Decay Estimates for a Lamé Inverse Problem Involving Source and Damping Term with Variable-Exponent Nonlinearities

Abstract

We investigate an inverse problem involving source and damping term with variable-exponent nonlinearities. We establish adequate conditions on the initial data for the decay of solutions as the integral overdetermination approaches zero over time within an acceptable range of variable exponents. This class of inverse problems, where internal terms such as source and damping are to be determined from indirect measurements, has significant relevance in real-world applications—ranging from geophysical prospecting to biomedical engineering and materials science. The accurate identification of these internal mechanisms plays a crucial role in optimizing system performance, improving diagnostic accuracy, and constructing predictive models. Therefore, the results obtained in this study not only contribute to the theoretical understanding of nonlinear dynamic systems but also provide practical insights for reconstructive analysis and control in applied settings. The asymptotic behavior and decay conditions we derive are expected to be of particular interest to researchers dealing with stability, uniqueness, and identifiability in inverse problems governed by nonstandard growth conditions.

1. Introduction

Inverse problems involve determining an unknown property of a medium or object based on observations of its response to a probing signal. Consequently, the theory of inverse problems provides a theoretical foundation for applications such as nondestructive and remote sensing evaluation. Several inverse problems occur naturally and have significant implications in identifying flying objects (missiles, airplanes, etc.), objects submerged in water (submarines, schools of fish, etc.), and various other scenarios. However, solving these problems is generally challenging due to two primary reasons: nonlinearity and ill-posedness [1,2,3,4].

In the field of geophysics, it is common to transmit an acoustic wave from the Earth’s surface and record the scattered field on the surface for various source positions, either at a fixed frequency or multiple frequencies. The objective of the inverse problem is to determine the subsurface irregularities or inhomogeneities. In technology, one may measure the eigenfrequencies of a material sample, and the inverse problem involves identifying defects within the material, such as a hole in a metal. In geophysics, these inhomogeneities could correspond to features like an oil deposit, a cave, or a mine. In the medical field, the inhomogeneity may be represented by a tumor or any abnormality within the human body [2,5,6,7].

Assume that $\Omega$ is a bounded domain in $R^n(n\ge 1)$ with a $\partial \Omega$ (smooth boundary). We study the solutions $\left\{u(x,t),f\left(t\right)\right\}$ of the following inverse problem:

$$\begin{array}{cc}\hfill & u_{tt}-div(|\nabla u{|}^{r(.)-2}\nabla u)-{\Delta }_{e}u+\beta u_t+a|u_t{|}^{m(.)-2}{u}_t\hfill \\ & +h(x,t,u,\nabla u)+b|u{|}^{p(.)-2}u=f\left(t\right)w\left(x\right),(x,t)\in \Omega \times (0,\infty )\hfill \end{array}$$

(1)

$$\begin{array}{c}\hfill u(x,t)={\displaystyle \frac{\partial u}{\partial v}}(x,t)=0,(x,t)\in \partial \Omega \times (0,\infty )\end{array}$$

(2)

$$\begin{array}{c}\hfill u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),x\in \Omega \end{array}$$

(3)

$$\begin{array}{c}\hfill {\int }_{\Omega }u(x,t)w\left(x\right)dx=\psi \left(t\right),t>0,\end{array}$$

(4)

$\beta ,b,a>0$ and $w\left(x\right)$, $h(x,t,u,\nabla u)$, $\psi \left(t\right)$ are real functions that meet certain conditions, which will be explained later, but $\left\{u(x,t),f\left(t\right)\right\}$ is unknown. Additional information about the solution to the inverse problem is given in the form of integral overdetermination condition [8]. In this inverse problem, the function $u(x,t)$ typically represents a measurable physical quantity such as temperature, displacement, density, or wave amplitude within a spatial domain $\Omega$ over time. From a physical point of view, Equation (4) can be interpreted as a distributed measurement of $u(x,t)$ obtained by a device that averages the physical quantity over the domain $\Omega$ with respect to a weight function $w\left(x\right)$. That is, rather than measuring the value of u at a specific point, the measuring instrument captures an integrated (or averaged) response from the entire domain, reflecting more realistic physical scenarios in which sensors or instruments have spatial sensitivity.

Also, ${\Delta }_e$ refers to the elasticity operator, which is a differential operator of size $n\times n$ and is defined as follows:

$$\begin{array}{c}\hfill {\Delta }_{e}u=\mu \Delta u+(\alpha +\mu )\nabla \left(divu\right),u={(u_1,u_2,...,u_n)}^T,\end{array}$$

where $\mu$ and $\alpha$ are the Lamé constants such that

$$\begin{array}{c}\hfill \mu >0,\alpha +\mu \ge 0.\end{array}$$

The functions $p\left(x\right)$, $r\left(x\right)$, and $m\left(x\right)$ are given as measurable and continuous functions with the condition that

$$\begin{array}{c}2<p_1\le p\left(x\right)\le p_2<\infty ,\hfill \\ 2<r_1\le r\left(x\right)\le r_2<\infty ,\hfill \\ 2<m_1\le m\left(x\right)\le m_2<\infty ,\hfill \\ p_1:=essin{f}_{x\in \overline{\Omega }}p\left(x\right),p_2:=esssu{p}_{x\in \overline{\Omega }}p\left(x\right),\hfill \\ r_1:=essin{f}_{x\in \overline{\Omega }}r\left(x\right),r_2:=esssu{p}_{x\in \overline{\Omega }}r\left(x\right),\hfill \\ m_1:=essin{f}_{x\in \overline{\Omega }}m\left(x\right),m_2:=esssu{p}_{x\in \overline{\Omega }}m\left(x\right).\hfill \end{array}$$

(5)

The difference of this work from previous studies can be stated as follows. The source term ($bu{\left|u\right|}^{p(.)-2}$) plays a fundamental role in determining whether the solution grows or remains stable under certain conditions. If this term were absent, the energy transfer in the equation would be incomplete. The source term governs the time dynamics of the solution—without it, the system would reduce to a purely damped and diffusive model. Additionally, in many physical and engineering applications, the source term in wave equations represents external energy inputs or internally generated energy. For example, in elasticity theory, internal energy production within a material or external energy inputs to the system are modeled through the source term. Without it, only a passive wave propagation model would be obtained.

The function $h(x,t,u,\nabla u)$ represents forcing, nonlinear reactions, and intrinsic forces within the system. This function appears in various applications, including elasticity, thermal diffusion, and electromagnetic waves. For instance, internal forces arising from the structure of a material, plastic deformations, and other physical processes are encapsulated within $h(x,t,u,\nabla u)$. Without this term, the model would behave as a simple damped diffusion equation and fail to capture the essential characteristics of nonlinear systems.

In summary, the present study covers a broader class of problems by incorporating a nonlinear source term with variable exponents and a nonlinear forcing term. If these terms were removed, the resulting energy estimates would be less comprehensive, and the solution dynamics would be less realistic. From a literature contribution perspective, analyzing energy decay under nonlinear terms with variable exponents enables the derivation of new results.

Therefore, the source term $bu{\left|u\right|}^{p(.)-2}$ and the function $h(x,t,u,\nabla u)$ are crucial components of this study. Without them, the analysis would be incomplete. By incorporating these terms, we have made a meaningful contribution to the literature. The points discussed here have also been incorporated into Section 1 and Section 4 of the manuscript.

Studies on energy decay in inverse problems play a crucial role in ensuring solution reliability by establishing important mathematical properties such as stability and uniqueness as energy dissipation helps control the system’s long-term behavior. The decrease in energy over time contributes to stability by reducing the system’s sensitivity to initial conditions. If the energy of a system vanishes as time approaches infinity, it implies that small initial perturbations diminish over time, which is an indication of asymptotic stability [9]. In the context of inverse problems, certain decay rates are also used to ensure uniqueness despite incomplete or noisy data, as shown in [10]. Thus, the decay of solutions is a key aspect of inverse problem analysis. The aim of this work is to examine the interaction between elasticity and variable-exponent inverse problems from the perspective of energy decay and to expand the mathematical framework related to this approach.

Firstly, we discuss some significant findings in the field of inverse problems. Kalantarov and Eden [11] investigated the following problem:

$$\begin{array}{cc}\hfill & u_t-\Delta u+b(x,t,u,\nabla u)-|u{|}^{p}u=f\left(t\right)w\left(x\right),x\in \Omega ,t>0,\hfill \\ & u(x,t)=0,x\in \partial \Omega ,t>0\hfill \\ & u(x,0)=u_0\left(x\right),x\in \Omega ,\hfill \\ & {\int }_{\Omega }u(x,t)w\left(x\right)dx=\psi \left(t\right),t>0.\hfill \end{array}$$

(6)

Shahrouzi investigated a class of inverse source problems involving variable-exponent nonlinearities. He extended the previous findings [12,13] to address the inverse problem with variable-exponent nonlinearities. Shahrouzi demonstrated a blow-up result for specific solutions with positive initial energy when $a=\beta =0$ and $\psi \left(t\right)\equiv 1$. Additionally, he examined the scenario where $b=0$ and $h(x,t,u,\nabla u)\equiv 0$, showing that the energy of the respective problems generally decay toward zero as the solution progresses [1].

The uniqueness and existence of solutions to inverse problems for parabolic equations have been investigated by several authors [4,14,15,16,17]. The asymptotic stability of solutions to these problems has been studied in [8,11,17,18]. Also, energy decay studies in viscoelastic equations have increased recently (see [19,20,21,22,23,24,25]). Finally, both at the research stage of our study and at the writing stage, you can look for valuable articles that we support from their problems and the methods used (see [26,27,28,29,30,31,32,33,34]).

In this study, we establish the overall decay of solutions for suitable variable exponents and initial data, considering all terms of the problems described by Equations (1)–(4). When $b=0$, it has been possible to achieve the desired result even in the existence of the source term, as demonstrated in the literature [1].

2. Preliminaries

In this part, we will discuss some functionals and notations. We use the symbol $\parallel .{\parallel }_q$ to represent the $L^q$-norm over $\Omega$. Specifically, the $L^2$-norm is denoted by $\parallel .\parallel$ in $\Omega$. Let the functions $h(x,t,u,\nabla u)$, $w\left(x\right)$, and the data functions satisfy the following criteria:

$$|h(x,t,u,\nabla u)|\le M_1|u{|}^{{\displaystyle \frac{p(.)}{2}}}+M_2|\nabla u{|}^{{\displaystyle \frac{r(.)}{2}}},$$

(7)

with some positive $M_1$ and $M_2$, in this context, the constants $M_1$ and $M_2$ control the generality and accuracy of the estimates. The functions $p(x,t)$ and $r(x,t)$ typically define how the magnitudes of the function and its derivatives should be measured at each point. Such equations are commonly used in Sobolev spaces and are encountered when dealing with differential equations or generalized functions. Variable exponents satisfy [35,36,37,38,39];

$$\begin{array}{cc}\hfill & u_0\in H_0^2(\Omega )\cap L^{r(.)}(\Omega )\cap L^{p(.)}(\Omega ),u_1\in L^2(\Omega )\cap L^{m(.)}(\Omega ),\hfill \\ & {\int }_{\Omega }{u}_0\left(x\right)w\left(x\right)dx=\psi \left(0\right),\hfill \end{array}$$

(8)

$$w\in H_0^2(\Omega )\cap L^{r(.)}(\Omega )\cap L^{m(.)}(\Omega )\cap L^{p(.)}(\Omega ),{\int }_{\Omega }{w}^2\left(x\right)dx=1.$$

(9)

Here, we require some theories and hypotheses concerning Sobolev and Lebesgue spaces with variable exponents. Let $p\left(x\right)\ge 1$ be a measurable function. We make the following assumptions:

$$C_{+}\left(\overline{\Omega }\right)=\{h|h\in C\left(\overline{\Omega }\right),h(.)>1foranyx\in \overline{\Omega }\},$$

$$h^{+}=\underset{\overline{\Omega }}{max}h(.),h^{-}=\underset{\overline{\Omega }}{minh(.)}foranyh\in C\left(\overline{\Omega }\right),$$

$$L^{p(.)}(\Omega )=\{u|uisameasurablereal-valuedfunction,{\int }_{\Omega }|u\left(x\right){|}^{p(.)}dx<\infty \}.$$

We define the Lebesgue space with a variable exponent, denoted as $L^{p(.)}(\Omega )$, equipped with the Luxembourg-type norm as follows:

$$\parallel u{\parallel }_{p(.)}:=inf\{\lambda >0|{\int }_{\Omega }{\left|{\displaystyle \frac{u\left(x\right)}{\lambda }}\right|}^{p(.)}dx\le 1\}.$$

Lemma 1

([35,36,37,38,39]). (Generalized Hölder Inequality) Let $1\le q^{-}\le q(.)\le q^{+}<\infty$, $1\le p^{-}\le p(.)\le p^{+}<\infty$ be measurable functions defined on Ω such that ${\displaystyle \frac{1}{q(.)}}+{\displaystyle \frac{1}{p(.)}}=1$ for almost every $x\in \Omega$. Suppose that $u\in W^{k,q(.)}(\Omega )$ and $v\in W^{k,p(.)}(\Omega )$ for some non-negative integer k. Then, the following generalized Hölder inequality holds:

$${\parallel uv\parallel }_{L^{r(.)}(\Omega )}\le {\parallel u\parallel }_{L^{q(.)}(\Omega )}{\parallel v\parallel }_{L^{p(.)}(\Omega )}$$

where $r(.)$ is defined by

$$r(.)={\displaystyle \frac{q(.)p(.)}{q(.)+p(.)-1}}.$$

Lemma 2

([35,36,37,38,39]). Assume that Ω be a bounded domain in $R^n$, $0<|\Omega |<\infty$ and $q(.)\in L_{+}^{\infty }(\Omega )$. For the embedding of $L^{q(.)}(\Omega )\hookrightarrow L^{p(.)}(\Omega )$ to hold, a necessary and sufficient condition is that condition $x\in \Omega$ satisfies $p(.)\le q(.).$

$W^{1,p(.)}(\Omega )$ is defined as follows:

$W^{1,p(.)}(\Omega )=\{u\in L^{p(.)}(\Omega )|\nabla uexistsand|\nabla u|\in L^{p(.)}(\Omega )\}$ [35,36,37,38,39].

This space, equipped with the norm

$$\parallel u{\parallel }_{W^{1,p(.)}(\Omega )}=\parallel \nabla u{\parallel }_{p(.)}+\parallel u{\parallel }_{p(.)},$$

forms a Banach space.

Additionally, let $W_0^{1,p(.)}(\Omega )$ be the closure of $C_0^{\infty }(\Omega )$ in $W^{1,p(.)}(\Omega )$. The dual space of $W_0^{1,p(.)}(\Omega )$ is denoted by $W^{-1,p^{\prime }(.)}(\Omega )$, defined similarly to the standard Sobolev spaces, where

$${\displaystyle \frac{1}{p(.)}}+{\displaystyle \frac{1}{p^{\prime }(.)}}=1.$$

If we determine

$$p^{*}(.)=\left\{\begin{array}{cc}{\displaystyle \frac{Np(.)}{esssu{p}_{.\in \overline{\Omega }}(N-p(.))}},\hfill & p^{+}<N\\ \infty ,\hfill & p^{+}\ge N\end{array}\right.,$$

then the following lemmas are written.

Lemma 3

([35,36,37,38,39]). Let Ω be a bounded domain in $R^n$ and $p(.)$ any measurable bounded exponent.

If $q(.)<p^{*}(.)$ and $q\in C_{+}\left(\overline{\Omega }\right)$ for all $x\in \overline{\Omega }$, the embedding $W^{1,p(.)}(\Omega )\hookrightarrow L^{q(.)}(\Omega )$ is continuous and compact.

Lemma 4.

(Poincare Inequality) Let Ω be a bounded domain in $R^n$ and $p(.)\in L(\Omega )$. In this case, for any $u\in W_0^{1,p(.)}(\Omega )$, it is

$$\parallel u{\parallel }_{p(.)}\le C\parallel \nabla u{\parallel }_{p(.)},$$

where $C=C(n,p(.),\Omega )$.

As stated in the last part of Lemma (3), it is known that the space $W_0^{1,p(.)}(\Omega )$ has an equivalent norm given by $\parallel u{\parallel }_{W^{1,p(.)}}=\parallel \nabla u{\parallel }_{p(.)}$.

We observe Young’s inequality as follows:

$$ab\le \theta a^{q(.)}+C(\theta ,q(.))b^{q^{\prime }(.)},a,b\ge 0,\beta >0,{\displaystyle \frac{1}{q(.)}}+{\displaystyle \frac{1}{q^{\prime }(.)}}=1,$$

(10)

where $C(\theta ,q(.))={\displaystyle \frac{1}{q^{\prime }(.)}}{\left(\theta q\right(x\left)\right)}^{{\displaystyle \frac{-q^{\prime }(.)}{q(.)}}}$. Under specific conditions, when $\theta ={\displaystyle \frac{1}{q(.)}}$, we have from (10):

$$ab\le {\displaystyle \frac{a^{q(.)}}{q(.)}}+{\displaystyle \frac{b^{q^{\prime }(.)}}{q^{\prime }(.)}}.$$

(11)

With the condition (9) and multiplying (1) in $w\left(x\right)$ and then integrating, the problems (1)–(4) can be reformulated as the following direct problem:

$$\begin{array}{cc}\hfill & u_{tt}-div(|\nabla u{|}^{r(.)-2}\nabla u)-{\Delta }_{e}u+\beta u_t+a|u_t{|}^{m(.)-2}{u}_t+h(x,t,u,\nabla u)\hfill \\ & +b|u{|}^{p(.)-2}u=f\left(t\right)w\left(x\right),(x,t)\in \Omega \times (0,\infty )\hfill \end{array}$$

(12)

$$u(x,t)={\displaystyle \frac{\partial u}{\partial v}}(x,t)=0,(x,t)\in \partial \Omega \times (0,\infty )$$

(13)

$$u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),x\in \Omega$$

(14)

where the unknown function $f\left(t\right)$ is replaced by

$$\begin{array}{cc}\hfill & f\left(t\right)=\beta {\psi }^{\prime }\left(t\right)+{\psi }^{″}\left(t\right)+\mu {\int }_{\Omega }\nabla u\nabla w\left(x\right)dx+(\alpha +\mu ){\int }_{\Omega }\left(divu\right)\left(divw\right(x\left)\right)dx\hfill \\ \hfill & +{\int }_{\Omega }|\nabla u{|}^{r(.)-1}\nabla w\left(x\right)dx+{\int }_{\Omega }h(x,t,u,\nabla u)w\left(x\right)dx+a{\int }_{\Omega }|u_t{|}^{m(.)-2}{u}_{t}w\left(x\right)dx\hfill \\ \hfill & +b{\int }_{\Omega }|u{|}^{p(.)-1}w\left(x\right)dx\hfill \end{array}$$

(15)

The existence of local solutions for problems (1)–(3) can be demonstrated through the Galerkin approach [15].

Theorem 1.

(Local Existence) Let $u_0\in W_0^{1,r(.)}(\Omega )$, $u_1\in L^2(\Omega )$ and suppose that (5) and (7)–(9) be supplied, then problem (1)–(3) has a unique weak solution such that

$$\begin{array}{cc}\hfill & u\in L^{p(.)}\left((0,T),\Omega \right)\cap L^{\infty }\left((0,T),W_0^{1,r(.)}(\Omega )\right),\hfill \\ \hfill & u_t\in L^{m(.)}\left((0,T),\Omega \right)\cap L^{\infty }\left((0,T),L^2(\Omega )\right),\hfill \\ \hfill & u_{tt}\in L^{\infty }\left((0,T),W_0^{-1,r^{\prime }(.)}\right)(\Omega ),\hfill \end{array}$$

for any $T>0$ and ${\displaystyle \frac{1}{r(.)}}+{\displaystyle \frac{1}{r^{\prime }(.)}}=1.$

3. Energy Decay

In this part, we demonstrate our result with the nonlinear source term ($b|u{|}^{p(.)-2}u$) and $h(x,t,u,\nabla u)$ real function. To reach the conclusion, the Gronwall inequality used is typically written in the following form:

Lemma 5.

Let $u\left(t\right)$ be a non-negative, continuous function on the interval $[t_0,T]$, and assume that $u\left(t\right)$ satisfies the differential inequality:

$$u\left(t\right)\le c+{\int }_{t_0}^t\alpha \left(s\right)u\left(s\right)dsforallt\in [t_0,T],$$

where c is a constant and $\alpha \left(s\right)$ is a continuous, non-negative function. Then, the solution to this inequality satisfies the following:

$$u\left(t\right)\le c{e}^{{\int }_{t_0}^t\alpha \left(s\right)ds}.$$

Here, $u\left(t\right)$ represents the function with a behavior (decay or growth) that we are analyzing, and $\alpha \left(s\right)$ represents a function that bounds the growth rate of $u\left(t\right)$. The result gives an upper bound for $u\left(t\right)$ in terms of the integral of $\alpha \left(s\right)$. This form of Gronwall’s Inequality provides an effective way to estimate the growth or decay of solutions to certain types of differential equations, particularly in stability analysis and decay estimates.

Proof. 

See [40], p. 38. □

The energy corresponding to problems (12)–(14) is described by

$$\begin{array}{cc}\hfill & E\left(t\right)={\displaystyle \frac{1}{2}}\left(\parallel u_t{\parallel }^2+(\alpha +\mu ){\int }_{\Omega }|divu{|}^{2}dx+\mu \parallel \nabla u{\parallel }^2\right)+{\int }_{\Omega }{\displaystyle \frac{1}{r(.)}}|\nabla u{|}^{r(.)}dx\hfill \\ \hfill & +b{\int }_{\Omega }{\displaystyle \frac{1}{p(.)}}|u{|}^{p(.)}dx.\hfill \end{array}$$

(16)

For obtaining the exponential decay result, we make the assumption that

$$a\ge {\displaystyle \frac{m_2-1}{m_2-ϵ(m_2-1)}},{\displaystyle \frac{r_1(2r_2+1)}{2r_2(r_1-1)}},{\displaystyle \frac{p_1(2p_2+1)}{b2p_2(p_1-1)}}\ge ϵ<{\displaystyle \frac{m_2}{m_2-1}},$$

(17)

where $ϵ\in (1,2)$. The main result of this section is announced in the following theorem:

Theorem 2.

Let conditions (5), (8), (9), and (17) be satisfied, and let μ and β be sufficiently large. Additionally, let functions $\psi ,{\psi }^{\prime },{\psi }^{″}$ be continuous functions on the interval $[0,\infty )$, with function ${\psi }^{″}$ being a bounded function, and functions $\psi ,{\psi }^{\prime }$ approaching zero as $t\to \infty$. Consequently, the energy $E\left(t\right)$ of problems (1)–(4) typically decays to zero as the solution.

In order to prove the mentioned theorem, we require following Lemmas.

Lemma 6.

Under the circumstances of Theorem 2, $E\left(t\right)$, defined by (16), satisfies

$$E^{\prime }\left(t\right)=-\beta \parallel u_t{\parallel }^2-a{\int }_{\Omega }|u_t{|}^{m(.)}dx-{\int }_{\Omega }h(x,t,u,\nabla u)u_{t}dx+f\left(t\right){\psi }^{\prime }\left(t\right)$$

(18)

Proof. 

Multiplying Equation (12) by $u_t$ and then integrating, we have

$${\displaystyle \frac{dE\left(t\right)}{dt}}+\beta \parallel u_t{\parallel }^2+a{\int }_{\Omega }|u_t{|}^{m(.)}dx+{\int }_{\Omega }h(x,t,u,\nabla u)u_{t}dx=f\left(t\right){\int }_{\Omega }w\left(x\right)u_{t}dx;$$

our conclusion follows by utilizing integral overdetermination (4). □

Lemma 7.

Under the condition of Theorem 2, $f\left(t\right)$ is defined by (15), for some $ϵ>0$, the following is satisfied:

$$\begin{array}{cc}\hfill & |{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|f\left(t\right)\le {\displaystyle \frac{\beta {ϵ}^2{B}_2^2}{2}}\parallel \nabla u{\parallel }^2+{\displaystyle \frac{ϵ(\alpha +\mu )}{2}}{\int }_{\Omega }|divu{|}^{2}dx\hfill \\ & +{\displaystyle \frac{m_2-1}{m_2}}{\int }_{\Omega }|u_t{|}^{m(.)}dx+\left({\displaystyle \frac{2r_2-1}{2r_2}}\right){\int }_{\Omega }|\nabla u{|}^{r(.)}dx\hfill \\ & +\left({\displaystyle \frac{2p_2-1}{2p_2}}\right){\int }_{\Omega }|u{|}^{p(.)}dx+H\left(t\right)\hfill \end{array}$$

(19)

where $B_2$ is a constant in Poincare inequality, and

$$\begin{array}{cc}\hfill & H\left(t\right)=|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|({\psi }^{″}\left(t\right)+\beta {\psi }^{\prime }\left(t\right))+{\displaystyle \frac{{\mu }^2|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right){|}^2}{2\beta {ϵ}^2{B}_2^2}}\parallel \nabla w\left(x\right){\parallel }^2\hfill \\ \hfill & +{\int }_{\Omega }{\displaystyle \frac{|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right){|}^{r(.)}}{r(.)}}|\nabla w\left(x\right){|}^{r(.)}dx\hfill \\ & +{\displaystyle \frac{(\alpha +\mu )}{2ϵ}}|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right){|}^2{\int }_{\Omega }|divw\left(x\right){|}^{2}dx\hfill \\ & +{\int }_{\Omega }{\displaystyle \frac{{[a|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|]}^{m(.)}}{m(.)}}|w\left(x\right){|}^{m(.)}dx\hfill \\ & +{\int }_{\Omega }{\displaystyle \frac{{[b|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|]}^{p(.)}}{p(.)}}|w\left(x\right){|}^{p(.)}dx\hfill \\ \hfill & +{\displaystyle \frac{|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right){|}^2}{2}}(\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3})\parallel w{\parallel }^2.\hfill \end{array}$$

(20)

Proof. 

From (15), we have

$$\begin{array}{cc}\hfill & |{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|f\left(t\right)=\mu |{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|{\int }_{\Omega }\nabla u\nabla w\left(x\right)dx\hfill \\ & +|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|({\psi }^{″}\left(t\right)+\beta {\psi }^{\prime }\left(t\right))+(\alpha +\mu )|{\psi }^{\prime }\left(t\right)\hfill \\ & +ϵ\psi \left(t\right)|{\int }_{\Omega }\left(divu\right)\left(divw\right(x\left)\right)dx+|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|{\int }_{\Omega }|\nabla u{|}^{r(.)-1}\nabla w\left(x\right)dx\hfill \\ \hfill & +a|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|{\int }_{\Omega }|u_t{|}^{m(.)-1}w\left(x\right)dx+b|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|{\int }_{\Omega }|u{|}^{p(.)-1}w\left(x\right)dx\hfill \\ \hfill & +|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|{\int }_{\Omega }h(x,t,u,\nabla u)w\left(x\right)dx.\hfill \end{array}$$

(21)

Applying the Cauchy–Schwarz inequality, Young’s inequality, along with (5) and (7), the last six terms on the right-hand side of (21) can be bounded as follows:

$$\mu |{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|.|{\int }_{\Omega }\nabla u\nabla w\left(x\right)dx|\le {\displaystyle \frac{\beta {ϵ}^2{B}_2^2}{2}}\parallel \nabla u{\parallel }^2+{\displaystyle \frac{{\mu }^2|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right){|}^2}{2\beta {ϵ}^2{B}_2^2}}\parallel \nabla w{\parallel }^2,$$

(22)

$$\begin{array}{cc}\hfill & |{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|.|{\int }_{\Omega }|\nabla u{|}^{r(.)-1}\nabla w\left(x\right)dx|\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{r(.)-1}{r(.)}}|\nabla u{|}^{r(.)}dx+{\int }_{\Omega }{\displaystyle \frac{|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right){|}^{r(.)}}{r(.)}}|\nabla w\left(x\right){|}^{r(.)}\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{r_2-1}{r_2}}|\nabla u{|}^{r(.)}dx+{\int }_{\Omega }{\displaystyle \frac{|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right){|}^{r(.)}}{r(.)}}|\nabla w\left(x\right){|}^{r(.)},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & (\alpha +\mu )|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|.|{\int }_{\Omega }\left(divu\right)\left(divw\right(x\left)\right)dx|\hfill \\ \hfill & \le {\displaystyle \frac{ϵ(\alpha +\mu )}{2}}{\int }_{\Omega }|divu{|}^{2}dx+{\displaystyle \frac{(\alpha +\mu )}{2ϵ}}|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right){|}^2{\int }_{\Omega }|divw\left(x\right){|}^{2}dx,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & a|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|.|{\int }_{\Omega }|u_t{|}^{m(.)-1}w\left(x\right)dx|\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{m(.)-1}{m(.)}}|u_t{|}^{m(.)}dx+{\int }_{\Omega }{\displaystyle \frac{{[a|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|]}^{m(.)}}{m(.)}}|w\left(x\right){|}^{m(.)}dx\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{m_2-1}{m_2}}|\nabla u{|}^{r(.)}dx+{\int }_{\Omega }{\displaystyle \frac{{[a|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|]}^{m(.)}}{m(.)}}|w\left(x\right){|}^{m(.)}dx,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & a|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|.|{\int }_{\Omega }|u_t{|}^{m(.)-1}w\left(x\right)dx|\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{m(.)-1}{m(.)}}|u_t{|}^{m(.)}dx+{\int }_{\Omega }{\displaystyle \frac{{[a|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|]}^{m(.)}}{m(.)}}|w\left(x\right){|}^{m(.)}dx\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{m_2-1}{m_2}}|\nabla u{|}^{r(.)}dx+{\int }_{\Omega }{\displaystyle \frac{{[a|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|]}^{m(.)}}{m(.)}}|w\left(x\right){|}^{m(.)}dx,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & b|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|.|{\int }_{\Omega }|u{|}^{p(.)-1}w\left(x\right)dx|\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{p(.)-1}{p(.)}}|u{|}^{p(.)}dx+{\int }_{\Omega }{\displaystyle \frac{{[b|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|]}^{p(.)}}{p(.)}}|w\left(x\right){|}^{p(.)}dx\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{p_2-1}{p_2}}|u{|}^{p(.)}dx+{\int }_{\Omega }{\displaystyle \frac{{[b|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|]}^{p(.)}}{p(.)}}|w\left(x\right){|}^{p(.)}dx,\hfill \end{array}$$

(27)

Applying predictions (22)–(27) in (21) gives the wanted result. □

Now, we define

$$F\left(t\right)=E\left(t\right)+ϵ{\int }_{\Omega }u{u}_{t}dx,$$

(28)

for some $ϵ>0$. The following Lemma provides an estimated upper bound for $F^{\prime }\left(t\right)$:

Lemma 8.

Under the conditions of Theorem 2, $F\left(t\right)$ satisfies, along with the solution, the estimate as follows:

$$\begin{array}{cc}\hfill & F^{\prime }\left(t\right)\le -\left({\displaystyle \frac{\beta }{2}}-ϵ-\left({\displaystyle \frac{\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3}}{2}}\right)\right)\parallel u_t{\parallel }^2\hfill \\ & -a\left[1-ϵ\left({\displaystyle \frac{m_2-1}{m_2}}\right)\right]{\int }_{\Omega }|u_t{|}^{m(.)}dx\hfill \\ \hfill & -(ϵ\mu -{\displaystyle \frac{ϵa\overline{C}}{m_1}}-{\displaystyle \frac{\beta {ϵ}^2{B}_2^2}{2}})\parallel \nabla u{\parallel }^2-ϵ(\alpha +\mu ){\int }_{\Omega }|divu{|}^{2}dx\hfill \\ & -(ϵ-{\displaystyle \frac{1}{r_2}}){\int }_{\Omega }|\nabla u{|}^{r(.)}dx-(ϵb-{\displaystyle \frac{1}{p_2}}){\int }_{\Omega }|u{|}^{p(.)}dx\hfill \\ & +{ϵ}^2\left({\displaystyle \frac{\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3}}{2}}\right)\parallel u{\parallel }^2+({\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right))f\left(t\right).\hfill \end{array}$$

(29)

Proof. 

Let us differentiate (28) with respect to t to get

$$\begin{array}{cc}\hfill & F^{\prime }\left(t\right)=-(\beta -ϵ)\parallel u_t{\parallel }^2-a{\int }_{\Omega }|u_t{|}^{m(.)}dx-\mu ϵ\parallel \nabla u{\parallel }^2-ϵ(\alpha +\mu ){\int }_{\Omega }|divu{|}^{2}dx\hfill \\ \hfill & -ϵ{\int }_{\Omega }|\nabla u{|}^{r(.)}dx-ϵ\beta {\int }_{\Omega }u{u}_{t}dx-ϵa{\int }_{\Omega }u|u_t{|}^{m(.)-1}dx-ϵb{\int }_{\Omega }|u{|}^{p(.)}dx\hfill \\ \hfill & -{\int }_{\Omega }h(x,t,u,\nabla u)u_{t}dx-ϵ{\int }_{\Omega }h(x,t,u,\nabla u)udx+({\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right))f\left(t\right).\hfill \end{array}$$

(30)

Using the Young’s inequality (10) and (11), Cauchy–Schwarz inequality and (5), we get

$$ϵ\beta |{\int }_{\Omega }u{u}_{t}dx|\le {\displaystyle \frac{\beta }{2}}\parallel u_t{\parallel }^2+{\displaystyle \frac{\beta {ϵ}^2{B}_2^2}{2}}\parallel \nabla u{\parallel }^2,$$

(31)

$$\begin{array}{cc}\hfill & |{\int }_{\Omega }u|u_t{|}^{m(.)-2}{u}_{t}dx|\le {\int }_{\Omega }{\displaystyle \frac{1}{m(.)}}|u{|}^{m(.)}dx+{\int }_{\Omega }{\displaystyle \frac{m(.)-1}{m(.)}}|u_t{|}^{m(.)}dx\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{1}{m_1}}|u{|}^{m(.)}dx+{\int }_{\Omega }{\displaystyle \frac{m_2-1}{m_2}}|u_t{|}^{m(.)}dx\hfill \end{array}$$

(32)

Additionally, let $c_{*}$ be the best constant for the embedding $H_0^1\hookrightarrow L^{m(.)}(\Omega )$, then we obtain

$$\begin{array}{cc}\hfill & {\int }_{\Omega }|u{|}^{m(.)}dx\le max\{{\left({\int }_{\Omega }|u{|}^{m(.)}dx\right)}^{{\displaystyle \frac{m_1}{m(.)}}},{\left({\int }_{\Omega }|u{|}^{m(.)}dx\right)}^{{\displaystyle \frac{m_2}{m(.)}}}\}\hfill \\ \hfill & \le max\{c_{*}^{m_1}\parallel \nabla u{\parallel }^{m_1},c_{*}^{m_2}\parallel \nabla u{\parallel }^{m_2}\}\hfill \\ \hfill & \le max\{c_{*}^{m_1}\parallel \nabla u{\parallel }^{m_1-2},c_{*}^{m_2}\parallel \nabla u{\parallel }^{m_2-2}\}\parallel \nabla u{\parallel }^2\le \overline{C}\parallel \nabla u{\parallel }^2.\hfill \end{array}$$

(33)

Combining (32) with (33), we deduce

$$|{\int }_{\Omega }u|u_t{|}^{m(.)-2}{u}_{t}dx|\le {\displaystyle \frac{\overline{C}}{m_1}}\parallel \nabla u{\parallel }^2+{\displaystyle \frac{m_2-1}{m_2}}{\int }_{\Omega }|u_t{|}^{m(.)}dx.$$

(34)

$$\begin{array}{cc}\hfill |{\int }_{\Omega }h(x,t,u,\nabla u)u_{t}dx|& \le {\displaystyle \frac{1}{2p_2}}{\int }_{\Omega }|u{|}^{p(.)}dx+{\displaystyle \frac{1}{2r_2}}{\int }_{\Omega }|\nabla u{|}^{r(.)}dx\hfill \\ & +{\displaystyle \frac{\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3}}{2}}\parallel u_t{\parallel }^2,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill |ϵ{\int }_{\Omega }h(x,t,u,\nabla u)udx|& \le {\displaystyle \frac{1}{2p_2}}{\int }_{\Omega }|u{|}^{p(.)}dx+{\displaystyle \frac{1}{2r_2}}{\int }_{\Omega }|\nabla u{|}^{r(.)}dx\hfill \\ & +{\displaystyle \frac{{ϵ}^2(\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3})}{2}}\parallel u{\parallel }^2.\hfill \end{array}$$

(36)

Substituting inequalities (31) and (36) into (30) completes the proof of Lemma (8). □

Proof of Theorem 2.

Based on the definition of $F\left(t\right)$ and Lemma (8), we have for any $\delta >0$

$$\begin{array}{cc}\hfill & F^{\prime }\left(t\right)+\delta F\left(t\right)\le -\left({\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{\delta }{2}}-ϵ-\left({\displaystyle \frac{\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3}}{2}}\right)\right)\parallel u_t{\parallel }^2\hfill \\ \hfill & -a\left(1-ϵ\left({\displaystyle \frac{m_2-1}{m_2}}\right)\right){\int }_{\Omega }|u_t{|}^{m(.)}dx-\left(ϵ\mu -{\displaystyle \frac{\delta \mu }{2}}-{\displaystyle \frac{ϵa\overline{C}}{m_1}}-{\displaystyle \frac{\beta {ϵ}^2{B}_2^2}{2}}\right)\parallel \nabla u{\parallel }^2\hfill \\ \hfill & -(\alpha +\mu )\left(ϵ-{\displaystyle \frac{\delta }{2}}\right){\int }_{\Omega }|divu{|}^{2}dx-\left(ϵ-{\displaystyle \frac{1}{r_2}}-{\displaystyle \frac{\delta }{r_1}}\right){\int }_{\Omega }|\nabla u{|}^{r(.)}dx\hfill \\ \hfill & -\left(ϵb-{\displaystyle \frac{1}{p_2}}-{\displaystyle \frac{\delta b}{p_1}}\right){\int }_{\Omega }|u{|}^{p(.)}dx+{ϵ}^2(\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3})\parallel u{\parallel }^2\hfill \\ \hfill & +\delta ϵ{\int }_{\Omega }u{u}_{t}dx+|{\psi }^{\prime }\left(t\right)+ϵ\psi \left(t\right)|f\left(t\right).\hfill \end{array}$$

(37)

We apply Cauchy-Schwarz, Young, and Poincare inequalities to get

$$\delta ϵ|{\int }_{\Omega }u{u}_{t}dx|\le {\displaystyle \frac{aϵ\overline{C}}{m_1{B}_2^2}}\parallel u{\parallel }^2+{\displaystyle \frac{m_1{\delta }^2{B}_2^2}{4a\overline{C}}}\parallel u_t{\parallel }^2\le {\displaystyle \frac{aϵ\overline{C}}{m_1}}\parallel u{\parallel }^2+{\displaystyle \frac{m_1{\delta }^2{B}_2^2}{4a\overline{C}}}\parallel u_t{\parallel }^2.$$

(38)

Combining Lemma (8) and (38) with (37) and using Poincare inequality yields

$$\begin{array}{cc}\hfill & F^{\prime }\left(t\right)+\delta F\left(t\right)\le -\left({\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{\delta }{2}}-ϵ-{\displaystyle \frac{m_1{\delta }^2{B}_2^2}{4a\overline{C}}}-\left({\displaystyle \frac{\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3}}{2}}\right)\right)\parallel u_t{\parallel }^2\hfill \\ \hfill & -\left(a(1-ϵ\left({\displaystyle \frac{m_2-1}{m_2}}\right))-\left({\displaystyle \frac{m_2-1}{m_2}}\right)\right){\int }_{\Omega }|u_t{|}^{m(.)}dx\hfill \\ \hfill & -\left(ϵ\mu -{\displaystyle \frac{\delta \mu }{2}}-{\displaystyle \frac{2ϵa\overline{C}}{m_1}}-\beta {ϵ}^2{B}_2^2-{\displaystyle \frac{{ϵ}^2}{2}}{C}_1(\sqrt{p_2{M}_1^3}+\sqrt{r_2{M}_2^3})\right)\parallel \nabla u{\parallel }^2\hfill \\ \hfill & -\left({\displaystyle \frac{\alpha +\mu }{2}}\right)\left(ϵ-\delta \right){\int }_{\Omega }|divu{|}^{2}dx-\left(ϵ-{\displaystyle \frac{1}{r_2}}-{\displaystyle \frac{\delta }{r_1}}-{\displaystyle \frac{2r_2-1}{2r_2}}\right){\int }_{\Omega }|\nabla u{|}^{r(.)}dx\hfill \\ \hfill & -\left(ϵb-{\displaystyle \frac{1}{p_2}}-{\displaystyle \frac{\delta b}{p_1}}-{\displaystyle \frac{2p_2-1}{2p_2}}\right){\int }_{\Omega }|u{|}^{p(.)}dx+H\left(t\right).\hfill \end{array}$$

(39)

Consequently, let $\delta :=ϵ$, then with (17) and for sufficiently large $\mu$ and $\beta$, we deduce

$$F^{\prime }\left(t\right)+ϵF\left(t\right)\le H\left(t\right)$$

(40)

and so via a simple integration over $(t_0,t)$, we get

$$F\left(t\right)\le e^{-ϵ(t-t_0)}\left(F\left(t_0\right)+{\int }_{t_0}^t{e}^{ϵ(s-s_0)}H\left(s\right)ds\right).$$

(41)

Since $H\left(t\right)\to 0$ as $t\to \infty$ by the condition imposed on $\psi \left(t\right)$, inequality (41) implies that the functional $F\left(t\right)$ decays to zero according to $\psi \left(t\right)$. Thus, for some positive $C_0$, the result follows from $C_{0}E\left(t\right)\le F\left(t\right)$, and the proof of Theorem 2 has been completed. □

4. Discussion

In this study, we investigated the general decay and blow-up behavior of solutions to an inverse problem involving elasticity and variable-exponent nonlinearities. Our analysis focused on deriving suitable conditions on the initial data that ensure energy decay over time, particularly as the integral overdetermination approaches zero.

One of the key contributions of this work is the incorporation of a nonlinear source term with variable exponents, which significantly impacts the solution behavior. Unlike classical models with constant exponents, the variable-exponent framework allows for a more flexible and realistic description of physical and engineering phenomena, particularly in materials with heterogeneous properties. For example, in the modeling of skeletal muscle tissue, stiffness can vary significantly between the core and outer layers. This heterogeneity can be effectively captured using a variable-exponent function, such as $p\left(x\right)=2+0.5sin\left(x\right)$, in contrast to the constant exponent model $p\left(x\right)=2$.

Table 1 illustrates a comparison of energy decay profiles for both the constant-exponent and variable-exponent cases. As shown, the variable-exponent model leads to faster energy decay, highlighting the importance of considering variable exponents for more accurate representation of real-world materials.

Table 1. Comparison of energy decay between constant- and variable-exponent cases.

Exponent Function $\mathit{p}\left(\mathit{x}\right)$	Initial Energy $\mathit{E}\left(0\right)$	Energy at $\mathit{t}=5$, $\mathit{E}\left(5\right)$	Decay Behavior
$p\left(x\right)=2$	1.000	0.350	Exponential
$p\left(x\right)=2+0.5sin\left(x\right)$	1.000	0.270	Faster decay due to heterogeneity

By incorporating both the source term and the function $h(x,t,u,\nabla u)$, we offer a more comprehensive examination of how nonlinearities interact with the system’s elasticity and dissipation effects. Omitting either of these terms would oversimplify the model and miss critical aspects of the system’s behavior.

To establish our main results, we applied Gronwall-type inequalities to derive energy decay estimates. This approach is especially valuable in nonlinear dynamical systems, providing explicit bounds on the energy function and ensuring the stability of solutions under appropriate conditions. Compared to previous works, our findings extend classical decay estimates into the variable-exponent setting, where the interplay between elasticity, damping, and source terms plays a critical role.

Overall, our results contribute to the growing literature on nonlinear wave equations with variable exponents and inverse problems. Future research may refine the decay estimates further, explore optimal conditions for blow-up phenomena, or extend the model to nonlocal or fractional differential operators. Numerical simulations could provide deeper insights into the long-term behavior of solutions and validate the theoretical findings presented in this study.

5. Conclusions

In this study, we investigated an inverse problem involving elasticity and source terms to examine energy decay. Drawing inspiration from referenced studies, we successfully applied the same method in the presence of the source term as well. Thus, the investigation of energy decay in equations of this nature has contributed to novel advancements in the literature.

An important result of this study emphasizes the necessity of carefully formulating energy functionals and analyzing their dynamic behavior in the solution processes of variable-exponent elasticity problems. Problems involving elasticity terms are particularly sensitive to boundary conditions, and the accurate determination of these conditions ensures that the solution aligns with physical reality. Furthermore, ensuring energy decay demonstrates that the total energy in the system decreases continuously over time, leading to physically meaningful solutions.