Asymptotic behavior of the wave equation solution with nonlinear boundary damping and source term of variable exponent-type

1.   Introduction

The wave equation with internal and boundary damping, along with a source term, is described by the system:

In this problem, the functions $\mathcal{F}_1$ and $\mathcal{F}_2$ are nonlinear source terms on the domain $\Omega \subseteq \mathcal{R}^n$ and the boundary $\partial\Omega = \Gamma_0 \cup \Gamma_1$, respectively, where $\Gamma_{0}$ and $\Gamma_{1}$ are closed and disjoint and meas.($\Gamma_{0}) > 0$. The vector $\eta$ is the unit outer normal to $\partial \Omega$. The functions $\omega_0$ and $\omega_1$ are given data. The functions $\psi_1$ is a nonlinear damping acting on the domain $\Omega$, while $\psi_2$ is a nonlinear damping acting on the boundary $\partial\Omega$.

The study of the existence, blow-up, and stability of solutions to wave equations has been extensively explored in previous research. For example, Lasiecka and Tataru ^[1] studied the following semilinear model of the wave equation with nonlinear boundary conditions and nonlinear boundary velocity feedback:

Assuming that the velocity boundary feedback is dissipative and the other nonlinear terms are conservative, uniform decay rates for the solutions are derived. Georgiev and Todorova ^[2] studied system (1.1) with $\psi_1(\omega_t) = \vert \omega_t \vert^{\vartheta-2} \omega_t$, $\psi_2(\omega_t) = \mathcal{F}_2(\omega) = 0$ and $\mathcal{F}_1(\omega) = \vert \omega \vert^{q-2} \omega$, proving global existence for $q\le \vartheta$ and a blow-up result when $q > \vartheta$. Levine and Serrin ^[3] expanded on this by investigating the case of negative energy with $\vartheta > 1$. Rivera and Andrade ^[4] examined a nonlinear wave equation with viscoelastic boundary conditions, showing the existence and uniform decay under certain initial data restrictions. Santos ^[5] focused on a one-dimensional wave equation with viscoelastic boundary feedback, demonstrating that under specific assumptions on $g^{\prime}$ and $g^{\prime \prime}$, sufficient dissipation leads to exponential or polynomial decay if the relaxation function follows the same pattern. Vitillaro ^[6] explored system (1.1) with $\psi_1(\omega_t) = \mathcal{F}_1(\omega) = 0$ and $\psi_2(\omega_t) = \vert \omega_t \vert^{\vartheta-2} \omega_t$ and $\mathcal{F}_2(\omega) = \vert \omega \vert^{q-2} u$, establishing local and global existence under appropriate conditions on the initial data and exponents. Cavalcanti et al. ^[7] studied the following problem

where $\psi:\mathcal{R}\to \mathcal{R}$ is a nondecreasing $C^1$ function such that

and there exist $C_{i} > 0, \quad i = 1, 2, 3, 4,$ such that

where $p\ge 1.$ They proved global existence of both strong and weak solutions, along with uniform decay rates, under restrictive conditions on the damping function $\psi$ and the kernel $g$. After that, Cavalcanti et al. ^[8] relaxed these conditions on $\psi$ and $g$, demonstrating uniform stability based on their behavior. Al-Gharabli et al. ^[9] extended this work by considering a large class of relaxation functions and establishing general and optimal decay results. Messaoudi and Mustafa ^[10] focused on system (1.3), exploring more general relaxation functions, and achieved a general decay result without assuming growth conditions on $\psi,$ with the results depending on both $g$ and $\psi$. Cavalcanti and Guesmia ^[11] analyzed the following hyperbolic problem involving memory terms

showing that under certain conditions, the memory term dissipation is sufficient to ensure system stability. Specifically, they demonstrated that if the relaxation function decays exponentially or polynomially, the solution follows the same decay rate.

Liu and Yu ^[12] investigated the following viscoelastic equation with nonlinear boundary damping and source terms

proving global existence and general decay of energy under suitable assumptions on the relaxation function and the initial data. Al-Mahdi et al. ^[13] extended this work by considering system (1.1) with modified terms: $\mathcal{F}_1(u) = 0$, $\mathcal{F}_2(\omega) = \vert \omega \vert^{q(x)-2} \omega$, $\psi_1(\omega_t)$ is replaced by $\int_0^t g(t-s) \Delta \omega(s) ds$, $\psi_2(\omega_t)$ is replaced by $\int_0^t g(t-s) \frac{\partial \omega}{\partial n} ds+ \vert \omega_t \vert^{\vartheta(x)-2} \omega_t$, proving global existence and establishing general and optimal decay estimates under specific conditions on the relaxation function and variable exponents $\vartheta(x)$ and $q(x)$. They also provided numerical tests to validate their theoretical decay results.

Zhang and Huang ^[14] studied a nonlinear Kirchhoff equation described by the system:

where $\Omega$ is a bounded domain of $\mathcal{R}^n$ with a smooth boundary $\partial\Omega = \Gamma_0 \cup \Gamma_1$, and $\alpha$ is a positive real constant. The functions $M(s), \chi(\omega), \psi(\omega_t)$ are satisfy some conditions, while $\eta$ represents the unit outward normal vector. Using the Galerkin approximation, Zhang and Huang established the global existence and uniqueness of the solution. They also addressed challenges posed by the nonlinear terms $M(s)$ and $\psi(\omega_t)$ through a transformation to zero initial data and employed compactness, monotonicity, and perturbed energy method to resolve the problem. Zhang and Ouyang ^[15] examined a viscoelastic wave equation with a memory term, nonlinear damping, and a source term:

where $\Omega$ is a bounded domain of $\mathcal{R}^n$ with a smooth boundary $\partial\Omega$, $\rho, \alpha > 0$, $p\geq2$, $q > 2$, and $g(t)$ is a positive function that represents the kernel of the memory term. Using the potential well method combined with the Galerkin approximation, they demonstrated the existence of global weak solutions. Additionally, under certain conditions on the damping coefficient and the relaxation function, they established the optimal decay of solutions via the perturbed energy method. They further showed that the solution can blow up for both positive and negative initial energy conditions.

For further results on wave equations, see the works of Aassila ^[16], Wang and Chen ^[17], Zuazua ^[18], Soufyane et al. ^[19], Zhang et al. ^[20].

There has been increasing interest among researchers in replacing constant exponents with variable exponents, driven by their practical applications ^[21] and related references. Variable exponents are commonly used in mathematical models and equations, particularly in damping terms, to better represent a system's diverse behaviors or properties. Damping, which helps dissipate energy and regulate a system's response to external forces, can be more accurately modeled using variable exponents. This allows for a more flexible representation of damping effects tailored to the specific characteristics of the system in question.

Inspired by these studies and the significance of mathematical models involving nonlinear damping and/or source terms with variable exponents, we consider problem (1.1) with $\psi_1(\omega_t) = \psi(\omega_t)$, $\mathcal{F}_1(\omega) = 0$ and $\psi_2(\omega_t) = \vert \omega_t \vert^{\vartheta(x)-2} \omega_t$, and $\mathcal{F}_2(\omega) = \vert \omega \vert^{\theta(x)-2} \omega$.

More precisely, we consider the following nonlinear wave equation with internal and boundary damping, along with a source term of variable exponent type:

We aim to study the global existence and stability of solutions to problem (1.9). We investigate the interaction between the internal nonlinear frictional damping and the nonlinear boundary damping of variable exponent type. Additionally, we derive general decay rates, including optimal exponential and polynomial decay rates as the special cases.

This paper is organized into five sections. In Section 2, we introduce the notation and necessary background material. In Section 3, we prove the global existence of the solution to the problem. In Sections 4 and 5, we present technical lemmas and decay results, respectively.

2.   Preliminaries

In this section, we outline some necessary materials for proving our results. Throughout the paper, we denote a generic positive constant by $c$. We consider the following assumptions:

$(A1)$ $\vartheta: \Gamma_1 \rightarrow [1, \infty)$ is a continuous function such that

where

$(A2)$ $\theta: \Gamma_1 \rightarrow [1, \infty)$ is a continuous function such that

where

Moreover, the variable functions $\vartheta(x)$ and $\theta(x)$ satisfy the log-Hölder continuity condition.

For more details about the Lebesgue and Sobolev spaces with variable exponents (see ^[22,23,24]).

$(A3)$ $\psi:\mathcal{R} \rightarrow \mathcal{R}$ is a $C^{0}$ nondecreasing function satisfying, for $c_1, c_2 > 0$,

where $\Psi: (0, \infty)\rightarrow (0, \infty)$ is $C^{1}$ function which is a linear or strictly increasing and strictly convex $C^{2}$ function on $(0, r]$ with $\Psi(0) = \Psi^{\prime}(0) = 0$.

Remark 2.1. Condition $(A3)$ was introduced for the first time in 1993 by Lasiecka and Tataru ^[1]. Examples of such functions satisfying Condition $(A3)$ are the following:

$(1)$ If $\psi(s) = c s^q$ and $q\ge 1$, then $\Psi(s) = c s^{\frac{q+1}{2}}$ satisfies $(A3)$.

$(2)$ If $\psi(s) = e^{-\frac{1}{s}}$, then $(A3)$ is satisfied for $\Psi(s) = \sqrt{\frac{s}{2}}\; e^{-\sqrt{\frac{2}{s}}}$ near zero.

We define the energy functional $E(t)$ associated to system (1.9) as follows:

Lemma 2.1. The energy functional $E(t)$ satisfies

Proof. Multiplying $(1.9)_1$ by $\omega_t$ integrating over the interval $\Omega$, we have

Using integration by parts, we obtain

Now, using $(1.9)_3$, and doing some modifications, we get

which gives (2.2). □

For completeness, we present the following existence result, which can be established using the Faedo-Galerkin method and the Banach fixed point theorem, similar to the approaches taken in ^[2,25,26] for analogous problems.

Theorem 2.1. (Local existence) Given $(\omega_0, \omega_1)\in H^1_{\Gamma_0}(\Omega)\times L^2(\Omega)$ and assume that $(A1)- (A3)$ hold. Then, there exists $T > 0$, such that problem (1.9) has a weak solution

3.   Global existence

In this section, we state and prove a global existence result under smallness conditions on the initial data $(\omega_0, \omega_1)$. For this purpose, we define the following functionals:

and

Clearly, we have

Lemma 3.1. Suppose that $(A1)-(A3)$ hold and $(\omega_0, \omega_1)\in H^1_{\Gamma_0}(\Omega)\times L^2(\Omega)$, such that

then

Proof. Since $\mathcal{I}$ is continuous and $\mathcal{I}(\omega_0) > 0,$ then there exists $T_m < T$ such that

which gives

Now,

Using Young's and Poincaré's inequalities and the trace theorem, we get $\forall t\in [0, T_m],$

where

Therefore,

□

Proposition 3.1. Suppose that $(A1)-(A3)$ hold. Let $(\omega_0, \omega_1)\in H^1_{\Gamma_0}(\Omega)\times L^2(\Omega)$ be given, satisfying (3.4). Then, the solution of (1.9) is global and bounded.

Proof. It suffices to show that $\|\nabla \omega\|_2^2+\|\omega_t\|_2^2$ is bounded independently of $t$. To achieve this, we use (2.2), (3.2) and (3.5) to get

Since $\mathcal{I}(t)$ is positive, Therefore

where $C$ is a positive constant, which depends only on $\theta_1$ and the proof is completed. □

Remark 3.1. Using (3.6), we have

4.   Technical lemmas

In this section, we present and prove several essential lemmas for demonstrating the main results.

Lemma 4.1. The functional defined by

satisfies, along the solutions of (1.9),

where $\Gamma_{*} = \{x \in \Gamma_1: \vartheta(x) < 2 \}.$

Proof.

The use of Young's and Poincaré's inequalities and choosing $\varepsilon_1 = \frac{1}{4c_p}$ give

Define the following partition of $\Gamma_{1}$:

Now, using Young's and Poincaré's inequalities, we obtain

choosing $\lambda = \frac{1}{8 c_p}$, then we have

Using Young's inequality with $p(x) = \frac{\vartheta(x)}{\vartheta(x)-1}$ and $p^{\prime}(x) = \vartheta(x)$ so, for all $x\in \Omega,$ we have

where

Hence, Young's inequality gives

Choosing $\varepsilon_2 = \frac{1}{8c \left(1+\left(\frac{2\theta_1}{\theta_1-2}E(0) \right)^{\frac{\vartheta_2-2}{2}}\right)}$, then $C_{\varepsilon_2}(x)$ is bounded and noting that $\Gamma_{**} \subset \Gamma_1$, then we have

By combining the above estimates, the proof is completed. □

Lemma 4.2. Let us introduce perturbed energy functional as follows:

satisfies, for all $t \geq 0$ and for a positive constant $N$,

Proof. We establish the proof by means of perturbed energy method. Taking the derivative of $\mathcal{M}$ with respect to $t$, and using the estimates in (4.2), and (2.2), we obtain

Choosing $N$ large enough such that $\mathcal{M}\sim E$, and recalling (2.2), therefore the proof of (4.9) is completed. □

Lemma 4.3. If $1 < \vartheta_1 < 2$, then the following estimate holds:

Proof. First, we define the following partition:

and use the fact that $\frac{2\vartheta(x)-2}{\vartheta(x)} \geq \frac{2\vartheta_1-2}{\vartheta_1}$, and Jensen's inequality to obtain

Using Young's inequality, we find that

Choosing $\varepsilon$ small enough, the proof of (4.11) is completed. □

Remark 4.1. If $\vartheta_1 \geq 2$ and since $meas({\Gamma_*}) = 0$ then

Lemma 4.4. Under assumption $(A3)$, the following estimates hold:

where $\Lambda (t)$ is defined in the proof.

Proof. Case 1: $\psi$ is linear, then

Case 2: $\psi$ is nonlinear, we define the following partition of $\Omega$

where $r$ is small enough such that

We also define

Now, using hypothesis $(A3)$ and Jensen's inequality, we get

□

5.   Stability result

In this section, we state and prove the stability result of system (1.9).

Theorem 5.1. Assume that $\vartheta_1\ge 2$ and $\psi$ is linear. Then

for some positive constants $\kappa_1$ and $\kappa_2$.

Proof. Combining (4.9), (4.15) with (4.14), we obtain,

Therefore, $\mathcal{M}+cE\sim E$ and a simple integration over $(0, t)$ yields, for some $\kappa_1, \kappa_2 > 0,$

□

Theorem 5.2. Assume that $1 < \vartheta_1 < 2$ and $\psi$ is linear. Then

where $\alpha = \frac{2-\vartheta_1}{2\vartheta_1-2} > 0.$

Proof. From (4.9), (4.11) and (4.15), we have

where $\mathcal{M}_1 = \mathcal{M} + cE \sim E$. Multiply both sides of (5.13) by $\left(E(t)\right)^{\alpha}$ where $\alpha = \frac{2-\vartheta_1}{2\vartheta_1-2}$, to obtain

where $\mathcal{M}_2 = \left(E(t)\right)^{\alpha} \mathcal{M}_1 +c E \sim E$. Integrating over $(0, t)$ and using the equivalence relation lead to (5.2). □

Theorem 5.3. Assume that $\vartheta_1 \geq 2$ and $\psi$ is nonlinear. Then, for some positive constants $\varrho_1$ and $\varrho_2$, we have

where $\Psi_{1}(t) = \int_{t}^{1}\frac{1}{\Psi_{2}(s)}ds$ and $\Psi_{2}(t) = t \Psi^{\prime}(\varepsilon_0 t)$

Proof. From (4.9), (4.1) and (4.15), we have

Now, for $\varepsilon_0 < r$, using the fact that $E^{\prime}\le 0$, $\Psi^{\prime} > 0, \Psi^{\prime\prime} > 0$ on $(0, r],$ we find that the functional $\tilde{\mathcal{M}},$ by

satisfies, for some $\alpha_{1}, \alpha_{1} > 0,$

and

Let $\Psi^{*}$ be the convex conjugate of $\Psi$ in the sense of Young with $A = \Psi^{\prime}\left(\varepsilon_0\frac{E(t)}{E(0)}\right)$ and $B = \Psi^{-1}(\Lambda(t))$, we arrive at

Consequently, with a suitable choice of $\varepsilon_0$ and $c_{0},$ we obtain, for all $t\ge 0,$

where $\Psi_{2}(t) = t \Psi^{\prime}(\varepsilon_0 t).$ Since $\Psi^{\prime}_{2}(t) = \Psi^{\prime}(\varepsilon_0 t)+\varepsilon_0 t \Psi^{\prime\prime}(\varepsilon_0 t),$ then, using the strict convexity of $\Psi$ on $(0, r],$ we find that $\Psi_{2}^{\prime}(t), \Psi_{2}(t) > 0$ on $(0, 1].$ Thus, with

taking in account (5.7) and (5.9), we have

and then

Then, a simple integration gives, for some $\varrho_{1}, \varrho_2 > 0,$

where $\Psi_{1}(t) = \int_{t}^{1}\frac{1}{\Psi_{2}(s)}ds.$ A combination of (5.10) and (5.11) gives (5.5). □

Theorem 5.4. Assume that $1 < \vartheta_1 < 2$ and $\psi$ is nonlinear. Then, for some positive constants $\varrho_3$ and $\varrho_4$, we have

where $\chi_{1}(t) = \int_{t}^{1}\frac{1}{\Psi_{2}(s)}ds$, $\chi_{2}(t) = t \chi^{\prime}(\varepsilon_0 t)$, $\chi = \left(\mathcal{G}^{-1}+\Psi^{-1}\right)^{-1}$ and $\mathcal{G}(t) = t^{\frac{\vartheta_1}{2\vartheta_1-2}}$.

Proof. From (4.9) and (4.13), we have

where $\mathcal{M} = E^\alpha \mathcal{M} +c E \sim E$. Let $\mathcal{G}(t) = t^{\frac{\vartheta_1}{2\vartheta_1-2}}$. Then the last inequality can be written as

Therefore, (5.14) becomes

where $\chi = \left(\mathcal{G}^{-1}+\Psi^{-1}\right)^{-1}$ and $\mathcal{\xi}(t) = \max\{{-E^{\prime}(t), \Lambda(t)}\}.$ Define the following functional

satisfies, for some $\alpha_{2}, \alpha_{3} > 0,$

Combining (5.15) and (5.16), we obtain

Let $\chi^{*}$ be the convex conjugate of $\chi$ in the sense of Young, then

and $\chi^{*}$ satisfies the following generalized Young inequality

Thus, with $A = \chi^{\prime}\left(\varepsilon_0\frac{E(t)}{E(0)}\right)$ and $B = \chi^{-1}(\mathcal{\xi}(t))$, we arrive at

Choosing $c_0, \varepsilon_0$ small enough, we get

where $\chi_{2}(t) = t \chi^{\prime}(\varepsilon_0 t).$ Letting

and taking in account (5.7) and (5.9), we have

and then

Then, a simple integration gives, for some $\varrho_{3}, \varrho_4 > 0,$

where $\chi_{1}(t) = \int_{t}^{1}\frac{1}{\chi_{2}(s)}ds,$ which finishes the proof. □

Examples 5.1. The following examples illustrate our results:

$(1)$ If $\psi (t) = ct$ and $\vartheta(x) = 2$, then

which is an exponential decay.

$(2)$ If $\psi (t) = ct$ and $\vartheta(x) = 2-\frac{3}{4+x}$, then $\vartheta_1 = \frac{5}{4}$ and $\vartheta_2 = \frac{7}{5}$, then the energy functional satisfies

$(3)$ If $\psi (t) = ct^2$ and $\vartheta(x) = 2+\frac{1}{1+x}$, then $\vartheta_1 = \frac{5}{2}$, $\vartheta_2 = 3$ and $\psi (t) = ct^{\frac{3}{2}}$. Then,

Therefore, we obtain

$(4)$ If $\psi (t) = c t^5$ and $\vartheta(x) = 2-\frac{3}{4+x}$, then $\vartheta_1 = \frac{5}{4}$, $\vartheta_2 = \frac{7}{5}$ and $\psi (t) = c t^3$. Then,

and

Therefore, we obtain

6.   Conclusions

In this work, we consider a nonlinear wave equation with internal and boundary damping and a source term of variable exponent type. We prove the global existence and stability of solutions to this problem problem. We study the interaction between the internal nonlinear frictional damping and the nonlinear boundary damping of variable exponent type. In addition, we establish general decay rates, including optimal exponential and polynomial decay rates as the special cases.

Author contributions

Adel M. Al-Mahdi: Conceptualization, methodology, formal analysis, writing-original draft; Mohammad M. Al-Gharabli: Formal analysis, validation, writing-reviewing and editing; Mohammad Kafini: Conceptualization, methodology, formal analysis, reviewing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The authors would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. The support provided by the Interdisciplinary Research Center for Construction & Building Materials (IRC-CBM) at King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia, for funding this work through Project (No. INCB2402), is also greatly acknowledged.

Funding

This work is funded by KFUPM, Grant No. INCB2402.

Conflict of interest

The authors declare no competing interests.