Decay result in a problem of a nonlinearly damped wave equation with variable exponent

1.   Introduction

In this work we are concerned with the decay rate of the following problem with nonlinear damping of variable exponent

where $T > 0$ and $\Omega$ is a bounded domain of $\mathbb{R}^{n}(n\geq 1).$ The functions $u_{0}$, $u_{1}$ are initial data and the variable exponent $m(\cdot)\in C\left(\overline{\Omega }\right)$ is a given functions satisfying

where

and also satisfies the log-Hölder continuity condition:

for $x, y\in \Omega,$ with $\left\vert x-y\right\vert < \delta$, $A > 0$ and $0 < \delta < 1.$ The function $\alpha :[0, \infty)\rightarrow \left(0, \infty \right)$ is a bounded nonincreasing $C^{1}-$function and

Problems with variable exponents appear as a direct consequence of the advancement of science and technology. Many physical and engineering models require more sophisticated mathematical functional spaces to be studied and well understood. For example, in fluid dynamics, the electrorheological fluids (smart fluids) have the property that the viscosity changes when exposed to an electrical field. More examples are found in studying models of the image processing and filtration processes through a porous media. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools to study such problems. More details on applications of these problems can be found in (^[1,2,3]).

A lot of papers in the literature dealt with stabilization of wave equations with different types of nonlinearities such as linear, polynomial and logarithmic. For instance, the following problem was studied by Nakao ^[4].

where $m, p > 2$ and $\Omega \subset$ $\mathbb{R}^{n}$ $(n\geq 1)$ is a bounded domain. He showed that, with Dirichlet-boundary conditions, the problem has a unique global weak solution if $2\leq p\leq 2\left(n-1\right) /\left(n-2\right)$, $n\geq 3$ and a global unique strong solution if $p > 2\left(n-1\right) /\left(n-2\right)$, $n\geq 3.$ In both cases, he proved that the energy of the solution decays algebraically if $m > 2$ and decays exponentially if $m = 2$. Benaissa and Messaoudi ^[5] considered

where $m, p > 2$ and showed, for small initial data in an appropriate function space, that the problem has a global weak solution which decays exponentially even if $m > 2$. We also mention here the work of Mustafa and Messaoudi ^[6], where they considered

and established an explicit and general decay rate result, without imposing any restrictive growth assumption on the frictional damping term.

As we mentioned earlier, modern technology and engineering required the use of variable exponents nonlinearities and the Lebesgue and Sobolev spaces with variable exponents as well. In this regard, we mention the work of Ghegal et al. ^[7] where, in a bounded domain, the following equation is considered

Under suitable conditions on the initial data and the variable exponents, the authors used stable-set method to prove a global existence result. Then, by applying an integral inequality due to Komornik, they obtained the stability result. More results can be found in (^[8,9,10]).

Hyperbolic problems involving variable-exponent nonlinearities with delay are also considered. For instance, Kafini and Messaoudi ^[11] studied the problem

For $b > 0,$ they established a global nonexistence result under suitable conditions on $\mu _{1}, \mu _{2}, m(\cdot), p(\cdot)$ and the initial data. While, for $b = 0,$ they obtained a decay result which is of either polynomial or exponential type depending on the nature of $m(\cdot)$.

Recently, Messaoudi in ^[12] considered the problem

and established several decay results depending on the nature of variable exponents $r\left(\cdot \right)$ and $m\left(\cdot \right)$. See ^{[13,14,15,16,17]}, for more results on the local existence and blow up for some problems with variable exponent nonlinearities.

Fractional derivatives have been also influenced by variable orders. One can see variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications as in ^[18]. Analyzing a variable-order time-fractional wave equation, which models, e.g., the vibration of a membrane in a viscoelastic environment examined in ^[19]. See also^[20,21,22] for more details.

In our work, we aim to study the nonlinear wave Eq $\left(1.1 \right)$ with nonlinear feedback having a variable exponent $m(x)$ and a time-dependent coefficient $\alpha (t)$. We establish a decay result of an exponential and polynomial type under specific conditions on both $m(\cdot)$ and $\alpha \left(t\right)$ and the initial data. This paper consists of three sections in addition to the introduction. In Section 2, we recall the basic definitions of the variable exponent Lebesgue spaces $L^{p(\cdot)}(\Omega),$ the Sobolev spaces $W^{1, p({\cdot })}(\Omega)$, as well as some of their properties. Section 3 is devoted to the existence and uniqueness of a weak global solution. In the last section, we show the decay result.

2.   Preliminaries

In this section, we present some materials needed for the statement and the proof of our results. In what follows, we give definitions and properties related to Lebesgue and Sobolev spaces with variable exponents, see ^[23,24] for more details.

Let $\Omega$ be a domain of $\mathbb{R}^{n}$ with $n\geq 2$ and $p:\Omega \longrightarrow \lbrack 1, \infty]$ be a measurable function. The Lebesgue space $L^{p(\cdot)}(\Omega)$ with a variable exponent $p(\cdot)$ is defined by

where

The Luxembourg-type norm is given by

The space $L^{p(\cdot)}(\Omega),$ equipped with the above norm, is a Banach space.

Lemma 2.1. (Hölder's inequality) $\rm{Let }$ $p, q, s\geq 1$ be measurable functions defined on $\Omega$ ${\rm{such}}\; {\rm{that }}$

$\rm{If }$ $f\in L^{p(\cdot)}(\Omega)$ and $g\in L^{q(\cdot)}(\Omega)$, then $fg\in L^{s(\cdot)}(\Omega)$ $\rm{and }$

Lemma 2.2. $\rm{If }$ $p:\Omega \longrightarrow \lbrack 1, \infty)$ $\text{is a measurable function and}\;1 \leq p_{1}\leq p(x)\leq p_{2} < \infty$, $\rm{then}$

${\rm{for}}\; {\rm{a.e. }}$ $x\in \Omega$ and for any $v\in L^{p(\cdot)}(\Omega).$

Lemma 2.3. ^[12] $\rm{If }$ $p:\Omega \longrightarrow \lbrack 1, \infty)$ is a measurable function and $1 \leq p_{1}\leq p(x)\leq p_{2} < \infty$, $\rm{then}$

The variable-exponent Sobolev space $W^{1, p(\cdot)}(\Omega)$ is defined as

This space is a Banach space with respect to the norm

Suppose $p(\cdot)$ satisfies (1.3). Then the space $W_{0}^{1, p(\cdot)}(\Omega)$ is defined to be the closure of $C_{0}^{\infty }(\Omega)$ in $W^{1, p(\cdot)}(\Omega).$ The definition of the space $W_{0}^{1, p(\cdot)}(\Omega)$ is usually different from the constant exponent case. However, under condition $\left(1.3\right)$ both definitions coincide. The dual space of $W_{0}^{1, p(\cdot)}(\Omega)$ is $W_{0}^{-1, p^{\prime }(\cdot)}(\Omega)$ defined in the same way as in the classical Sobolev spaces, where

Lemma 2.4. (Poincaré's inequality) $\textrm{Let }$$\Omega$$\ \textrm{be a bounded domain of }$$\mathbb{R}^{n}$$\ \textrm{and }$$p(\cdot)$$\ \textrm{satisfies }$$\left(1.2\right)$ $\textrm{and}$ $\left(1.3\right)$$\textrm{, then}$

$\textrm{where }$$C$$\ \textrm{is a positive constant depends on }$$p(\cdot)$$\ \textrm{and }$$\Omega.$$\ \textrm{In particular, the space }$$W_{0}^{1, p(\cdot)}(\Omega)$$\ \textrm{has an equivalent norm given by}$

Lemma 2.5. $\textrm{If }$$p:\overline{\Omega }\longrightarrow \lbrack 1, \infty)$ $\textrm{is continuous and }$

$\textrm{then the embedding }$$H^{1}(\Omega)\hookrightarrow L^{p(\cdot)}(\Omega)$$\ \textrm{is continuous and compact.}$

Lemma 2.6. ^[27] $\textrm{Let }$$E:\mathbb{R}^{+}\longrightarrow \mathbb{R}^{+}$$\ \textrm{be a nonincreasing function and }$$\phi :\mathbb{R}^{+}\longrightarrow \mathbb{R}$$\ \textrm{be an increasing }$$C^{1}$ -$\textrm{function satisfying}$

$\textrm{Assume further, that there exist }$$q\geq 0,$ $A > 0$$\ \textrm{such that}$

$\textrm{Then}$, $\forall t\ \geq \ 0,$

$\textrm{where }$$C$$\ \textrm{and }$$\omega$$\ \textrm{are positive constants independent of the initial energy}$ $E(0)$$\textrm{.}$

Definition 2.7. Given the initial data $(u_{0}, u_{1})\in H_{0}^{1}\left(\Omega \right) \times L^{2}\left(\Omega \right),$ a function $u$ defined on $\Omega \times \left(0, T\right)$ is called a weak solution of problem $\left(1.1\right)$ if

and it verifies the variational equation

We introduce the energy functional associated to problem $\left(1.1 \right)$ as

Lemma 2.8. $\textrm{Let }$$u$$\ \textrm{be the solution of }$$\left(1.1\right).$$\ \textrm{Then, }$

Proof. Multiplying Eq $\left(1.1\right)$ by $u_{t}$ and integrating over $\Omega,$ the result follows.

Remark 2.9. In the sequel, we use $C$ to denote a generic constant which may differ from one place to another.

3.   Existence and uniqueness

The following theorem states our existence and uniqueness results, which are the main focus of this section.

Theorem 3.1. $\textrm{Assume that the variable exponent }$$m(\cdot)$ $\textrm{satisfies conditions }$$\left(1.2\right)$$\ \textrm{and }$$\left(1.3\right)$$\textrm{. Then, for any initial data }$$u_{0}\in H_{0}^{1}(\Omega), u_{1}\in L^{2}(\Omega)$, $\textrm{problem }$$\left(1.1\right)$$\textrm{admits a unique global weak solution.}$

Proof. To prove the existence of a weak solution to $\left(1.1\right),$ we make use of the Galerkin approximation method. For that reason we assume $\left\{ v_{j}\right\} _{j\geq 1}$ is an orthogonal basis for $H_{0}^{1}(\Omega)$ and orthonormal in $L^{2}(\Omega).$ We find a solution of the form

to the approximate problem

where

This system, by the standard ODE theory has a unique solution guaranteed on $[0, t_{k})$, $0 < t_{k}\leq T.$ Next, we need to show that this solution can be extended to the maximal interval $[0, T),$ $\forall k\geq 1$ and for any $T > 0.$

Replace $v_{j}$ by $u_{t}^{k}$ in $\left(3.1\right)$ to get

and integrate over $\left(0, t\right)$ for $t\in \left(0, t_{k}\right)$ to arrive at

Hence, the solution can be extended to $[0, T),$ for any given $T > 0.$

Using $\left(1.4\right)$, we arrive at

where we can conclude that

Therefore, we can extract subsequences, still denoted by $u^{k}$ and $u_{t}^{k},$ such that

As $u_{t}^{k}$ is bounded in $L^{m(\cdot)}\left(\Omega \times \left(0, T\right) \right),$ then $\left\vert u_{t}^{k}\right\vert ^{m(\cdot)-2}u_{t}^{k}$ is bounded in $L^{\frac{m(\cdot)}{m(\cdot)-1}}\left(\Omega \times \left(0, T\right) \right).$ Hence,

To show that $\psi = \left\vert u_{t}\right\vert ^{m(\cdot)-2}u_{t},$ we integrate $\left(3.1\right)$ over $\left(0, t\right)$ to get, $\forall j = 1, ..., k$,

Now, letting $k\rightarrow +\infty$ and differentiating the latter result with respect to $t$ gives

Hence,

If we define

and

then we have

Using Eq $\left(3.3\right),$ we get

As $k\rightarrow +\infty,$

Integration of $\left(3.4\right)$ over $\left(0, T\right)$ after replacing $v$ by $u_{t}$ give

Adding $\left(3.5\right)$ and $\left(3.6\right)$ give

Thus, by the density of $H_{0}^{1}(\Omega)$ in $L^{m(\cdot)}(\Omega)$ we have

If we let $v = \lambda w+u_{t}$ for $w\in L^{m(\cdot)}\left(\Omega \times \left(0, T\right) \right)$ then

As $0 < \lambda \rightarrow 0,$ we have,

Similarly, if $0 > \lambda \rightarrow 0,$ we have,

This implies that $\psi = A\left(u_{t}\right) = \left\vert u_{t}\right\vert ^{m(\cdot)-2}u_{t}.$

To handle the initial conditions, we use Lions' Lemma ^[25], to obtain, up to a subsequence, that

Therefore, $u^{k}(\cdot, 0)$ makes sense and $u^{k}(\cdot, 0)\rightarrow u(\cdot, 0)$ in $L^{2}(\Omega)$. Also, by density we have

hence $u(\cdot, 0) = u_{0}$.

For the other condition, as in ^[26], we obtain from $\left(3.1 \right)$ and for any $j\leq k$ and $\phi \in C_{0}^{\infty }\left(0, T\right)$,

As $k\rightarrow +\infty$, we obtain that, for all $v\in H_{0}^{1}(\Omega),$

This implies that

and $u$ solves the equation

Therefore,

where $u_{t}^{k}(\cdot, 0)$ makes sense and $u_{t}^{k}(\cdot, 0)\rightarrow u_{t}(\cdot, 0)$ in $H^{-1}(\Omega)$. But we have

So $u_{t}(\cdot, 0) = u_{1}$.

To prove the uniqueness, we assume $u$ and $v$ are two solutions of $\left(3.1\right)$. Then $w = u-v$ satisfies the following problem

Multiply the equation by $w_{t}$ and integrate over $\Omega$, to obtain

Integration over $\left(0, t\right),$ to get

Using the fact that

we obtain

This implies that $w = C = 0$, since $w = 0$ on $\partial \Omega$. Hence, the uniqueness. This completes the proof of Theorem 3.1.

4.   Decay of solutions

Theorem 4.1. $\textrm{Let }$$(u_{0}, u_{1})\in H_{0}^{1}\left(\Omega \right) \times L^{2}\left(\Omega \right)$$\ \textrm{be given. Assume that }$$\int_{0}^{\infty}\alpha \left(\tau \right) d\tau = \infty$ and $m(\cdot)\in C\left(\overline{\Omega }\right)$$\ \textrm{that satisfies }$

$\ \textrm{Then, the solution energy }$$\left(2.1\right)$ $\textrm{satisfies, for two positive constants }$$k_{1}, k_{2},$

Proof. Multiply $\left(1.1\right)$ by $\alpha uE^{q}(t)$ and integrate over $\Omega \times \left(s, T\right),$ $0 < s < T,$ to obtain

which gives

for $q\geq 0$ to be specified later.

Recalling the fact that $\int_{\Omega }\left(|\nabla u|^{2}+u_{t}^{2}\right) dx = 2E(t)$ and using the relation

equation $\left(4.2\right)$ becomes

The first term in the right side of $\left(4.3\right)$ is estimated, using Poincaré's inequality, $\left(2.2\right)$ and the fact that

to have

Using Young's inequality, the second term leads to

Taking $\delta = 1/2C,$ we get

Similar to the first term, we have

The fourth term:

The fifth term:

The last term in the right-hand side of $\left(4.3\right)$ is handled by using Young's inequality with

So, for a.e. $x\in \Omega,$ $\varepsilon > 0,$ and

we have

If$\ $we fix $\varepsilon = 1/2CE^{\frac{m_{1}}{2}-1}(0),$ noting that $c_{\varepsilon }(x)$ is bounded since $m(x)$ is bounded, then $\left(4.9\right)$ becomes

Combining $\left(4.3\right)$–$\left(4.10\right)$ and taking $T\rightarrow \infty$ we arrive at

Therefore, $\left(4.1\right)$ is established by the virtue of Lemma 2.6 for $q = 0$ and $\phi (t) = \int_{0}^{t}\alpha \left(s\right) ds.$

Example 1. If we take $\alpha \left(t\right) = 1$ and $m\left(x\right) = 2$ then we have $\phi (t) = t$ and hence, for two positive constants $k_{1}, k_{2},$

The next theorem handles the case: $1 < m_{1} < 2.$

Theorem 4.2. (Polynomial Decay) $\textrm{Let }$$(u_{0}, u_{1})\in H_{0}^{1}\left(\Omega \right) \times L^{2}\left(\Omega \right)$$\ \textrm{be given. Assume that }$$\int_{0}^{\infty }\alpha \left(\tau \right) d\tau = \infty$ $\textrm{and }$$m(\cdot)\in C\left(\overline{\Omega }\right)$$\ \textrm{and satisfies }$$\left(1.2\right).$ $\textrm{Assume further that}\ $$m_{1} < 2.$ $\textrm{Then, the solution energy }$$\left(1.1\right)$$\ \textrm{satisfies, for some positive constant }$$K,$

Proof. We follow the same steps in the proof of the previous theorem. But we have to re-estimate the last term in $\left(4.3\right)$. For this purpose, we define

Thus,

Then we use Young's inequality and Poincaré's inequality, to get

In order to estimate the last term of $\left(4.12\right)$, we define

Then Hölder's inequality and the embedding give

Thus,

Using Young's inequality, we obtain for any $\lambda > 0$,

If we let $q+1 = \frac{q}{2-m_{1}}$ hence $q = \frac{2-m_{1}}{m_{1}-1},$ then $\left(4.14\right)$ implies that

Then we choose $\delta = 1/4C.$ After $\delta$ is fixed, we choose $\lambda = 1/4c_{\delta }$ to obtain

Now over $\Omega _{2},$ we follow the same steps as in $\left(4.9\right)$ to conclude that

Combining $\left(4.15\right)$ and $\left(4.16\right)$, give

Consequently, from $\left(4.3\right)$–$\left(4.8\right)$ and $\left(4.17\right),$ we have

If we let $T\rightarrow \infty,$ then from Lemma 2.6 with $\phi \left(t\right) = \int_{0}^{t}\alpha \left(\tau \right) d\tau$ and $q = \frac{2-m_{1} }{m_{1}-1} > 0$, we arrive for some $K > 0,$

This completes the proof.

Example 2. If we take $\Omega = \left(0, 1\right),$ $\alpha \left(t\right) = \frac{2+t}{1+t}$ and $m\left(x\right) = 2-\frac{1}{2+x},$ then we have $\phi (t) = t+\ln \left(1+t\right)$, $m_{1} = 3/2$ and

for a positive constant $K.$

5.   Conclusions

In this paper, we have shown that the time varying coefficient appears in the problem has a direct effect in well posednesss and the decay rates. In fact, we investigated the nonlinear wave Eq $\left(1.1 \right)$ with nonlinear feedback having a variable exponent $m(x)$ and a time-dependent coefficient $\alpha (t)$. We established a decay result of an exponential and polynomial type under specific conditions on both $m(\cdot)$ and $\alpha \left(t\right)$ and the initial data.

Acknowledgments

The authors would like to express their sincere thanks to King Fahd University of Petroleum and Minerals (KFUPM)/Interdisciplinary Research Center (IRC) for Construction and Building Materials for its support. This work has been funded by KFUPM under Project # SB191048.

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.