Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities

1.   Introduction

In this paper we are concerned with the following problem

where $\Omega$ is a bounded domain of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, $\nu$ is the unit outer normal to $\partial \Omega$, $u_0$ and $u_1$ are the given data, $b$ is a relaxation function and $\gamma(.)$ is a variable exponent.

Problem (1.1) contains three class of problems:

I. Viscoelasticity with wide class of relaxation functions.

The importance of the viscoelastic properties of materials has been realized because of the rapid developments in rubber and plastics industry. Many advances in the studies of constitutive relations, failure theories and life prediction of viscoelastic materials and structures were reported and reviewed in the last two decades ^[1]. There is an extensive literature on the stabilization of viscoelastic wave equations and many results have been established. There are a lot of contributions to generalize the decay rates by allowing an extended class of relaxation functions and give general decay rates. In fact, the journey of generalization of relaxation functions passed through several steps, we mention here the following stages:

$1)$ As in ^[2], the relaxation function $b$ satisfies, for two positive constants $a_{1}$ and $a_{2},$

$2)$As in ^[3,4], the relaxation function $b$ satisfies

where $a:\mathbb{R}^+ \to \mathbb{R}^+$ is a nonincreasing differentiable function.

$3)$ As in ^[5], the relaxation function $b$ satisfies

where $\chi$ is a positive function, $\chi(0) = \chi^{\prime}(0) = 0$, and $\chi$ is strictly increasing and strictly convex near the origin.

$4)$ As in ^[6], the relaxation function $b$ satisfies

$5)$As in ^[7], the relaxation function $b$ satisfies

where $B\in C^1(\mathbb{R})$, with $B(0) = 0$ and $B$ is linear or strictly increasing and strictly convex function $C^2$ near the origin.

II. Variable-exponent nonlinearity.

With the advancement of sciences and technology, many physical and engineering models required more sophisticated mathematical functional spaces to be studied and well understood. For example, in fluid dynamics, the elecrtorheological fluids (smart fluids) have the property that the viscosity changes (often drastically) when exposed to an electrical field. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools to study such problems as well as other models like fluids with temperature-dependent viscosity, nonlinear viscoelasticity, filtration processes through a porous media and image processing. More details on these problems can be found in ^[8,9]. For hyperbolic problems involving variable-exponent nonlinearities, we refer to ^{[10,11,12,13,14,15]}. For more results of other problems with the nonlinearity of power type, we refer the interested reader to see ^[16,17,18].

III. Logarithmic source term.

The logarithmic nonlinearity appears naturally in inflation cosmology and supersymmetric filed theories, quantum mechanics and nuclear physics ^[19,20]. Problems with logarithmic nonlinearity have a lot of applications in many branches of physics such as nuclear physics, optics and geophysics ^[21,22,23].

In this paper, we consider problem (1.1) and prove the global existence of solutions, using the well-depth method. We then establish explicit and general decay results of the solution under suitable assumptions on the variable exponent $\gamma(.)$ and very general assumption on the relaxation function. To the best of our knowledge, such a problem has not been discussed before in the context of nonlinearity with variable exponents.

In addition to the introduction, this paper has four other sections. In Section 2, we present some preliminaries. The Existence is given in Section 3. In Section 4, we establish some technical lemmas needed for the proof of the main results. Our stability results and their proof are given in Section 5.

2.   Preliminaries

In this section, we present some preliminaries about the logarithmic nonlinerity and the Lebesgue and Sobolev spaces with variable exponents (see ^{[24,25,26,27]}). Throughout this paper, c is used to denote a generic positive constant.

Definition 2.1. Let $\beta:\Omega\rightarrow [1, \infty]$ be a measurable function, where $\Omega$ is a bounded domain of $\mathbb{R}^n$, then we have the following definitions:

1) The Lebesgue space with a variable exponent $\beta(\cdot)$ is defined by

where $\varrho_{\beta(\cdot)}(v) = \int_{\Omega}\frac{1}{\beta(x)}|v(x)|^{\beta(x)}dx$ is a modular.

2) The variable-exponent Sobolev space $W^{1, \beta(\cdot)}(\Omega)$ is:

3) $W_{0}^{1, \beta(\cdot)}(\Omega)$ is the closure of $C_{0}^{\infty}(\Omega)$ in $W^{1, \beta(\cdot)}(\Omega)$.

Remark 2.2. ^[9]

1) $L^{\beta(\cdot)}(\Omega)$ is a Banach space equipped with the following Luxembourg-type norm

2) $W^{1, \beta(\cdot)}(\Omega)$ is a Banach space with respect to the norm

Definition 2.3. Let $K$ be a convex function on $(0, r]$, then the convex conjugate of $K$, in the sense of Young (see ^[32]), is defined as follows:

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

Let

Lemma 2.4. ^[9] If $\beta:\Omega\rightarrow [1, \infty)$ is a measurable function with $\beta_2 < \infty,$ then $C_{0}^{\infty}(\Omega)$ is dense in $L^{\beta(\cdot)}(\Omega)$.

Remark 2.5 (Log-Hölder continuity condition). The exponent $p(\cdot): \Omega \rightarrow [1, \infty]$ is said to be satisfying the log-Hölder continuity condition; if there exists a constant $c > 0$ such that, for all $\delta$ with $0 < \delta < 1,$

Lemma 2.6. ^[9][Poincaré's Inequality] Let $\Omega$ be a bounded domain of $\mathbb{R}^n$ and $p(\cdot)$ satisfies $(2.3),$ then

In particular, the space $W_{0}^{1, p(\cdot)}(\Omega)$ has an equivalent norm given by $\|v\|_{W_{0}^{1, p(\cdot)}(\Omega)} = \|\nabla v\|_{p(\cdot)}.$

Lemma 2.7. ^[9][Embedding Property] Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with a smooth boundary $\partial \Omega.$ Assume that $p, k\in C(\overline{\Omega})$ such that

and $k(x) < p^*(x)$ in $\overline{\Omega}$ with

then we have continuous and compact embedding $W^{1, p(.)}(\Omega)\hookrightarrow L^{k(.)}(\Omega).$ So, there exists $c_e > 0$ such that

Lemma 2.8. ^[27] Let $\epsilon \in (0, 1).$ Then there exists $\beta_{\varepsilon} > 0$ such that

We consider the following hypotheses:

$(A1)$ The relaxation function $b:\mathbb{R^+}\to \mathbb{R^+}$ is a $C^1$ nonincreasing function satisfying

and there exists a $C^1$ function $B:(0, \infty)\to (0, \infty)$ which is strictly increasing and strictly convex $C^2$ function on $(0, r],$ $r\le b(0),$ with $B(0) = B'(0) = 0,$ such that

where $a$ is a positive nonincreasing differentiable function.

$(A2)$ ${\gamma}: \overline{\Omega} \rightarrow [1, \infty)$ is a continuous function satisfies the log-Hölder continuity condition (Remark 2.5) such that

and $1 < {\gamma}_1 < {\gamma}(x) \leq {\gamma}_2,$ where

$(A3)$ The constant ${\alpha}$ in (1.1) satisfies $0 < {\alpha} < {\alpha}_0$, where ${\alpha}_0$ is the positive real number satisfying

where $\|. \|_2 = \|.\|_{L^{2}(\Omega)}$.

Lemma 2.9. ^[28,29] (Logarithmic Sobolev inequality) Let $u$ be any function in $H^{1}_{0}(\Omega)$ and $d$ be any positive real number. Then

Lemma 2.10. There exists a unique ${\alpha}_0 > 0$ such that

Proof. Let $g(s) = \sqrt{\frac{2\pi \bar{b}}{ s}}-e^{-\frac{3}{2}-\frac{1}{s}}$, then $g$ is a continuous and decreasing function on $(0, \infty),$ with

Then, there exists a unique ${\alpha}_0 > 0$ such that $g({\alpha}_0) = 0$ and (2.9) holds

Remark 2.11. Lemma 2.10 shows that the selection of ${\alpha}$ in $(A3)$ is possible.

Remark 2.12. Using the facts that $B(0) = 0$ and $B$ is strictly convex on $(0, r]$, then

Remark 2.13. ^[7] If $B$ is a strictly increasing and strictly convex $\boldsymbol{C}^2$ function on $(0, r]$, with $B(0) = B'(0) = 0$, then there is a strictly convex and strictly increasing $\boldsymbol{C}^2$ function $\overline{B}:[0, +\infty)\longrightarrow[0, +\infty)$ which is an extension of $B$. For simplicity, in the rest of this paper, we use $B$ instead of $\overline{B}$.

3.   Existence

In this section, we state the local existence theorem whose proof can be established by combining the arguments of ^[10,30,31]. Also, we state and prove a global existence result under smallness conditions on the initial data $(u_0, u_1)$.

Theorem 3.1 (Local Existence). Suppose conditions $(A1)$-$(A3)$ hold and $(u_0, u_1)\in H^1_0(\Omega)\times L^2(\Omega).$ Then, there exists $T > 0$, such that problem $(1.1)$ has a weak solution

Definition 3.2. We define the following functionals which are needed for establishing the global existence

where for $v\in L^{2}_{loc}(\mathbb{R}^+; L^2(\Omega)),$

$E(t)$ represents the modified energy functional associated to problem (1.1).

then

Notation: We define

and

where $0 < d < \sqrt{\frac{2\pi \bar{b}}{{\alpha}}}.$

Lemma 3.3. Assume that $(u_{0}, u_{1})\in H^1_0(\Omega)\times L^{2}(\Omega)$, $(A1)$ holds,

Then, $I(u(t))\ge 0$ for all $t\in [0, T).$

Proof. First, we show that $\|u\|_2 < \rho_{*}$, $\forall t\in [0, T).$ By (2.5), (3.4) and (2.9), we obtain

If we select $d < \sqrt{\frac{2\pi \bar{b}}{{\alpha}}}$, then (3.6) becomes

where $D_0 = \frac{{\alpha}+2}{2}+{\alpha}(1+\ln{d})$ and ${\rho} = \|u\|_2.$ Using (3.7), we can deduce that that $Z$ is increasing on $(0, {\rho}_{*})$, decreasing on $({\rho}_{*}, +\infty)$ and $Z({\rho})\to -\infty$ as ${\rho} \to +\infty.$ Moreover,

Suppose that $\|u(x, t)\|_2 < {\rho}_{*}$ is not true in $[0, T).$ Therefore, using the continuity of $u(t),$ it follows that there exists $0 < t_0 < T$ such that $\|u(x, t_0)\|_2 = \rho_{*}.$ From Eq (3.7), we can see that

which is a contradiction with $E(t)\le E(0) < E_1$ for all $t \ge 0.$ Recalling the definition of $I(u(t)),$ and using (2.9) with $d < \sqrt{\frac{2\pi \bar{b}}{{\alpha}}},$ for all $t\in [0, T)$, lead to

This completes the proof.

Remark 3.4. We can see that if $\|u_0\|_2 < \rho_*$ and $E(0) < E_1,$ then $J(u(t))\ge 0$ and consequently $E(t)\ge 0$ for all $t\in [0, T).$ Therefore, from $(3.8)$, for $t\in [0, T)$ we have

which shows that the soultion is global and bounded in time.

4.   Technical lemmas

In this section, we establish several lemmas needed for the proof of our main result.

Lemma 4.1. The energy functional associated to problem $(1.1)$ satisfies, for any $t\ge 0$,

Proof. Multiplying (1.1) by $u_t$, integrating over $\Omega$ and using the boundary conditions, imply (4.1).

Lemma 4.2. ^[31] Assume that $b$ satisfies $(A1)$. Then, for $u\in H^{1}_{0}(\Omega),$

and

Lemma 4.3. ^[7] Assume $(A1)$ holds. Then, for any $t \ge t_0,$, we have

Lemma 4.4. Assume that $(A1)$-$(A3)$ and $(3.5)$ hold, then the functional

satisfies, along with the solution of (1.1), the estimates:

and

Proof. Differentiate $I_1$ and use the differential equation in (1.1), to get

Young's inequality and (4.2) give

Estimation of the term $-\int_{\Omega}u|u_t|^{{\gamma}(x)-2}u_t dx$:

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

where

Hence,

Now, using (3.1), (4.1), (3.9) and Lemma 2.7, we obtain

where

and

Then, (4.6) and (4.7) yield

Combining the above results with fixing $\delta_0 = \frac{\bar{b}}{2}$ and $\delta = \frac{\bar{b}}{4 c}$ completes the proof of (4.2).

For the proof of (4.3), we re-estimate the fifth term in (4.4) as follows:

First, we define

Then, we get

Using the definition of $\Omega_1$, we have

Therefore, using Young's and Poincaré's inequalities and (4.10), we obtain

where

After setting $\theta = \frac{\bar{b}}{8 c^2_*}$, (4.11) becomes

Next, for any $\delta$ we have, by the case ${\gamma}(x) \ge 2,$

Therefore, by combining (4.9)-(4.14), we arrive at

By fixing $\delta = \frac{\bar{b}}{8c}$, $c_{\delta}(x)$ remains bounded and, consequently, we obtain (4.3).

Lemma 4.5. Assume that $(A1)$-$(A3)$ and $(3.5)$ hold, then for any $\delta > 0$, the functional

satisfies, along the solution of (1.1), the estimates:

and for $1 < \gamma_1 < 2$, we have the following estimate

Proof. Direct differentiation of $I_2$, using (1.1), yields

Using Young's inequality and Lemma 4.2, we obtain

The use of Lemma 4.2, Young's and Poincaré's inequalities leads to

Exploiting Lemma (4.2) and Young's inequality, we obtain

Next, for almost every $x\in \Omega$ fixed, we have

Therefore, for almost every $x\in \Omega,$ we have

By using Young's, Hölder's and Poincaré's inequalities and Lemma 4.2, we get

where

Similarly, we have

Therefore,

where $c = \left[c_e^{{\gamma}_2}\left(\frac{4\pi}{2\pi\bar{b}-{\alpha} d^2}E(0)\right)^{\frac{{\gamma}_2-2}{2}}+c_e^{{\gamma}_1}\left(\frac{4\pi}{2\pi\bar{b}-{\alpha} d^2}E(0)\right)^{\frac{{\gamma}_1-2}{2}}\right].$

For the last term in (4.17), the use of (2.4), Young's, Cauchy-Schwarz' and Poincaré's inequalities, the embedding theorem and Lemma 4.2 leads to, for any $\delta > 0$,

Combining the above estimates with (4.17), we obtain (4.15).

For the proof of (4.16), we re-estimate the fifth term in (4.17) as follows:

Then (4.16) is established.

Lemma 4.6. Given $t_0 > 0.$ Assume that $(A1)$-$(A3)$ and $(3.5)$ hold. Then,

satisfies, for a suitable choice of $N_1, \, N_2 > 0$ and for some positive constants $\lambda_0$ and $c$, the estimates, for any $\, t \geq t_0$,

and

Proof. Since $b$ is positive and $b(0) > 0$ then, for any $t_0 > 0$, we have

By using (4.1), (4.2) and (4.15), then, for $t\ge t_0$ and any $\lambda_0 > 0$, we have

Using the Logarithmic Sobolev inequality, for $0 < \lambda_0 < \frac{1}{2}$, we get

At this point, we select $\lambda_0$ and ${\alpha}$ so small that

Then, we choose $N_2$ large enough so that:

and

and then $N_1$ large enough that

Therefore, we arrive at the desired result (4.27). On the other hand, we can choose $N_{1}$ even larger (if needed) so that

5.   Decay results

In this section, we establish our main decay results. For this purpose, we need the following remarks and lemma.

Remark 5.1. Using $(3.6)$ and $(4.1)$, we get

Remark 5.2. In the case of $B$ is linear and since $a$ is nonincreasing, we have

Lemma 5.3. If $(A1)$-$(A2)$ are satisfied, then we have the following estimate

where $\varepsilon_0$ is small enough and the functional $\psi$ is defined by

Proof. To establish (5.3), let us define the following functional

Then, using (3.1), (4.1) and the dentition of $\Lambda(t)$, we have

Thus, $\varepsilon_0$ can be chosen so small so that, for all $t > 0$,

Without loss of the generality, for all $t > 0$, we assume that $\Lambda(t) > 0$, otherwise we get an exponential decay from (4.27). The use of Jensen's inequality and using (5.4), (2.10) and (5.7) gives

hence (5.3) is established.

Theorem 5.4 (The case: $\bf \gamma_1\geq 2$). Assume that $(A1)$-$(A3)$ and $(3.5)$ hold. Let $(u_0, u_1)\in H^1_0(\Omega)\times L^2(\Omega).$ Then, there exist positive constants $c$, $t_0$ and $t_1$ such that the solution of $(1.1)$ satisfies,

and

where ${\mathcal{B}_2}(s) = s \mathcal{B}^{\prime}(\varepsilon_{1}s)$ and $\mathcal{B}(t) = \left(\left[B^{-1}\right]^{\frac{1}{1+\epsilon}}\right)^{-1}(t)$.

Proof. Case 1: $B$ is linear

We multiply (4.27) by $a(t)$ and use (5.1) and (5.2) to get

Multiply (5.11) by $a^{\epsilon}(t) E^{\epsilon}(t)$, and recall that $a^{\prime} \le 0,$ to obtain

Use of Young's inequality, with $q = \epsilon+1$ and $q^{*} = \frac{\epsilon+1}{\epsilon}$, gives, for any $\varepsilon^{\prime} > 0$,

We then choose $0 < \varepsilon^{\prime} < \frac{\lambda_0 }{c}$ and use that $a^{\prime} \le 0$ and $E^{\prime} \le 0$, to get, for $c_1 = \lambda_0 -\varepsilon^{\prime} c$,

which implies

where $L_1 = a^{\epsilon +1}E^{\epsilon}L +cE$. Then $L_1\sim E$ (thanks to (4.29)) and

Integrating over $(t_0, t)$ and using the fact that $L_1 \sim E$, we obtain (5.9).

Case 2: $B$ is non-linear.

Using (4.27), (5.1) and (5.3), we obtain, $\forall t\ge t_0$,

Combining the strictly increasing property of $\overline{B}$ and the fact that $\frac{1}{t} < 1$ whenever $t > 1$, we obtain

then, (5.12) becomes, for $\forall t \ge t_1 = \max{\{t_0, 1\}},$

Set

Using the facts that $\mathcal{B}^{\prime} > 0$ and $\mathcal{B}^{\prime\prime} > 0$ on $(o, r]$, (5.14) reduces to

Now, for $\varepsilon_{1} < r$ and using (5.16) and the fact that $E^{\prime}\le 0$, $\mathcal{B}^{\prime} > 0, \mathcal{B}^{\prime\prime} > 0$ on $(0, r],$ we find that the functional $L_{2},$ defined by

satisfies, for some $c_{1}, c_{2} > 0.$

and, for all $t\ge t_1$,

So, using (2.1) and (2.2) with $\alpha_1 = \mathcal{B}^{\prime}\left(\frac{\varepsilon_{1}}{t^{\frac{1}{1+\epsilon}}} \cdot\frac{E(t)}{E(0)}\right)$ and $\alpha_2 = \mathcal{B}^{-1}(\chi(t)),$ we arrive at

Then, multiplying (5.19) by $a(t)$ and using (5.4), (5.15), we get

Using the non-increasing property of $a$, we obtain, for all $t \ge t_1,$

Therefore, by setting $L_{3}: = a L_{2}+cE \sim E$, we conclude that

This gives, for a suitable choice of $\varepsilon_1$,

or

An integration of (5.20) yields

Using the facts that $\mathcal{B}', \mathcal{B}'' > 0$ and the non-increasing property of $E$, we deduce that the map $t \mapsto E(t)\mathcal{B}^{\prime}\left(\frac{\varepsilon_{1}}{t^{\frac{1}{1+\epsilon}}} \cdot \frac{E(t)}{E(0)}\right)$ is non-increasing and consequently, we have

Multiplying each side of (5.22) by $\frac{1}{t^{\frac{1}{1+\epsilon}}}$, we have

Next, we set ${\mathcal{B}_2}(s) = s \mathcal{B}^{\prime}(\varepsilon_{1}s)$ which is strictly increasing, and consequently we obtain,

Finally, we infer

This finishes the proof.

The following examples illustrate the results of Theorem 5.4:

Example 1. Let $b(t) = c_1 e^{-c_2(1+t)},$ where $c_2 > 0$ and $c_1 > 0$ is small enough so that $(A1)$ holds. Then $b^{\prime}(t) = -a(t) B(b(t))$ where $B(t) = t$ and $a(t) = c$. Therefore, we can use $(5.9)$ to deduce

Example 2. Let $b(t) = \frac{c_1}{(1+t)^q}$, where $q > 1+\epsilon$ and $c_1$ is chosen so that hypothesis $(A1)$ is satisfied. Then

where $a$ is a fixed constant. Then, $(5.10)$ gives,

To establish the stability result in the case $\bf 1 < \gamma_1 < 2$, we need the following lemma:

Lemma 5.5. The energy functional $E(t)$ satisfies the following estimate:

where $\gamma_{\varepsilon} = \min\{\gamma_1-1, \frac{1}{1+\varepsilon}\}$.

Proof. Using (2.5), (3.1), (3.3), (3.6) and Lemma 3.3, we have

then, using (4.1),

So, from (4.1), (4.7) and using Young's inequality, we get

Setting $\gamma_{\varepsilon} = \min\{\gamma_1-1, \frac{1}{1+\varepsilon}\}$ and using (5.30), we obtain

which completes the proof of Lemma 5.5.

Theorem 5.6 (The case: $\bf 1 < \gamma_1 < 2$). Assume that $(A1)$-$(A3)$ and $(3.5)$ hold. Let $(u_0, u_1)\in H^1_0(\Omega)\times L^2(\Omega).$ Then, there exist positive constants $C$, $k_2, \, k_3$ such that the energy functional associated to problem (1.1) satisfies

and, if $B$ is nonlinear, we have

where $\gamma_{\varepsilon} = \min\{\gamma_1-1, \frac{1}{1+\varepsilon}\}$, ${\mathcal{B}_3}(s) = s \mathcal{B}^{\prime}(\varepsilon_{3}s)$and $\mathcal{B}(s) = \left(\left[B^{-1}\right]^{\frac{1}{1+\epsilon}}\right)^{-1}(s)$.

Proof. Case B is linear.

Multiplying (4.28) by $a(t)$ and combining (2.6), (3.1), (5.2) and (5.28), we obtain, for some $m_1 > 0$,

Let $\mathcal{L}: = aL+cE\sim E$, multiply both sides of the above estimate by $a^q E^q$, with $q = \frac{1}{\gamma_{\varepsilon}}-1$ and apply Young's inequality, to get,

Set $\mathcal{L}_1: = a^qE^q \mathcal{L}+cE \sim E$, take $\epsilon_2$ small enough and use the non-increasing property of $E$ we obtain, for some $m_2, m_3 > 0,$

A simple integration over $(t_0, t)$ and using the equivalence $L \sim E,$ we obtain,

Case B is nonlinear.

Using (4.27), (5.1) and (5.3), we obtain, $\forall t\ge t_0$,

Using (5.13)-(5.15), (5.35) reduces to

Now, for $\varepsilon_{3} < r$ and using (5.16) and the fact that $E^{\prime}\le 0$, $H^{\prime} > 0, H^{\prime\prime} > 0$ on $(0, r],$ we find that the functional $\mathcal{F},$ defined by

satisfies

and, for all $t\ge t_1$,

After applying with the generalized Young inequality

we arrive at

Using the facts that $\frac{1}{2-\gamma_1} > 1$ and $\mathcal{B}^{\prime}\left(\frac{\varepsilon_{3}}{t^{\frac{1}{1+\epsilon}}} \cdot \frac{E(t)}{E(0)}\right)$ is bounded, we have

Then, multiplying (5.39) by $a(t)$, using (5.15), (5.40) and the fact that $E(t) > 0$, we get

where $\mathcal{F}_1 = \mathcal{F}+c_{\varepsilon}E^{\prime}$. Using the non-increasing property of $a$, we obtain, for all $t \ge t_1,$

Therefore, by setting $\mathcal{F}_{2}: = a \mathcal{F}_{1}+cE \sim E$, we conclude that

This gives, for a suitable choice of $\varepsilon_3$ and $\varepsilon$,

or

An integration of (5.41) yields

Using the facts that $\mathcal{B}', \mathcal{B}'' > 0$ and the non-increasing property of $E$, we deduce that the map $t \mapsto E(t)\mathcal{B}^{\prime}\left(\frac{\varepsilon_{3}}{t^{\frac{1}{1+\epsilon}}} \cdot \frac{E(t)}{E(0)}\right)$ is non-increasing and consequently, we have

Multiplying each side of (5.43) by $\frac{1}{t^{\frac{1}{1+\epsilon}}}$, we have

Using the fact that ${\mathcal{B}_3}(s) = s \mathcal{B}^{\prime}(\varepsilon_{3}s)$ is strictly increasing, we obtain

Finally, we infer

This finishes the proof.

The following examples illustrate the results of Theorem 5.6:

Example 3. Let $b(t) = c_1 e^{-c_2(1+t)},$ where $c_2 > 0$ and $c_1 > 0$ is small enough so that $(A1)$ holds. Then $b^{\prime}(t) = -a(t) B(b(t))$ where $B(t) = t$ and $a(t) = c$. Therefore, $(5.32)$ gives for $t > t_0$ and $\epsilon \in (0, 1)$,

Example 4. Let $b(t) = \frac{c_1}{(1+t)^q}$, where $q > 1+\epsilon$ and $c_1$ is chosen so that hypothesis $(A1)$ is satisfied. Then

where $a$ is a fixed constant. Then, $(5.33)$ gives, for $t > t_1$ and $\epsilon \in (0, 1)$,

Remark 5.7. The classical power-type nonlinearity term in ^[33] provides a canonical description for the dynamics analysis of a quasi-wave propagation in a nonlinear process, therefore, the fast cumulative of such nonlinear interactions results in a significant effect to the solution under large spatial and temporal scales. However, the logarithmic nonlinearity in $(1.1)$ only expresses slowly cumulative of nonlinear, thus giving another kind of description for dynamic process. Let us note here that though the logarithmic nonlinearity is somehow weaker than the polynomial nonlinearity, both the existence and stability result are not obtained by straightforward application of the method used for polynomial nonlinearity.

Acknowledgment

The authors would like to express their profound gratitude to King Fahd University of Petroleum and Minerals (KFUPM)- Interdisciplinary Research Center (IRC) for Construction and Building Materials for their continuous supports. The authors also thank the referee for her/his very careful reading and valuable comments. This work is funded by KFUPM under Project #SB191037.

Conflict of interest

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