Search Results (220)

This paper investigates the well-posedness of the inverse problem for the time-fractional Burgers equation, which aims to reconstruct initial conditions from terminal observations. Such equations are crucial for the modeling of hydrodynamic phenomena with memory effects. The inverse problem involves inferring initial conditions from terminal observation data, and such problems are typically ill-posed. A framework based on optimal control theory is proposed, addressing the ill-posedness via $H^1$ regularization. Three substantial results are achieved: (1) a rigorous mathematical framework transforming the ill-posed inverse problem into a well-posed optimization problem with proven existence of solutions; (2) theoretical guarantee of solution uniqueness when the regularization parameter is $\alpha >0$ and the stability is of order O($\delta$) with respect to observation noise ($\delta$); and (3) the discovery of a “super-stability” phenomenon in numerical experiments, where the actual stability index (0.046) significantly outperforms theoretical expectations (1.0). Finally, the theoretical framework is validated through comprehensive numerical experiments, demonstrating the accuracy and practical effectiveness of the proposed optimal control approach for the reconstruction of hydrodynamic initial conditions. Full article

This paper proposes a Distributed Adaptive Regularization Algorithm (DARN) for solving composite non-convex and non-smooth optimization problems in multi-agent systems. The algorithm employs a three-phase iterative framework to achieve efficient collaborative optimization: (1) a local regularized optimization step, which utilizes proximal mappings to enforce strong convexity of weakly convex objectives and ensure subproblem well-posedness; (2) a consensus update based on doubly stochastic matrices, guaranteeing asymptotic convergence of agent states to a global consensus point; and (3) an innovative adaptive regularization mechanism that dynamically adjusts regularization strength using local function value variations to balance stability and convergence speed. Theoretical analysis demonstrates that the algorithm maintains strict monotonic descent under non-convex and non-smooth conditions by constructing a mixed time-scale Lyapunov function, achieving a sublinear convergence rate. Notably, we prove that the projection-based update rule for regularization parameters preserves lower-bound constraints, while spectral decay properties of consensus errors and perturbations from local updates are globally governed by the Lyapunov function. Numerical experiments validate the algorithm’s superiority in sparse principal component analysis and robust matrix completion tasks, showing a 6.6% improvement in convergence speed and a 51.7% reduction in consensus error compared to fixed-regularization methods. This work provides theoretical guarantees and an efficient framework for distributed non-convex optimization in heterogeneous networks. Full article

This study presents an image inpainting model based on an energy functional that incorporates the $L^p$ norm of the fractional Laplacian operator as a regularization term and the $H^{-1}$ norm as a fidelity term. Using the properties of the fractional Laplacian operator, the $L^p$ norm is employed with an adjustable parameter p to enhance the operator’s ability to restore fine details in various types of images. The replacement of the conventional $L^2$ norm with the $H^{-1}$ norm enables better preservation of global structures in denoising and restoration tasks. This paper introduces a diffusion partial differential equation by adding an intermediate term and provides a theoretical proof of the existence and uniqueness of its solution in Sobolev spaces. Furthermore, it demonstrates that the solution converges to the minimizer of the energy functional as time approaches infinity. Numerical experiments that compare the proposed method with traditional and deep learning models validate its effectiveness in image inpainting tasks. Full article

We advance a mathematical framework for collective conviction by deriving a continuum theory from the network-based model introduced by us recently. The resulting equation governs the evolution of belief through a degenerate anisotropic logistic–diffusion process, where diffusion slows as conviction saturates. In one spatial dimension, we prove global well-posedness, demonstrate spectral front pinning that arrests the spread of influence at finite depth, and construct explicit traveling-wave solutions. In two dimensions, we uncover a geometric mechanism of curvature–induced quenching, where belief propagation halts along regions of low effective mobility and curvature. Building on this insight, we formulate a variational principle for optimal control under resource constraints. The derived feedback law prescribes how to spatially allocate repression effort to maximize inhibition of front motion, concentrating resources along high-curvature, low-mobility arcs. Numerical simulations validate the theory, illustrating how localized suppression dramatically reduces transverse spread without affecting fast axes. These results bridge analytical modeling with societal phenomena such as protest diffusion, misinformation spread, and institutional resistance, offering a principled foundation for selective intervention policies in structured populations. Full article

Dengue fever remains a major global health threat, and mathematical models are crucial for predicting its spread and evaluating control strategies. This study introduces a highly flexible dengue transmission model using a novel piecewise fractional derivative framework, which can capture abrupt changes in epidemic dynamics, such as those caused by public health interventions or seasonal shifts. We conduct a rigorous comparative analysis of four widely used but distinct mechanisms of disease transmission (incidence rates): Harmonic Mean, Holling Type II, Beddington–DeAngelis, and Crowley–Martin. The model’s well-posedness is established, and the basic reproduction number (${\Re }_0$) is derived for each incidence function. Our central finding is that the choice of this mathematical mechanism critically alters predictions. For example, models that account for behavioral changes (Beddington–DeAngelis, Crowley–Martin) identify different key drivers of transmission compared to simpler models. Sensitivity analysis reveals that vector mortality is the most influential control parameter in these more realistic models. These results underscore that accurately representing transmission behavior is essential for reliable epidemic forecasting and for designing effective, targeted intervention strategies. Full article

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness and Ulam–Hyers stability are then derived using Banach’s contraction principle. By introducing a novel singular-kernel Gronwall inequality, we extend the analysis to Ulam–Hyers–Rassias stability and continuous dependence on initial data. The theoretical framework is unified for general fractional orders and validated through examples, demonstrating its applicability to implicit systems with memory effects. Key contributions include weighted-space analysis and stability criteria for this class of equations. Full article

In this paper, we focus on mean-field stochastic differential equations driven by G-Brownian motion (G-MFSDEs for short) with a drift coefficient satisfying the local one-sided Lipschitz condition with respect to the state variable and the global Lipschitz condition with respect to the law. We are concerned with the well-posedness and the numerical approximation of the G-MFSDE. Probability uncertainty leads the resulting expectation usually to be the G-expectation, which means that we cannot apply the numerical approximation for McKean–Vlasov equations to G-MFSDEs directly. To numerically approximate the G-MFSDE, with the help of G-expectation theory, we use the sample average value to represent the law and establish the interacting particle system whose mean square limit is the G-MFSDE. After this, we introduce the modified stochastic theta method to approximate the interacting particle system and study its strong convergence and asymptotic mean square stability. Finally, we present an example to verify our theoretical results. Full article

The current paper is devoted to the dynamical property of the stochastic Cox–Ingersoll–Ross (CIR) model with pure jump noise, which is an extension of the CIR model. Firstly, we characterize the existence and 2-moment of the CIR process with a pure jump process. Consequently, we provide sufficient conditions for the compensated Poisson random measure under which the CIR process with a pure jump process is ergodic. Moreover, the stationary solution can be constructed from the invariant measure. Some numerical simulations are provided to visualize the theoretical results. Full article

We consider an evolutionary inclusion associated with a time-dependent convex in an abstract Hilbert space. We recall a unique solvability result obtained based on arguments of nonlinear equations with maximal monotone operators combined with a penalty method. Then, we state and prove two well-posedness results. Next, we provide three examples of such inclusions that arise in mechanics. The first one concerns an elastic–perfectly plastic constitutive law, while the last two examples are mathematical models that describe the equilibrium of an elastic body and an elastic–perfectly plastic body, respectively, in frictional contact with an obstacle. The contact is bilateral and the friction is modeled with the Tresca friction law. We use our abstract results in the study of these examples to provide the convergence of the solution with respect to the data. Full article

In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For this, we first established the existence and uniqueness of a global mild solution for small and divergence-free initial data. Moreover, our approach is based on proving sharp bilinear convolution estimates in critical Sobolev norms, which in turn guarantee the uniform analyticity of both the velocity and magnetic fields with respect to time. Furthermore, leveraging the decay properties of the associated fractional heat semigroup and a bootstrap argument, we derived algebraic decay rates and established the long-time dissipative behavior of FrMHD solutions. These results extended the existing literature on fractional Navier–Stokes equations by fully incorporating magnetic coupling and Coriolis effects within a unified fractional-dissipation framework. Full article

In this study, we develop a fractional-order mathematical model for investigating the transmission dynamics of monkeypox (Mpox), accounting for interactions between humans, rodents, and environmental reservoirs. The model uniquely integrates two key control strategies—public health awareness and environmental sanitation—often overlooked in previous models. We analyze the model’s well-posedness by establishing the existence, uniqueness, and positivity of solutions using the fixed-point theorem. Using data from the Democratic Republic of Congo, we estimate the model parameters and demonstrate that the fractional-order model ($\varphi =0.5$) fits real-world data more accurately than its integer-order counterpart ($\varphi =1$). The sensitivity analysis using partial rank correlation coefficients highlights the key drivers of disease spread. Numerical simulations reveal that the memory effects inherent in fractional derivatives significantly influence the epidemic’s trajectory. Importantly, our results show that increasing awareness ($ϵ$) and sanitation efforts ($\eta$) can substantially reduce transmission, with sustained suppression of Mpox when both parameters exceed $90\%$. These findings highlight the synergistic impact of behavioral and environmental interventions in controlling emerging zoonotic diseases. Full article

This paper investigates the Cauchy problem for the full compressible Navier–Stokes–Korteweg equations, which model fluid dynamics with capillary properties in ${\mathbb{R}}^d(d\ge 3)$. And the global well-posedness and Gevrey analytic of strong solutions for the system are established in the $L^2-L^p$ type critical hybrid Besov space with $2\le p\le \frac{2d}{d-2}$ and $p<d$. Full article

The white-noise-driven KdV-type Boussinesq system is a class of stochastic partial differential equations (SPDEs) that describe nonlinear wave propagation under the influence of random noise—specifically white noise—and generalize features from both the Korteweg–de Vries (KdV) and Boussinesq equations. We consider a Cauchy problem for two stochastic systems based on the KdV-type Boussinesq equations. For these systems, we determine sufficient conditions to ensure that this problem is locally and globally well posed for initial data in Sobolev spaces by the linear and bilinear estimates and their modification together with the Banach fixed point. Full article

In this paper, we study a class of nonlocal Schrödinger–Poisson–Slater equations: $-\Delta u+u+\lambda \left(I_{\alpha }\ast {\left|u\right|}^q\right){\left|u\right|}^{q-2}u={\left|u\right|}^{p-2}u$, where $q,p>1$, $\lambda >0$, and $I_{\alpha }$ is the Riesz potential. We obtain the existence, stability, and symmetry-breaking of solutions for both radial and nonradial cases. In the radial case, we use variational methods to establish the coercivity and weak lower semicontinuity of the energy functional, ensuring the existence of a positive solution when p is below a critical threshold $\overline{p}={\displaystyle \frac{4q+2\alpha }{2+\alpha }}$. In addition, we prove that the energy functional attains a minimum, guaranteeing the existence of a ground-state solution under specific conditions on the parameters. We also apply the Pohozaev identity to identify parameter regimes where only the trivial solution is possible. In the nonradial case, we use the Nehari manifold method to prove the existence of ground-state solutions, analyze symmetry-breaking by studying the behavior of the energy functional and identifying the parameter regimes in the nonradial case, and apply concentration-compactness methods to prove the global well-posedness of the Cauchy problem and demonstrate the orbital stability of the ground state. Our results demonstrate the stability of solutions in both radial and nonradial cases, identifying critical parameter regimes for stability and instability. This work enhances our understanding of the role of nonlocal interactions in symmetry-breaking and stability, while extending existing theories to multiparameter and higher-dimensional settings in the Schrödinger–Poisson–Slater model. Full article

In this paper, we prove the local well-posedness of classical solutions $(\psi ,A,\varphi )$ to the $nD(n\ge 3)$ time-dependent Ginzburg–Landau model in superconductivity with the choice of Coulomb gauge and the main assumptions ${\psi }_0,A_0\in H^s\left({\mathbb{R}}^n\right)$ with $\mathrm{div}{A}_0=0$ in ${\mathbb{R}}^n$ and $s>{\displaystyle \frac{n}{2}}$. This result can be used in the proof of regularity criterion and global-in-time well-posedness of the strong solution. Full article