Delayed wave equation with logarithmic variable-exponent nonlinearity

1.   Introduction

In this work, we are concerned with the following delayed nonlinear wave problem

where $\tau$, $k$, $\mu_1 > \, 0$ and $\mu_2$ is a real number. The functions $u_0$, $u_1$, $f_0$ are the initial and history data to be determined later. The domain $\Omega$ $\subset \mathbb{R}^{n}$ is an open bounded domain with a smooth boundary $\partial \Omega.$ The variable exponents $m(\cdot)$ and $p(\cdot)$ are given measurable functions on $\Omega$ and satisfy

where

and the log-Hölder continuity condition:

for a.e., $x, y\in \Omega,$ with $\left\vert x-y\right\vert < \delta,$ $A > 0$ and $0 < \delta < 1.$

In the absence of delay ($\mu _{2} = 0$), the hyperbolic equation in (1.1) is well studied and many blow-up and decay results have been proved. Relaxation or viscoelastic term also were added. See ^{[1,2,3,4,5,6,7]}.

Like physical, chemical, biological, and thermal processes, time delays are frequent occurrences. It is well established that the delay term, if no extra stabilization techniques are included, can be a source of instability. Nicaise and Pignotti ^[8] did in fact analyze to the following wave equation

where $\Omega$ $\subset \mathbb{R}^{n}$ is a bounded domain and $a,$ $a_{0}$ are positive real parameters. They proved that the system is exponentially stable under the condition ($0\leq a < a_{0}$). In the case $a\geq a_{0}$, they produced a sequence of delays for which the corresponding solution is instable. After that, various types of delay were considered and similar stability results were established. See, in this regard^[9,10,11].

Recently, equations with variable exponents of nonlinearity have been used to model a variety of physical phenomena, including flows of electro-rheological fluids or fluids with temperature-dependent viscosity, nonlinear viscoelasticity, filtration processes through porous media, and image processing. The references in ^{[12,13,14,15,16,17]} provide additional information on these issues.

For instance, a hyperbolic problems with nonlinearities of variable-exponent type presented in the work of Antontsev ^[18], where he considered the equation

and demonstrated numerous blow-up results on the variables $a$, $b$, $p$, and $\sigma$ for some non-positive initial energy solutions. Similarly, in ^[19], Antontsev used Galerkin approximations in spaces of the Orlicz-Sobolev type to demonstrate the presence of local and global weak solutions of (1.4). The blow up for weak solutions with nonpositive energy functional was then established. Guo and Gao ^[20] demonstrated that solutions to quasilinear hyperbolic equations with positive initial energy and $p(x, t)$-Laplacian are blow up. We recommend reading Antontsev and Shmarev ^[21] and Galaktionov ^[22] for other problems involving variable-exponent nonlinearities. S. Park ^[23] thought about issues of a similar nature but with constant exponents.

Most literary works impose the Sobolev Logarithmic Inequality (SLI) when a logarithmic source term is present in order to establish specific decay or blow up results. Observe ^{[24,25,26,27,28,29,30]}, for instance. Authors were constrained by the usage of (SLI) by using terms like $u\ln \left\vert u\right\vert$ or $u^{2}\ln \left\vert u\right\vert$ or by imposing additional weaken requirements if the nonlinearity is more challenging.

The presence of a local weak solution for constant exponents was established in ^[28]. With variable exponent nonlinearity of the logarithmic kind and a delay term present in problem (1.1), our goal is to examine the stability of any strong solution without the use of (SLI). To put it more precisely, we seek to demonstrate a global existence result under sufficient assumptions using only the initial data, variables $\mu _{1}, \mu _{2}, m,$ and $p$. After that, we developed decay results of polynomial and exponential types based on certain exponent $m(\cdot)$ values. This paper has an introduction and four more sections to serve that aim. In Section 2, we reviewed the meanings of the variable exponent Lebesgue spaces $L^{p(\cdot)}(\Omega)$ and the Sobolev spaces $W^{1, p(\cdot)}(\Omega)$. We demonstrate a global existence result in Section 3. For the decay results, see Section 4. While the numerical analysis is covered in Section 5.

2.   Preliminaries

The materials required for the assertion and the demonstration of our results are provided in this part. We now provide definitions and characteristics for Lebesgue and Sobolev spaces with varying exponents. See ^[21,29].

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

for some $\lambda > 0.$

Definition 2.1. The Luxembourg-type norm is given by

The space $L^{p(\cdot)}(\Omega),$ equipped with this norm, is a Banach space (see ^[21]). The variable-exponent Sobolev space $W^{1, p(\cdot)}(\Omega)$ is defined as

This space is a Banach space with respect to the norm

The definition of the space $W_{0}^{1, p(\cdot)}(\Omega)$ is the closure of $C_{0}^{\infty }(\Omega)$ in $W^{1, p(\cdot)}(\Omega).$ In contrast to the case with constant exponents, the specification of the space $W_{0}^{1, p(\cdot)}(\Omega)$ is typically different. Both definitions, however, match up under condition (1.3). In the same way as in the classical Sobolev spaces, where $\frac{1}{p(\cdot)}+\frac{1}{p^{\prime }(\cdot)} = 1$, the dual space of $W_{0}^{1, p(\cdot)}(\Omega)$ is $W_{0}^{-1, p^{\prime }(\cdot)}(\Omega)$.

Lemma 2.2. ^[19] (Poincaré's inequality). $\textrm{Let }$$\Omega$ $\textrm{be a bounded domain of }$$\mathbb{R}^{n}$ $\textrm{and }$$p(\cdot)$$\ \textrm{satisfies}$ (1.3)$\textrm{, then}$

$\textrm{where the positive constant }$$C$$\ \textrm{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 2.3. $\textrm{If }$$p:\overline{\Omega }\longrightarrow \lbrack 1, \infty)$$\ \textrm{is continuous and }$

$\textrm{then the embedding }$$H_{0}^{1}(\Omega)\hookrightarrow L^{p(\cdot)}(\Omega)$$\ \textrm{is continuous.}$

Lemma 2.4. $\textrm{If }$$p:\Omega \longrightarrow \lbrack 1, \infty)$$\ \textrm{is a measurable function and}$ $p_{2} < \infty,$ $\textrm{then}$ $C_{0}^{\infty }(\Omega)$ $\textrm{is dense in}$ $L^{p(\cdot)}(\Omega).$

Lemma 2.5. ^[19] (Hölder's inequality). $\textrm{Let }$$p, q, s\geq 1$$\textrm{ be measurable functions defined on }$$\Omega$$\ \textrm{such that }$

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

Lemma 2.6. (Unit Ball Property). $\textrm{Let }$$p$$\ \textrm{be a measurable function on }$$\Omega$$\textrm{. Then}$

$\textrm{where}$

Lemma 2.7. $\textrm{If }$$p$$\ \textrm{is a measurable function on }$$\Omega$$\ \textrm{satisfying}$ (1.1), then

$\textrm{for a.e., }$$x\in \Omega$$\ \textrm{and for any }$$u\in L^{p(\cdot)}(\Omega).$

3.   Global existence

We prove a global existence result, referring to the method used in ^[10]. We first introduce the new variable

Thus, we have

Then, problem (1.1) takes the form

Definition 3.1. For $T > 0$ fixed, we call $(u, z)$ a strong solution if

and satisfies the equations of (3.1) in $H^{-1}(\Omega)$ and $L^{2}(\Omega)$ respectively and the initial data.

The energy functional associated to (3.1) is given by

for $t\geq 0$ and $\zeta$ is a continuous function satisfying

One can take, for instance,

The following lemma shows that the associated energy of the problem is nonincreasing under the condition $\mu _{1} > \left\vert \mu _{2}\right\vert$.

Lemma 3.2. $\textrm{Let }$$(u, z)$$\ \textrm{be the solution of}$ (3.1). $\ \textrm{Then, for some }$$C_{0} > 0$$\textrm{, }$

Proof. Multiplying Eq $\left(3.1\right) _{1}$ by $u_{t}$ and integrating over $\Omega$ and multiplying $\left(3.1\right) _{2}$ by $\frac{1}{\tau }\zeta (x)\left\vert z\right\vert ^{m(x)-2}z$ and integrating over $\Omega \times (0, 1)$, then summing up, we get

We, now, estimate the last two terms of the right-hand side of (3.5) as follows,

For the last term, we use Young's inequality with $q = \frac{m(x)}{ m(x)-1}$ and $q^{\prime } = m(x)$ to get

Consequently, we arrive at

Hence, we obtain

Finally, the relation (3.3) yields, $\forall$ $x\in \overline{\Omega },$

Since $m(x)$ is bounded, hence $\zeta \left(x\right),$ we deduce that $f_{1}(x)$ and $f_{2}(x)$ are bounded. Therefore, if we define

and take $C_{0} = \inf_{{}\overline{\Omega }}C_{0}(x),$ then $C_{0}(x)\geq C_{0} > 0.$ Hence,

Now, we show that the solution of (3.1) is uniformly bounded and global in time.

For this purpose, we set

Hence,

Lemma 3.3. $\textrm{Suppose that the initial data }$$u_{0}, u_{1}\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ $\textrm{satisfying }$$I(0) > 0$ $\textrm{and }$

$\textrm{Then}$ $I(t) > 0,$ $\textrm{for any}$ $t\in \lbrack 0, T]$ $\textrm{and }$$\gamma > 0$$\ \textrm{to be specified later.}$

Proof. If

then the result is straightforward. So, we will assume

Since $I(0) > 0$ we deduce by continuity that there exists $T^{\ast }\leq T$ such that $I(t)\geq 0$ for all $t\in \lbrack 0, T^{\ast }].$ This implies that, for all $t\in \lbrack 0, T^{\ast }],$

Thus,

On the other hand, using the facts that $\ln \left\vert u\right\vert < \left\vert u\right\vert$ and $\left\vert u\right\vert > 1$, we get

If we choose $0 < k < \frac{2}{n-2},$ then the embedding $H_{0}^{1}(\Omega)\hookrightarrow L^{p_{2}+k}(\Omega)$ yields

where $C_{p_{2}+k}$ is the embedding constant. So,

Consequently, from (3.6) and (3.7) we deduce that

By repeating this procedure, $T^{\ast }$ can be extended to $T$.

Theorem 3.4. $\textrm{If the initial data }$$u_{0}, u_{1}$$\ \textrm{satisfy the conditions of Lemma 3.3, then the solution of}(3.1)\ \textrm{is uniformly bounded and global in time.}$

Proof. It suffices to show that $||\nabla u||_{2}^{2}+||u_{t}||_{2}^{2}$ is bounded independently of $t$. Clearly,

Therefore,

where $C$ is a positive constant depending only on $k, p_{1}\ $and $p_{2}$.

4.   Decay

Lemma 4.1. (Komornik ^[31] p. 103 and 124). $\textrm{Let }$$\mathrm{E}:\mathbb{R} ^{+}\longrightarrow \mathbb{R}^{+}$$\ \textrm{be a nonincreasing function. Assume that there exist }$$\sigma > 0, \omega > 0$$\ \textrm{such that}$

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

Before we state the main theorem, we need the following technical lemma.

Lemma 4.2. $\textrm{The functional}$

$\textrm{satisfies, along the solution of}$ (3.1),

Proof. A direct differentiation of $F(t),$ using $\left(3.1\right) _{2}$, leads to

Our main result reads as follows.

Theorem 4.3. $\textrm{Assume that the conditions}$ (1.2) $\ \textrm{and}$ (1.3) $\ \textrm{are satisfied}$. $\textrm{Then there exist two positive constants }$$c$ $\textrm{and }$$\alpha$$\ \textrm{such that any global solution of}(3.1)\ \textrm{satisfies}$

Proof. Multiply $\left(3.1 \right) _{1}$ by $u\mathrm{E}^{q}(t),$ for $q > 0$ to be specified later, and integrate over $\Omega \times \left(s, T\right),$ $s < T,$ to obtain

which gives

Recalling the definition of $\mathrm{E}(t)$ given in (3.2), adding and subtracting some terms and using the relation

Eq (4.1) becomes

The first term in the right hand side of (4.2) is estimated as follows.

where $C_{P}$ is the Poincaré constant. Using the fact that $\mathrm{E}(t)$ is decreasing, we deduce that

Similarly, we treat the term:

To handle the next term, we set

and use Hölder's and Young's inequalities, to get

For $m_{1} > 2,$ the choice of $q = \frac{m_{2}}{2}-1$ will make $\frac{qm_{1}}{ m_{1}-2} = q+1+\frac{m_{2}-m_{1}}{m_{1}-2}.$ Hence,

For the case $m_{1} = 2,$ the choice of $q = \frac{m_{2}}{2}-1,$ will give a similar result.

For the next term, we use Young's inequality. So, for a.e., $x\in \Omega,$ we have

where we used Young's inequality with

and, hence,

Therefore, using the embedding of $H_{0}^{1}(\Omega)\hookrightarrow L^{m_{1}}(\Omega)$ and $H_{0}^{1}(\Omega)\hookrightarrow L^{m_{2}}(\Omega)$, we arrive at

where $c(\varepsilon)$ is a finite constant depend on $\varepsilon$ whence it is fixed because $m(x)$ is bounded.

The next term of (4.2) can be estimated in a similar manner to reach

For the logarithmic term, we use the same idea of (3.6). We have

Thus the embedding $H_{0}^{1}(\Omega)\hookrightarrow L^{p_{2}-1+k}(\Omega),$ yields, for some $0 < \widetilde{\beta } < 1,$ to

Thus, we have

The last term of (4.2) can be estimated, using Lemma 4.2, as follows,

As $\zeta \left(x\right)$ is bounded, we obtain, for $c > 0,$

Combining (4.2) − (4.9), we arrive at

Recalling that $\widetilde{\beta } < 1$, then the choice of $\varepsilon$ small enough will make $\varepsilon +\widetilde{\beta } < 1.$ Therefore,

As $T\longrightarrow \infty$, we get

Therefore, Komornik's lemma is satisfied with $\sigma = q = \frac{m_{2}}{2}-1$ which implies the desired result.$\ \ \ \ \ \ \ \ \ \ \ $

5.   Numerical experiments

In this section, we devote some numerical experiments to illustrate the theoretical results in theorems (3.4) and (4.3) on a one-dimensional test problem of the form (3.1) with space variable $x$. For this purpose, we discretize the system (3.1) using a finite difference method (FDM) in both time and space with second-order accuracy in time and space over the time-space domain $(0, 1]\times[0, 1]$. The spatial space $\Omega = (0, 1)$ is divided into $M = 20$ subintervals in Test $1$ and $M = 100$ in Test $2$ with a step $\Delta x = \frac{1}{M}$, where the time interval $(0, 1)$ is divided into $N = 1000$ subintervals with a time step $\Delta t = \frac{1}{N}$. We take the initial conditions of the problem as $u_0 = sin(\pi x)$, $u_1 = 0$, $\mu_1 = 10$, $\mu_2 = -5$, and $\tau = 0.4$.

We compare the following numerical two tests based on the the value of $m(x)$:

● Test 1: Exponential decaying. We take $m(x) = 2$ with $p(x) = 6+x^2$ to verify the second case of theorem $4.3$,

where the notation $\sim$ means the two quantities have the same order.

● Test 2: Polynomial decaying. We take $m(x) = 2+x^2$ with $p(x) = 6+x^2$ to verify the first case,

5.1.   Numerical method

Now, we introduce our numerical scheme of the problem by using finite difference method for time and space discretization. For this purpose, we divide the special domain into $M$ subintervals and the time interval into $N$ subintervals:

In finite difference method, we find an approximate solution at the points (nodes): $x_1, x_2, \ldots , x_M$ for each time level $t_1, t_2, t_3, \ldots , t_N,$ such that $U(i, n)\approx u(x_i, t_n)$, where $u(x_i, t_n)$ is the exact solution at $(x_i, t_n)$.

The approximation of the problem (3.1) is accomplished by replacing the derivatives with appropriate difference quotients as the following:

Then we substitute the above difference quotients in the first equation of the system problem (3.1) and we solve the resultant equation for $U(i, n+1)$ to get the numerical scheme.

By iteration method, we find $U(i, n+1)$ for $n = 1, 2, 3, \ldots , N$ and $i = 1, 2, 3, \ldots , M$ with $U(i, 0) = u_0(x_i)$, $U(i, 1) = u_0(x_i)+\Delta t u_1(x_i)$ and $U(0, n) = U(M, n) = 0$.

5.2.   Numerical results

We use the MATLAB tool to perform the numerical scheme. In this section, we discuss the numerical results and compare them with the results of theorem (4.3):

Exponential decaying $(m(x) = 2)$: Figure 1 shows that the solution $u(t)$ is a function of $t$ at some fixed values of $x$ = 0.25, 0.5, 0.7, 0.9 which are oscillating and decaying at each cross section cut of $x$.

Figure 2 shows that the energy function $\mathrm{E}(t)$ is decaying exponentially. The left graph is for $\mathrm{E}(t)$ vs $t$ with linear scale axes, while the right one is for $\mathrm{E}(t)$ vs $t$ with log scale on the $y-$axis and linear scale on the $x-$axis. As we see, the graph is line (linear relation between $log[\mathrm{E}(t)]$ and $t$ with negative slope). This means that the energy is decaying exponentially of the form:

where by taking $log$ to both sides, we get:

This result agrees with the second case ($m(x) = 2$, exponential decay) of theorem (4.3).

Polynomial decay $(m(x) = 2+x^2)$: Figure 3 shows that the solution $u(t)$ as a function of $t$ at some fixed values of $x$ = 0.25, 0.5, 0.7, 0.9 which are oscillating and decaying at each cross section cut of $x$.

According to the first part of the theorem (4.3),

means that

So, the relation between $log[\mathrm{E}(t)]$ vs $log[1+t]$ is a decreasing linear relation. Hence, the energy is decaying polynomially.

This theoretical result coincides with the numerical result in Figure 4; the right graph of $\mathrm{E}(t)$ vs $(t+1)$ with $log-log$ scale axes is linear with negative slope. Also, if we compare the left graph of Figure 2 (Test 1) and the left graph of Figure 4 (Test 2), we observe that the first one (exponential decay) is decaying faster than the second one (polynomial decaying).

Finally, Figure 5 shows the solution $u(x, t)$ in $3D$ with $m(x) = 2+x$, $p(x) = 3+x$ and $\mu_1$, $\mu_2$, $\tau$, $u_0$, and $u_1$ are the same.

Acknowledgment

The authors would like to express their sincere thanks to the interdisciplinary research center in construction and building materials, King Fahd University of Petroleum and Minerals (KFUPM) for their support.

Conflict of interest

The authors declare that there is no conflict of interest.