Theoretical and computational results of a wave equation with variable exponent and time-dependent nonlinear damping

We study the following wave equation utt-Δu+α(t)utm(·)-2ut=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{tt}-\Delta u+\alpha (t)\left| u_{t}\right| ^{m(\cdot )-2}u_{t}=0$$\end{document} with a nonlinear damping having a variable exponent m(x) and a time-dependent coefficient α(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t)$$\end{document}. We use the multiplier method to establish energy decay results depending on both m and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. We also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme.

(1.1) This is a weakly damped wave equation associated with homogeneous Dirichlet boundary conditions and initial data in suitable function spaces. Here, is a bounded domain of R n (n ≥ 1) with a smooth boundary ∂ , α : [0, ∞) → (0, ∞) is a nonincreasing C 1 -function (1.2) and m ∈ C 1 ( ) is satisfying with and also satisfying the log-Hölder continuity condition: , for all x, y ∈ with |x − y| < δ, 0 < δ < 1, A > 0. (1.4) In the case when m is a constant satisfying 1 < m < 2 * , there have been many results concerning the existence and energy decay rates of the solutions, we refer the readers to [6,12,14,26,28] and the references therein. Similar results were also obtained for frictional dissipative boundary condition (see [15,17,29]). When the damping is competing with a source term of the form |u| p−2 u, p ≥ 2, results on finite-time blow-up for p > m and global existence for p ≤ m have been established in [10,18,20,21].
In recent years, more attention has been paid to the study of mathematical nonlinear models of hyperbolic, parabolic and elliptic equations with variable exponents of nonlinearity. Some models from physical phenomena like flows of electro-rheological fluids or fluids with temperature-dependent viscosity, filtration processes in a porous media, nonlinear viscoelasticity, and image processing, give rise to such problems. More details on the subject can be found in [4,5]. Regarding hyperbolic problems with nonlinearities of variable-exponent type, only few works have appeared. For instance, Antontsev [2,3] considered the equation in a bounded domain ⊂ R n , with non-positive initial energy. Under appropriate conditions on the functions a, b, p, σ , the local, global and blow-up solutions have been established. Guo and Gao [11] considered the same problem (1.5) and proved several blow-up results for certain solutions with positive initial energy. The following equation: was investigated by Messaoudi and Talahmeh [22,24]. They established the existence of a unique weak solution using the Faedo-Galerkin method and proved the finite-time blow up of solutions. Moreover, when r (x) ≡ 2, Sun et al. [27] gave lower and upper bounds for the blow-up time. Recently, Messaoudi et al. [25] looked at (1.6), with b = 0 and 2 ≤ m(x) < 2 * , and proved decay estimates for the solution under suitable assumptions on the variable exponents m, r and the initial data. We also refer to Gao and Gao [9] who studied a nonlinear viscoelastic equation with variable exponents and proved the existence of weak solutions. Our aim in this work is to investigate (1.1), in which the damping considered is modulated by a timedependent coefficient α(t) satisfying (1.2) and has a variable exponent m(x) satisfying (1.3) and (1.4). We study both cases when m 1 ≥ 2 and m 1 < 2 and establish explicit energy decay rates depending on both m and α. To the best of our knowledge, this latter case has never been discussed for variable-exponent nonlinearity even for α ≡ 1. The paper is organized as follows. In Sect. 2, we present some notations and material needed for our work. The statement and the proof of our main results will be given in Sect. 3. In Sect. 4, we give a numerical verification of the theoretical decay results.

Preliminaries
In this section, we present some preliminary facts about Lebesgue and Sobolev spaces with variable exponents (see [7,16]). Let p : → [1, ∞) be a measurable function, where is a domain of R n . We define the Lebesgue space with a variable exponent p(·) by Equipped with the following Luxembourg-type norm We, next, define the variable-exponent Sobolev space W 1, p(·) ( ) as follows: This space is a Banach space with respect to the norm u W 1, p(·) ( ) = u p(·) + ∇u p(·) . Furthermore, we Here we note that the space W 1, p(·) 0 ( ) is usually defined in a different way for the variable exponent case. However, both definitions are equivalent under (1.4).
• Hölder's inequality: Let p, q, s ≥ 1 be measurable functions defined on such that , for a.e. y ∈ .
• Poincaré's inequality: Let be a bounded domain of R n and p(·) satisfies (1.4), then where the positive constant C depends on p 1 , p 2 and only. In particular, the space W 1, p(·) 0 ( ) has an equivalent norm given by u W 1, p(·) 0 ( ) = ∇u p(·) . • Embedding property: Let be a bounded domain in R n with a smooth boundary ∂ . Assume that p, q ∈ C( ) such that , then there is a continuous and compact To establish our decay results, the following lemmas will be of essential use.
Assume that there exist q ≥ 0 and w > 0 such that Then there exists a positive constant k depending continuously on E(0) such that, ∀t ≥ 0, holds for some c > 0 and any p > 2 where At the end of this section, we state the following existence and regularity result, whose proof can be established similarly to [23,24].

The main results
We define the energy functional by which means that E(t) is a nonincreasing function. Let us mention that we will use c, throughout this paper, to denote a generic positive constant. Now, we state and prove our first result.
Then there exist positive constants k and w such that the solution of (1.1) satisfies, ∀t ≥ 0, Proof We multiply (1.1) 1 by α E q u, for q ≥ 0 to be specified later, and integrate over × (S, T ) to get Integrating by parts in the first term, using the definition of E(t), leads to We estimate the terms in the right-hand side as follows: Using Young's and Poincaré's inequalities, we have then, by the properties of E and α, we conclude that As in above, we conclude that If m 2 = 2, then If m 2 > 2, we consider the partition of (see [19]), 1 = {x ∈ : |u t | ≥ 1} and 2 = {x ∈ : |u t | < 1} and make use of Hölder's and Young's inequalities and (3.1) as follows: which gives, for all δ > 0, We choose δ = 1 2c , q = m 2 2 − 1, so 2(q+1) Here, we use Young's inequality with p(x) = m(x) m(x)−1 and p (x) = m(x). So, for all x ∈ , we have Therefore, for all ε, we get Inserting all the above estimates into (3.3) and taking T → ∞, we arrive at where q = m 2 2 − 1. Hence, using Lemma 2.1 with σ (t) = t 0 α(s)ds, the estimate (3.2) is established. Remark 3.2 Note that the decay result of [25] is only a special case of our result in Theorem 3.1. Now, to study the case 1 < m 1 < 2, we first need to obtain a uniform bound for the second-order energy defined by Differentiating (1.1) 1 with respect to t, multiplying by u tt and integrating over × (0, T ), we obtain Thus, But, by (1.2) and the condition that which means that Next, we state and prove our second result.

Numerical tests
In the light of our theoretical results, we present in this section four numerical tests. We discretize the system (1.1) using a second-order finite difference method in time and space for the space-time domain [0, L] × [0, T e ] = [0, 3] × [0, 60]. By implementing the conservative scheme of Lax-Wendroff, we computationally compare four tests, for similar construction see [1,8,13]. The first three tests examine the results of Theorem 3.1, while the fourth test is based on the results of Theorem 3.3: • TEST 1: We present the exponential decay case of the energy function given in (3.1), using the constant functions α(t) = 1 and m(x) = 2. • TEST 2: In the second numerical test, we examine the a polynomial-type energy decay rate case, using the functions α(t) = 1 1 + t and m(x) = 2.
• TEST 3: Here, we present a logarithmic-type energy decay, using the functions α(t) = 1   • TEST 4: In Test 4, we also present a logarithmic-type energy decay, using the functions α(t) = 1 1 + t and In order to ensure the numerical stability of the implemented method and the executed code, we use t = 0.5dx satisfying the stability condition according to the Courant-Friedrichs-Lewy (CFL) inequality, where dt represents the time step and dx the spatial step. The spatial interval [0, 3] is subdivided into 500 subintervals, whereas the temporal interval [0, T e ] = [0, 60] is deduced from the stability condition above. We run our code  For the first numerical Test 1, we examine the exponential decay case. Under the initial and boundary conditions above, we plot in Fig. 1 three cross sections at x = 0.75, 1.5, 2.25 (see Fig. 1a-c). In Fig. 1a, we plot the corresponding energy functional (3.1). In addition, we plot in Fig. 2 the decay behavior of the whole wave till time t = 20.
Under similar initial and boundary conditions, we present in Fig. 3 the results of the polynomial decay obtained for Test 2. We show the evolution of the cross section cuts at x = 0.75, x = 1.5 and at x = 2.25 (see   Fig. 3a-c). Moreover, we plot in Fig. 4 Fig. 3d, where the polynomial decay of the energy functional is clearly obtained. Consequently, we can clearly compare the energy decay rates obtained in Test 1 and in Test 2.
In Test 3, we examine a logarithmic decay case. Therefore, we display the results in Fig. 5. The selected cross section cuts at x = 0.75, x = 1.5 and at x = 2.25 are presented in Fig. 5a-c, d we plot the corresponding energy functional. Furthermore, we plot in Fig. 6 the two-dimensional wave in the same space-time domain  In Test 4, we examine the result of Theorem 3.3. We plot in Fig. 7, the selected cross section cuts at x = 0.75, x = 1.5 and at x = 2.25, see 7a-c. In Fig. 7d, we plot the corresponding energy functional. Furthermore, we plot in Fig. 8 the whole wave in the same space-time domain as above.