \sigma \left(t\right){\parallel }_{L^2\left(\Omega \right)}\le |\Omega {|}^{{\displaystyle \frac{1}{2}}}{M}_{\sigma }\}$$

and utilize the relation:

$$\parallel \sigma {\parallel }_{H^1\left(\Omega \right)}={\left(\parallel \sigma {\parallel }_{L^2\left(\Omega \right)}^2+\parallel \mathit{\nabla }\sigma {\parallel }_{L^2\left(\Omega \right)}^2\right)}^{1/2}$$

the bounds $\parallel \sigma \left(t\right){\parallel }_{L^2\left(\Omega \right)}\le |\Omega {|}^{{\displaystyle \frac{1}{2}}}{M}_{\sigma }$ and $\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2\left(\Omega \right)}^2\le {\displaystyle \frac{4C}{\lambda }}$ imply

$$\parallel \sigma \left(t\right){\parallel }_{H^1\left(\Omega \right)}^2\le |\Omega {|}^{{\displaystyle \frac{1}{2}}}{M}_{\sigma }+{\displaystyle \frac{4C}{\lambda }}$$

Hence B is bounded in X, implying $\mathbf{U}\left(t\right)$∈ B for all t ≥ T_D. This establishes dissipativity and ultimate boundedness. The identification of a bounded absorbing set in H¹ space is of significant importance for large-scale modeling, as it defines an admissible search space for optimization algorithms. By rigorously delimiting the phase space, this result ensures that any learning-based framework or gradient-descending process remains within physically meaningful bounds, preventing numerical divergence. □

Remark 3.

System dynamics and long-time asymptotics.

Global Existence and Long-Time Stability:

The a priori estimates established in Lemmas 2 and 3 ensure global well-posedness of the coupled KPP–RS system. In particular, the dissipative L²-estimate for the state variable V prevents finite-time blow-up and yields time-uniform boundedness, while the uniform-in-time H¹-estimate for the stress field σ reflects the intrinsic parabolic dissipation induced by stress diffusion and KPP-type saturation.

Role of Dissipative Rate-and-State Dynamics:

In the present formulation, the rate-and-state evolution law includes an intrinsic damping mechanism (e.g., an aging or slip law with $\beta >0$). Consequently, the $L^2$-dynamics of $V$ satisfy a genuine decay inequality of the form

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel V\left(t\right){\parallel }_{L^2}^2\le -{\lambda }_V\parallel V\left(t\right){\parallel }_{L^2}^2+C_V,{\lambda }_V>0$$

which implies the existence of a compact absorbing set for the state variable.

Consequences for the Coupled System:

Under these dissipative assumptions, the total system energy

$$E\left(t\right)={\displaystyle \frac{1}{2}}\left(\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{L^2}^2+\parallel V\left(t\right){\parallel }_{L^2}^2\right)$$

satisfies a net decay inequality of the form

$${\displaystyle \frac{d}{dt}}E\left(t\right)\le -{\lambda }_{\mathrm{n}\mathrm{e}\mathrm{t}}E\left(t\right)+C_{\mathrm{n}\mathrm{e}\mathrm{t}},{\lambda }_{\mathrm{n}\mathrm{e}\mathrm{t}}>0.$$

Together with the uniform L^∞-bound for σ (Lemma 1), this yields an absorbing set in the phase space X = H¹(Ω) × L²(Ω). Hence, the coupled KPP–RS system is not only globally well posed but also dissipative, i.e., asymptotically bounded in time in the presence of bounded external forces.

Remark 4.

Quantitative Interpretation of the Absorbing Set for Numerical Stability.

The absorbing set established in Theorem 2 provides an explicit upper bound R > 0 such that, after a finite transient time T₀, the solution satisfies:

$$\parallel \mathsf{\nabla }\sigma \left(t\right){\parallel }_{H^1\left(\Omega \right)}^2+\parallel V\left(t\right){\parallel }_{L^2\left(\Omega \right)}^2\le R \forall t\ge T_0$$

This bound has direct computational implications. Since the nonlinear reaction terms in the KPP–RS system are quadratic in $\sigma$ and linear in V, their magnitude in the L²-norm is controlled by:

$$\parallel N\left(\sigma ,V\right){\parallel }_{L^2}\le Q{R}^2$$

for some constant $Q$ > 0 depending on model coefficients. Consequently, the growth rate of residual terms in discrete or learning-based implementations remains uniformly bounded.

In the context of physics-informed neural networks (PINNs) or hybrid neural solvers, such a priori bounds limit the magnitude of residual gradients during optimization. This reduces the risk of instability mechanisms such as gradient explosion or numerical divergence caused by unbounded stress amplification. Therefore, the absorbing set serves as a mathematically certified stability envelope for numerical training procedures.

2.4. Numerical Stability and CFL Condition

We analyze the numerical stability of the explicit scheme applied to the reaction–diffusion system. Let $\mathbf{U}=\left(\sigma ,V\right)$. The fully discrete update is

$${\mathbf{U}}^{n+1}={\mathbf{U}}^n+\Delta t\left(D_{\sigma }{\Delta }_h{\mathbf{U}}^n+N_h\left({\mathbf{U}}^n\right)\right)$$

(42)

where $D_{\sigma }>0$ denotes the same diffusion coefficient introduced in Section 2.1 for the stress evolution equation, ${\Delta }_h$ is the discrete Laplacian, and $N_h\left(\mathbf{U}\right)=\left(F_h\left(\sigma \right),G_h\left(\sigma ,V\right)\right)$ represents the nonlinear reaction terms.

We assume:

• Homogeneous Neumann or Dirichlet boundary conditions;
• Standard second-order finite differences in $d$ dimensions;
• Nonlinear terms Lipschitz-continuous in the discrete $L^2$-norm

Our goal is to obtain a rigorous discrete energy estimate ensuring numerical stability. The CFL-restricted discrete system can be viewed as a finite-dimensional dynamical system whose stability follows from classical stability theory for ordinary differential equations [32]. The main analysis consists of seven steps:

2.4.1. Discrete Energy Functional

The discrete energy argument follows standard approaches for advection–diffusion–reaction schemes of [33]. Define the discrete energy

$$E^n=\parallel {\sigma }^n{\parallel }_h^2+\parallel V^n{\parallel }_h^2$$

where $\parallel \cdot {\parallel }_h$ is the standard cell-centered $L^2$-norm. We aim to prove that for sufficiently small $\Delta t$,

$$E^{n+1}\le C_T{E}^n+C_T,0\le n\Delta t\le T$$

which produces a uniform bound by discrete Gronwall’s inequality.

2.4.2. Discrete Laplacian Negativity

For second-order finite differences, the discrete Laplacian satisfies the discrete Green identity

$$({\Delta }_{h}u,u{)}_h=-\parallel {\nabla }_{h}u{\parallel }_h^2\le 0$$

(43)

and by spectral analysis,

$$\parallel {\Delta }_{h}u{\parallel }_h\le \mid {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\mid \parallel u{\parallel }_h,\mid {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\mid ={\displaystyle \frac{4d}{\Delta x^2}}+\mathcal{O}\left(1\right)$$

(44)

2.4.3. Energy Estimate for the $\mathit{\sigma }$-Equation

The discrete approximation of $\sigma$ equation is:

$${\sigma }^{n+1}-{\sigma }^n=\Delta t\left(D_{\sigma }{\Delta }_h{\sigma }^n+F_h\left({\sigma }^n,V^n\right)\right)$$

Take the discrete inner product with ${\sigma }^{n+1}$:

$${\left({\displaystyle \frac{{\sigma }^{n+1}-{\sigma }^n}{\Delta t}},{\sigma }^{n+1}\right)}_h=D_{\sigma }({\Delta }_h{\sigma }^n,{\sigma }^{n+1}{)}_h+(F_h({\sigma }^n,V^n),{\sigma }^{n+1}{)}_h$$

Use the algebraic identity:

$$\left(a-b,a\right)={\displaystyle \frac{1}{2}}\left(\parallel a{\parallel }^2-\parallel b{\parallel }^2+\parallel a-b{\parallel }^2\right)$$

(45)

to obtain:

$${\displaystyle \frac{1}{2\Delta t}}(\parallel {\sigma }^{n+1}{\parallel }_h^2-\parallel {\sigma }^n{\parallel }_h^2)\le D_{\sigma }({\Delta }_h{\sigma }^n,{\sigma }^{n+1}{)}_h+(F_h({\sigma }^n,V^n),{\sigma }^{n+1}{)}_h$$

Diffusion term:

Using the discrete Green identity,

$$\left({\Delta }_h{\sigma }^n,{\sigma }^{n+1}{)}_h=-({\nabla }_h{\sigma }^n,{\nabla }_h{\sigma }^{n+1}{)}_h\right.,$$

and applying Cauchy–Schwarz and Young’s inequalities, we obtain

$${(\Delta }_h{\sigma }^n,{\sigma }^{n+1}{)}_h=-({\nabla }_h{\sigma }^n,{\nabla }_h{\sigma }^{n+1}{)}_h\le {\displaystyle \frac{1}{2}}\left(\parallel {\nabla }_h{\sigma }^n{\parallel }_h^2+\parallel {\nabla }_h{\sigma }^{n+1}{\parallel }_h^2\right)$$

Because the diffusion operator is evaluated at time level $n$ while the test function is evaluated at time level $n+1$, the cross-term does not yield a negative-definite contribution to the discrete energy. Hence, unlike implicit or semi-implicit schemes, the explicit discretization of diffusion does not provide automatic damping, and its stability must be enforced through a CFL restriction derived from spectral (von Neumann) analysis.

Nonlinear term:

Assume Lipschitz nonlinearity:

$$\parallel F_h\left(\sigma ,V\right)-F_h\left(\tilde{\sigma },\tilde{V}\right){\parallel }_h\le L_F\parallel \left(\sigma ,V\right)-\left(\tilde{\sigma },\tilde{V}\right){\parallel }_h$$

Then, by Young’s inequality,

$$(F_h({\sigma }^n,V^n),{\sigma }^{n+1}{)}_h\le {\displaystyle \frac{\eta }{2}}\parallel {\sigma }^{n+1}{\parallel }_h^2+{\displaystyle \frac{1}{2\eta }}\parallel F_h({\sigma }^n,V^n){\parallel }_h^2$$

(46)

2.4.4. Energy Estimate for the $V$-Equation

Similarly:

$${\displaystyle \frac{1}{2\Delta t}}\left(\parallel V^{n+1}{\parallel }_h^2−\parallel V^n{\parallel }_h^2\right)\le \left(G_h\right({\sigma }^n,V^n),V^{n+1}{)}_h$$

With Lipschitz constant $L_G$,

$$(G_h({\sigma }^n,V^n),V^{n+1}{)}_h\le ϵ\parallel V^{n+1}{\parallel }_h^2+{\displaystyle \frac{1}{4ϵ}}\parallel G_h({\sigma }^n,V^n){\parallel }_h^2$$

(47)

Here η > 0 in Equation (46) and ϵ > 0 in Equation (47) are arbitrary small parameters introduced by Young’s inequality. They will be chosen later (independent of Δt) so that the terms:

${\displaystyle \frac{\eta }{2}}\parallel {\sigma }^{n+1}{\parallel }_h^2$ and $ϵ\parallel V^{n+1}{\parallel }_h^2$ can be absorbed into the left-hand side of the discrete energy inequality.

2.4.5. Combined Energy Inequality

Collecting the estimates from the $\sigma$- and $V$-equations and using the Lipschitz bounds for the nonlinear terms, we obtain the conservative inequality

$$E^{n+1}-E^n\le \Delta t\left(-2D_{\sigma }\parallel {\nabla }_h{\sigma }^n{\parallel }_h^2+C_1{E}^n+C_2\right)$$

(48)

where $C_1,C_2>0$ depend on the Lipschitz constant

$$L_N=L_F+L_G$$

The diffusion term, being non-dissipative in the explicit scheme, is absorbed into the constant $C_1$. This inequality forms the basis for applying the discrete Gronwall lemma once the CFL condition is imposed.

Using Equation (45), we obtain recursive inequality

$$E^{n+1}\le \left(1+C_1\Delta t\right)E^n+C_2\Delta t$$

(49)

which is the appropriate starting point for applying the discrete Gronwall inequality, after imposing a CFL-type restriction to control the amplification of high-frequency modes.

2.4.6. CFL Condition Ensuring Stability

We consider a uniform Cartesian grid with spatial step sizes Δx and Δy in the x- and y-directions, respectively. The spatial discretization of the Laplacian operator is performed using a standard second-order finite difference scheme. For simplicity, the time discretization is taken to be explicit.

In the following stability analysis, the reaction term is linearized around the current solution, and the Lipschitz constant L denotes an upper bound of the Jacobian of the nonlinear reaction operator with respect to the state variables. This constant may represent the Lipschitz bound of F alone or that of the coupled system (F, G), depending on the chosen splitting.

To control the growth of high-frequency modes, we apply a von Neumann analysis to the linearized scalar model. Assume a discrete Fourier mode

$${\sigma }_{j,k}^n={\xi }^n{e}^{i({\kappa }_x\mathrm{j}\Delta x+{\kappa }_{y}k\Delta y)}$$

where ${\kappa }_x$ and ${\kappa }_y$ denote the discrete wave numbers in the x- and y-directions, respectively. Since stability must hold uniformly for all admissible wave numbers, we consider the most restrictive case corresponding to the largest (in magnitude) eigenvalue of the discrete Laplacian operator, denoted by $\mid {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\mid$. For a two-dimensional second-order finite difference discretization, the largest eigenvalue satisfies $\mid {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\mid$ ≤ 2 (Δx⁻² + Δy⁻²), which is a standard bound for the discrete Laplacian. As a result, the explicit dependence on κ is eliminated, and the CFL condition is governed by the worst-case high-frequency mode. This reduction from a mode-dependent condition to a global time step restriction is standard in von Neumann stability analysis for explicit schemes. Substituting into the explicit update yields the amplification factor

$$G\left(\kappa \right)=1+\Delta t\left(D\lambda \left(\kappa \right)+\mu \right)$$

where $\lambda \left(\kappa \right)\le 0$ is the Laplacian eigenvalue and $\mid \mu \mid \le L_N$ is the Jacobian bound for the nonlinear term. Stability requires

$$\mid G\left(\kappa \right)\mid \le 1 \mathrm{for} \mathrm{all} \kappa$$

which is satisfied if and only if

$$\Delta t<{\displaystyle \frac{1}{D_{\sigma }\mid {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\mid +L_N}}$$

(50)

Since

$$\mid {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\mid ={\displaystyle \frac{4d}{\Delta x^2}}$$

Giving

$$\Delta t<{\displaystyle \frac{1}{\frac{4d{D}_{\sigma }}{\Delta x^2}+\left(L_N\right)}} , L_N=L_F+L_G$$

Multiplying numerator/denominator by $\Delta x^2$ and we obtain the explicit CFL condition

$$\Delta t<{\displaystyle \frac{\Delta x^2}{4d{D}_{\sigma }+\left(L_N\right)\Delta x^2}}$$

(51)

where $d$ is the spatial dimension, $D_{\sigma }$ is the diffusion coefficient, and $L_N$ is the Lipschitz constant of the nonlinear reaction term.

The explicit CFL condition derived in Equation (51) provides a theoretical benchmark for time-step selection in both classical numerical schemes and hybrid computational frameworks. This restriction ensures that the discrete diffusion–reaction operator does not introduce artificial amplification of high-frequency modes, thereby preserving the dissipative structure established at the continuous level.

In the context of PINNs and Neural ODE-based solvers, this stability condition offers guidance on the temporal resolution required to maintain consistency between the PDE backbone and the learning architecture. While standard PINNs implement residual constraints through collocation rather than explicit time marching, hybrid approaches that incorporate explicit or semi-explicit PDE solvers remain subject to CFL-type limitations. If such stability constraints are violated, the resulting numerical discretization may produce spurious oscillations in the residual dynamics or unstable iterative updates, even when the continuous system is globally dissipative. Therefore, enforcing the CFL condition ensures that the discrete model preserves the stability envelope of the continuous KPP–RS system. In this sense, stability of the PDE backbone directly supports the robustness and reliability of the learning-based implementation.

For weak nonlinearities ($L_N\ll {\displaystyle \frac{4d{D}_{\sigma }}{\Delta x^2}}$):

$$\Delta t<{\displaystyle \frac{\Delta x^2}{4d{D}_{\sigma }}}$$

(52)

Such stability criteria are particularly relevant for large-scale or data-assimilative simulations, where inappropriate time discretization may lead to spurious instabilities and unreliable training signals.

2.4.7. Final Stability Theorem

Theorem 3.

Explicit Time-Stepping Stability of the Discrete System.

Let

$${\mathbf{U}}^{n+1}={\mathbf{U}}^n+\Delta t\left(D_{\sigma }{\Delta }_h{\mathbf{U}}^n+N_h\left({\mathbf{U}}^n\right)\right)$$

with Lipschitz nonlinearities. If the time step satisfies

$$\Delta t<{\displaystyle \frac{1}{D_{\sigma }\mid {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\mid +L_N}}$$

then the numerical energy $E^n$ obeys

$$E^{n+1}\le \left(1+C\Delta t\right)E^n+C\Delta t$$

and hence is uniformly bounded for $0\le n\Delta t\le T$ by discrete Gronwall’s inequality:

$$E^n\le \left(E^0+CT\right)e^{CT}$$

(53)

Thus, the explicit scheme is numerically stable under the CFL restriction.

2.4.8. Convergence of the Explicit Scheme

Since the explicit finite-difference scheme derived above is consistent with the continuous reaction–diffusion system and satisfies the stability condition provided by Theorem 3, the convergence of the numerical solution follows from the classical Lax–Richtmyer equivalence theorem [34]. In particular, for linearized schemes, consistency together with stability implies convergence in the discrete L²-norm, establishing that

$$‖{\mathbf{U}}_{\mathrm{h}}^{\mathrm{n}}-\mathbf{U}\left(t\right)‖\to 0\mathrm{as}\Delta t,\Delta \mathit{x}\to 0.$$

This confirms that the explicit method is not only stable under the CFL restriction but also convergent.

3. Results

The dynamic stress evolution framework proposed here builds upon the long-standing development of rate-and-state (RS) friction and nonlinear reaction–diffusion formulations in fault mechanics. These results, particularly the identified bounded absorbing set and the derived CFL condition, provide the necessary mathematical guarantees for the stability and convergence of large-scale computational models and neural network architectures. Laboratory and theoretical studies have shown that RS laws successfully capture slip evolution, time-dependent healing, and nucleation processes under variable loading [19,20,21]. Meanwhile, Fisher–KPP-type reaction–diffusion equations represent universal mechanisms for nonlinear propagation and dissipative evolution [1,2,3,4]. The present article is devoted to the analytical foundation of the coupled KPP–RS system. Its primary objective is to establish well-posedness, dissipativity, and stability properties of the PDE system. The calibration of model parameters against real earthquakes (e.g., the 1999 Chi–Chi and 2018 Hualien events), numerical simulations under realistic geometries, and comparisons with observed aftershock sequences will be presented in a companion paper. Such separation between theoretical and computational contributions is standard for mathematically oriented studies and allows the present work to focus exclusively on analytical rigor.

3.1. Deterministic Stress Paths Versus Statistical Clustering

A major implication of the global uniqueness result is that post-mainshock stress evolution follows a deterministic trajectory dictated solely by the initial conditions. This stands in contrast with stochastic branching models such as ETAS [35,36], which successfully reproduce clustering but lack a deterministic physical stress path. Here, ETAS-type statistical models and physics-informed aftershock productivity studies [37,38] are cited for complementary perspectives. From a statistical perspective, earthquake occurrence and aftershock clustering have been extensively investigated using space–time point-process and ETAS-type models [39], inverse seismicity-rate analyses [40], complemented by observational and laboratory studies of aftershock clustering [41,42], and phenomenological laws such as the Omori decay [43]. While these approaches successfully reproduce observed clustering patterns across scales and settings, they do not prescribe a deterministic physical stress evolution pathway. Recent advances in data-driven aftershock modeling demonstrate that spatiotemporal clustering patterns can, to a large extent, be predicted from the evolving stress field surrounding the mainshock rupture [44]. These findings reinforce the interpretation that the deterministic PDE trajectory serves as a physical “backbone,” while stochastic processes operate on top of this evolving landscape. While statistical models like ETAS successfully reproduce observed clustering patterns across scales, the KPP-RS system supplies a unique, deterministic stress evolution pathway dictated by physical laws. This deterministic nature serves as a robust physical prior for modern computational architectures, a positioning we expand upon in Section 4.6 from a machine-learning perspective.

3.2. Connection to Coulomb Stress Transfer and Aftershock Physics

Since the early demonstrations of static Coulomb stress transfer following major earthquakes [45,46], many studies have attempted to integrate stress-based triggering into time-dependent seismicity models [47,48,49]. Recent studies further suggest that aftershock sequences may reflect long-term mechanical deactivation of the source region, rather than purely short-term triggering effects [50]. The dynamic trajectories derived in this study provide a mathematically defined realization of this concept, suggesting that aftershock clustering aligns with evolving stress gradients

3.3. Implications for Time-Dependent Hazard and Operational Forecasting

Time-dependent seismic hazard models have shown significant deviation from classical Poissonian PSHA [51]. Yet these models often lack a physically consistent stress evolution law. The KPP–RS system fills this gap by supplying a deterministic, globally defined stress evolution mechanism. Coupling this PDE backbone with stochastic earthquake generation aligns with the goals of operational earthquake forecasting (OEF) [52], enabling hybrid physical–statistical hazard models.

3.4. Limitations and Future Directions

Despite its mathematical completeness, the proposed framework remains idealized. The KPP-type reaction term and RS friction law necessarily simplify complex, multi-physics processes such as fluid migration, viscoelastic relaxation, and damage rheology. Furthermore, realistic faults are geometrically heterogeneous and embedded in anisotropic or layered media. Future work should, therefore, aim to:

• Calibrate model parameters using well-instrumented sequences.
• Examine sensitivity to initial stress uncertainty.
• Extend the PDE analysis to 3-D geometries and depth-dependent rheologies;
• Couple the deterministic PDE evolution with stochastic forecasting models to evaluate forecast skill improvements quantitatively.

3.5. Summary of Framework Contributions

The analysis presented here is carried out under simplifying assumptions such as a two–dimensional spatial domain, constant diffusion and friction coefficients, and homogeneous Neumann boundary conditions. These assumptions ensure analytical tractability and highlight the core dynamical mechanisms of the DSGF model. While the present framework focuses on a two-dimensional domain in order to isolate the fundamental dynamical mechanisms and maintain analytical transparency, the underlying methodological structure is not restricted to this setting. Nevertheless, the methodology developed in this paper can be extended to more realistic settings, including three–dimensional geometries, heterogeneous material coefficients, and mixed boundary conditions. Such extensions require additional technical ingredients—most notably variable-coefficient elliptic estimates—and will be addressed in future work. These extensions represent a natural continuation of the current work and are identified as directions for future investigation.

In summary, the KPP–RS dynamic stress system provides a rigorous mathematical foundation for viewing postseismic stress evolution as a unique, bounded, globally valid trajectory. This deterministic evolution forms a physically meaningful baseline that governs the spatial pattern in which aftershock-favorable zones emerge. Rather than competing with probabilistic models, the framework supplies the missing dynamical layer that such models lack. By linking nonlinear diffusion, RS friction, stress transfer physics, and probabilistic forecasting within a unified system, this work advances the conceptual integration necessary for next-generation stress-informed PSHA and OEF models capable of more faithfully representing the earthquake cycle.

4. Discussion (Physical Implications of Mathematical Results and Model Reliability)

In this work, we establish—via rigorous functional analytic arguments—the well-posedness, uniform a priori bounds, and global-in-time existence of the coupled KPP–RS dynamic stress system in the space

$$H^1\left(\Omega \right)\times L^2\left(\Omega \right)$$

By establishing a deterministic backbone with proven global dissipativity, this framework serves as a rigorous physical prior that can constrain black-box machine learning models, effectively enhancing their interpretability and preventing unphysical predictions in large-scale stress-driven simulations. These results provide a mathematically solid foundation for the model and give rise to several physically meaningful implications, summarized as follows.

4.1. Deterministic Stress Evolution and Physical Interpretation of Aftershock Sequences

The rigorous proof of global well-posedness and uniqueness established in this study provides a deterministic foundation for postseismic stress evolution. For a given initial condition, the coupled KPP–RS system admits a unique solution trajectory, implying that the evolution of stress following a mainshock is governed by a well-defined physical pathway rather than by purely stochastic fluctuations.

The proof of global asymptotic stability further demonstrates that the system possesses a well-defined terminal state, toward which all admissible solutions converge. Within this framework, postseismic stress evolution may be interpreted as a deterministic backbone constrained by dissipative dynamics. Aftershocks are therefore not treated as accidental point-triggered events, but as necessary energy-release processes that accompany the relaxation of the system toward its attractor.

This perspective motivates the concept of seismic energy debt as an interpretive framework. Because the total system energy is uniformly bounded and the evolution path is unique, the remaining energy available for dissipation through aftershocks or aseismic slip becomes a physically constrained state variable rather than a purely statistical probability. In contrast, classical stress-drop theory quantifies the energy released during rupture but does not specify how the system must subsequently return to equilibrium.

The derived energy inequalities exclude unphysical blow-up scenarios and guarantee uniform boundedness of the stress field. As a result, the post-mainshock energy imbalance is finite and, in principle, quantifiable. This implies that the cumulative energy to be dissipated during the relaxation phase is constrained by physical laws, providing a meaningful upper bound on the system’s overall seismic and aseismic response. While the macroscopic stress evolution is deterministic, the spatial and temporal manifestation of individual aftershocks may still appear irregular. In this framework, stress gradients identify regions of elevated instability, whereas microscopic heterogeneities—such as fault roughness, material defects, or pore-fluid variations—govern the precise locations and timing of discrete energy-release events. This separation reflects a characteristic feature of nonlinear dynamical systems: deterministic macroscopic evolution coexists with apparently stochastic microscopic behavior.

4.2. Exclusion of Finite-Time Blow-Up and Physical Self-Consistency

The uniform-in-time a priori bounds $\parallel \sigma \left(t\right){\parallel }_{L^{\mathrm{\infty }}}$ and $\parallel V\left(t\right){\parallel }_{L^2}$ play a central role in the analysis and immediately excludes the possibility of finite-time blow-up of the solution.

Physical interpretation:

This result confirms that the postseismic stress field σ and the frictional state variable V possess intrinsic regulatory mechanisms under the nonlinear reaction–diffusion–friction coupling. In particular, the stress evolution does not diverge to infinity, and no unphysical singularities appear in the system. Consequently, the KPP–RS system satisfies a fundamental requirement of physical self-consistency, ensuring that it reliably represents admissible geophysical processes.

4.3. Deterministic Evolution and Predictive Capability

The uniqueness of the solution which established through the Banach fixed-point theorem is a key analytical property for any predictive model.

Mathematical implication:

Uniqueness guarantees that a given initial postseismic stress configuration leads to a single, non-branching evolution trajectory of the coupled system.

Physical interpretation:

This property endows the model with strong predictive power. The stress evolution becomes fully deterministic and reproducible, eliminating numerical or analytical ambiguities that could compromise stability. Deterministic trajectories constitute an essential prerequisite for integrating the dynamic stress evolution framework into time-dependent forecasting schemes such as operational earthquake forecasting (OEF) models.

4.4. Global-in-Time Existence and Suitability Across Multiple Physical Time Scales

The proof of global existence ensures that the solution can be extended to arbitrary time horizons. Such a property is crucial for applications involving different phases of the seismic cycle.

Physical interpretation:

The framework naturally accommodates a broad range of time scales: rapid stress redistribution in the seconds to minutes following a mainshock, postseismic slip and frictional healing over hours to days, and slow relaxation processes evolving over years to decades. The mathematical guarantee of global-in-time regularity confirms that the KPP–RS system is inherently suited for modeling both short-term and long-term stress–friction interactions within a unified dynamic structure.

4.5. On the Absence of Spatial Diffusion in the $V$-Equation

Physical interpretation:

The rate-and-state variable $V$ evolves according to a purely local ordinary differential equation, which reflects the standard assumption in frictional fault mechanics that state evolution is governed by local slip processes rather than by spatial diffusion. This modeling choice is consistent with both the aging and slip laws of Dieterich–Ruina friction. While incorporating nonlocal or diffusive effects in $V$ may capture additional physics such as spatially correlated slip transfer, such generalizations would substantially increase the analytical complexity and are beyond the scope of the present study. The framework developed here, however, provides a natural starting point for such extensions.

4.6. Positioning Relative to Existing Physics-Constrained Learning Frameworks

Physical interpretation:

The rapid development of PINNs and hybrid machine-learning paradigms has demonstrated the value of embedding governing equations as soft constraints within data-driven architectures. In seismological applications, recent studies have incorporated RSF laws or stress-transfer equations into learning frameworks to improve interpretability and generalization. These approaches primarily emphasize parameter inversion and residual minimization during training, and have achieved promising empirical results.

However, relatively less attention has been devoted to establishing the global dynamical properties of the underlying nonlinear stress-evolution systems prior to their integration into learning architectures. In particular, questions of global boundedness, dissipativity, and stability-controlled discretization are seldom treated within a unified functional-analytic framework. The present work addresses this gap by constructing a rigorously well-posed, coupled KPP–RS system and establishing:

• Global well-posedness, including existence and uniqueness of solutions via analytic semigroup theory and Banach’s fixed-point theorem;
• Global dissipativity, through the identification of a bounded absorbing set in the H¹ phase space, ensuring intrinsic control of stress amplification;
• A priori numerical stability criteria, including an explicit CFL-type condition that constrains time stepping in discrete implementations;
• Energy-consistent convergence guarantees, linking continuous dynamics and discrete approximations within the Lax–Richtmyer stability framework.

Within the context of earthquake stress modeling, the provision of a fully analyzed dissipative structure for a nonlinear coupled stress-evolution system represents an important complement to existing physics-informed learning approaches. Rather than embedding physical equations solely as residual penalties, the proposed framework establishes a mathematically controlled dynamical backbone upon which learning-based methods may operate. Notably, the derived absorbing set is not merely of theoretical interest. It delineates a physically admissible parameter regime that excludes unbounded stress growth and provides an intrinsic stability envelope for hybrid or neural solvers. In this sense, the system contributes a stability-certified foundation for large-scale computational geophysics and physics-informed artificial intelligence.

5. Conclusions

This study develops and analyzes a coupled KPP–RS system, establishing a physically consistent and mathematically controlled framework for nonlinear stress evolution. In contrast to purely data-driven approaches, the proposed model guarantees stability, boundedness, and physical plausibility of the governing dynamics.

Using analytic semigroup theory, we prove the global existence and uniqueness of solutions, ensuring that the coupled stress–friction system is mathematically well posed. This property is a necessary prerequisite for any learning-based or pattern-recognition framework that relies on stable and reproducible spatiotemporal signals. We further demonstrate the global dissipativity of the system and identify a bounded absorbing set in the H¹ phase space, which provides intrinsic physical constraints on admissible system trajectories and prevents unphysical growth in large-scale optimization or black-box modeling.

On the numerical side, a CFL–type stability condition is derived for explicit finite-difference schemes. Together with a truncation-error analysis, this establishes consistency and convergence of the numerical discretization, providing a theoretically grounded benchmark for time-step selection in numerical simulations and physics-informed computational frameworks. The analytical properties obtained here directly correspond to physical mechanisms in fault systems: KPP saturation represents stress-limiting processes, diffusion governs postseismic relaxation, bounded frictional-state evolution reflects healing and stabilization, and energy decay explains the progressive reduction in aftershock activity. Applications to real earthquake sequences will be presented in a separate companion study. Overall, this work provides a rigorous theoretical foundation for DSGF-type models and supports their reliable application in dynamic stress modeling and provides a mathematically consistent backbone for future physics-informed or learning-assisted frameworks. Furthermore, the proposed KPP–RS system provides a rigorous physical prior that can be seamlessly integrated into modern computational architectures, such as PINNs or Neural Ordinary Differential Equations (Neural ODEs). By embedding these deterministic constraints, practitioners can enhance the interpretability of large-scale models, shifting from purely black-box approaches to physically consistent representations of stress-driven dynamics. By establishing global well-posedness, dissipativity, and CFL-consistent discretization within a unified analytical structure, the present work provides a stability-controlled foundation for future hybrid computational modeling of stress evolution.