Existence and stability results of a plate equation with nonlinear damping and source term

1.   Introduction

The importance of bridges is undeniable and their presence in human daily life goes back a long time. With the presence of the bridges, road and railway traffic runs without any interruption over rivers and hazardous areas, time and fuel are saved, congestion on roads is minimized, distances between places are reduced, and many accidents have been avoided, as the bridges have reduced the number of bends and zig-zags in roads. As a result, many economies have grown and many societies have become connected. However, bridges have brought some challenges, such as collapse and instability due to natural hazards such as wind, earthquakes, etc. To overcome these difficulties, engineers and scientists have made efforts to find the best designs and possible models. Our aim in this work is to investigate the following plate problem

where $\Omega = (0, \pi) \times (-d, d)$, $d, \beta > 0$, $g: \mathbb{R}\rightarrow \mathbb{R}$ and $\alpha: [0, +\infty) \rightarrow (0, + \infty)$ is a nonincreasing differentiable function, $u$ is the vertical displacement of the bridge and $\sigma$ is the Poisson ratio. This is a weakly damped nonlinear suspension-bridge problem, in which the damping is modulated by a time dependent-coefficient $\alpha(t)$. Firstly, we prove the local existence using the Faedo-Gherkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish an explicit and general decay result, depending on $g$ and $\alpha$, for which the exponential and polynomial decay rate estimates are only special cases. The proof is based on the multiplier method and makes use of some properties of convex functions, including the use of the general Young inequality and Jensen's inequality.

2.   Literature review

The famous report by Claude-Louis Navier ^[1] was the only mathematical treatise of suspension bridges for several decades. Another milestone theoretical contribution was the monograph by Melan ^[2]. After the Tacoma collapse, engineers felt the necessary to introduce the time variable in mathematical models and equations in order to attempt explanations of what had occurred. As a matter of fact, in Appendix VI of the Federal Report ^[3], a model of inextensible cables is derived and the linearized Melan equation was obtained. Other important contributions were the works by Smith-Vincent and the analysis of vibrations in suspension bridges presented by Bleich-McCullough-Rosecrans-Vincent ^[4]. In all these historical references, the bridge was modelled linearly as a beam suspended to a cable. Hence, all the equations were linear. Mathematicians have not shown any interest in suspension bridges until recently. McKenna, in 1987, introduced the first nonlinear models to study them from a theoretical point of view, and he was followed by several other mathematicians (see ^[5,6]). McKenna's main idea was to consider the slackening of the hangers as a nonlinear phenomenon, a statement which is by now well-known also among engineers ^[7,8]. The slackening phenomenon was analyzed in various complex beam models by several authors (see ^[9,10,11]). Motivated by the wonderful book of Rocard ^[12], where it was pointed out that the correct way to model a suspension bridge is through a thin plate, Ferrero-Gazzola ^[13] introduced the following hyperbolic problem:

where $\Omega = (0, \pi)\times (-\ell, \ell)$ is a planar rectangular plate, $\sigma$ is the well-known Poisson ratio, $\eta$ is the damping coefficient, $h$ is the nonlinear restoring force of the hangers and $f$ is an external force. After the appearance of the above model, many mathematicians showed interest in investigating variants of it, using different kinds of damping with the aim to obtain stability of the bridge modeled through the above problem. Messaoudi ^[14] considered the following nonlinear Petrovsky equation

and proved the existence of a local weak solution, showed that this solution is global if $m\ge p$ and blows up in finite time if $p > m$ and the energy is negative. Wang ^[15] considered the equation

where $a = a(x, y, t)$ together with the above initial and boundary conditions. After showing the uniqueness and existence of local solutions, he gave sufficient conditions for global existence and finite-time blow-up of solutions. Mukiawa ^[16] considered a plate equation modeling a suspension bridge with weak damping and hanger restoring force. He proved the well-posedness and established an explicit and general decay result without putting restrictive growth conditions on the frictional damping term. Messaoudi and Mukiawa ^[17] studied problem (2.3), where the linear frictional damping was replaced by nonlinear frictional damping and established the existence of a global weak solution and proved exponential and polynomial stability results. Audu et al. ^[18] considered a plate equation as a model for a suspension bridge with a general nonlinear internal feedback and time-varying weight. Under some conditions on the feedback and the coefficient functions, the authors established a general decay estimate. For more results related to the existence of work on similar problems, we mention the work of Xu et al. ^[19], in which they proved the local existence of a weak solution by the Galerkin method and the global existence by the potential well method. He et al. ^[20] considered the following Kirchhoff type equation

where $\Omega\subseteq \mathbb{R}^3$ is a bounded domain or $\Omega = \mathbb{R}^3$, $0 \leq h \in L^2 (\Omega)$ and $f \in C \left(\mathbb{R}, \mathbb{R}\right)$. The authors proved the existence of at least one or two positive solutions by using the monotonicity trick, and nonexistence criterion is also established by virtue of the corresponding Pohoaev identity. Recently, Wang et al. ^[21] considered the fractional Rayleigh-Stokes problem where the nonlinearity term satisfied certain critical conditions and proved the local existence, uniqueness and continuous dependence upon the initial data of $\varepsilon$-regular mild solutions. More results in this direction can be found in ^{[22,23,24,25,26,27]}. The paper is organized as follows. In Section 3, we present some preliminaries and essential lemmas. We prove the local existence in Section 4 and the global existence in Section 5. The statement and the proof of our stability result will be given in Section 6.

3.   Preliminaries and essential lemmas

In this section, we present some material needed in the proofs of our results. First, we introduce the following space

together with the inner product

It is well known that $(H^2_*(\Omega), (\cdot, \cdot)_{H^2_*})$ is a Hilbert space, and the norm $\Vert. \Vert ^2_{H^2_*}$ is equivalent to the usual $H^2$, see ^[13]. Throughout this paper, $c$ is used to denote a generic positive constant.

Lemma 3.1. ^[15]Let $u\in H^2_*(\Omega)$ and assume that $1\leq p < \infty$, then, there exists a positive constant $C_e = C_e(\Omega, p) > 0$ such that

Lemma 3.2. (Jensen's inequality)Let $\psi:[a, b]\longrightarrow\mathbb{R}$ be a convex function. Assume that the functions $f:(0, L)\longrightarrow[a, b]$ and $r:(0, L)\longrightarrow\mathbb{R}$ are integrable such that $r(x)\geq0$, for any $x\in (0, L)$ and $\int_{0}^{L}r(x)dx = k > 0$. Then,

We consider the following hypotheses:

(H1). The function $g:\mathbb{R}\rightarrow \mathbb{R}$ is nondecreasing $C^{0}$ function satisfying for $\varepsilon, c_{1}, c_{2} > 0$,

where $G:\mathbb{R}^+\rightarrow \mathbb{R}^+$ is a $C^1$ function which is linear or strictly increasing and strictly convex $C^2$ function on $[0, \varepsilon]$ with $G(0) = 0$ and $G^{\prime}(0) = 0$. In addition, the function $g$ satisfies, for $\vartheta > 0$,

(H2). The function $\alpha: \mathbb{R}^+\to \mathbb{R}^+$ is a nonincreasing differentiable function such that $\int_{0}^{\infty}\alpha(t)dt = \infty$.

Remark 3.3. Hypothesis (H1) implies that $s g(s) > 0,$ for all $s\ne 0$ and it was introduced and employed by Lasiecka and Tataru ^[28]. It was shown there that the monotonicity and continuity of $g$ guarantee the existence of the function $G$ with the properties stated in (H1).

Remark 3.4. As in ^[28], we use Condition (3.5) to prove the uniqueness of the solution.

The following lemmas will be of essential use in establishing our main results.

Lemma 3.5. ^[29] Let $E:\mathbb{R}^+\rightarrow \mathbb{R}^+$ be a nonincreasing function and $\gamma :\mathbb{R}^+\rightarrow \mathbb{R}^+$ be a strictlyincreasing $C^{1}$-function, with $\gamma (t)\rightarrow +\infty$ as $t\rightarrow +\infty.$ Assume that there exists $c > 0$ such that

Then there exist positive constants $k$ and $\omega$ such that

Lemma 3.6. ^[30] Let $E:\mathbb{R}^+\rightarrow \mathbb{R}^+$ be a differentiable and nonincreasing function and $\chi: \mathbb{R}^+\rightarrow \mathbb{R}^+$ be a convex and increasing function such that $\chi(0) = 0$. Assume that

Then, $E$ satisfies the following estimate

where $\psi(t) = \int_{t}^{1}\frac{1}{\chi(s)}ds$, and

4.   Local existence

In this section, we state and prove the local existence of weak solutions of problem (1.1). Similar results can be found in ^[31,32]. To this end, we consider the following problem

where $f \in L^2(\Omega \times (0, T))$ and $(u_0, u_1)\in H^2_{*}(\Omega)\times L^2(\Omega)$. Then, we prove the following theorem:

Theorem 4.1. Let $(u_0, u_1)\in H^2_{*}(\Omega)\times L^2(\Omega)$. Assume that(H1) and (H2) hold. Then, problem (4.1) has a unique local weaksolution

where $\mathcal{H}(\Omega)$ is the dual space of $H^2_{*}(\Omega)$.

Proof. Uniqueness: Suppose that (4.1) has two weak solutions $(u, v)$. Then, $w = u-v$ satisfies

Multiplying (4.2) by $w_t$ and integrating over $(0, t),$ we get

Integrating (4.3) over $(0, t),$ we obtain

Using Condition (3.5) and (H2), for $a.e. \; x\in\Omega$, we have

We conclude $u = v = 0$ on $\Omega \times(0, T),$ which proves the uniqueness of the solution of problem (4.1). Existence: To prove the existence of the solution for problem (4.1), we use the Faedo-Galerkin method as follows: First, we consider $\left\lbrace v_{j} \right\rbrace _{j = 1}^{\infty}$ an orthonormal basis of $H^2_{*}(\Omega)$ and define, for all $k \geq 1,$ a sequence $v^{k}$ in $\mathcal{V}_{k} = span \left\lbrace v_{1}, v_{2}, ..., v_{k} \right\rbrace \subset H^2_{*}(\Omega),$ given by

for all $x\in \Omega$ and $t\in (0, T)$ and satisfies the following approximate problem

for all $j = 1, 2, ..., k$,

such that

For any $k\geq 1,$ problem (4.6) generates a system of $k$ nonlinear ordinary differential equations. The ODE's standard existence theory assures the existence of a unique local solution $u^{k}$ for problem (4.6) on $[0, T_{k}),$ with $0 < T_{k} \leq T.$ Next, we have to show, by a priori estimates, that $T_{k} = T, \forall k\geq 1.$ Now, multiplying (4.6) by $a'_{j}(t)$, using Green's formula and the boundary conditions, and then summing each result over $j$ we obtain, for all $0 < t \leq T_{k},$

Then, integrating (4.9) over $(0, t)$ leads to

From the convergence (4.8), using the fact that $f\in L^{2}\left(\Omega \times \left(0, T\right) \right),$ and exploiting Young's inequality, then (4.10) becomes, for some $C > 0,$ and for any $t \in [0, t_k)$

Therefore, we obtain

Choosing $\varepsilon = \frac{1}{4},$ estimate (4.12) yields, for all $T_{k}\leq T$ and $C > 0$,

Consequently, the solution $u^{k}$ can be extended to $(0, T)$, for any $k\geq 1.$ In addition, we have

Therefore, we can extract a subsequence, denoted by $(u^{\ell})$ such that, when $\ell \rightarrow \infty,$ we have

Next, we prove that $g(u^\ell_t)$ is bounded in $L^2\left((0, T);L^{2}(\Omega)\right)$. For this purpose, we consider two cases:

Case 1. $G$ is linear on $[0, \varepsilon]$. Then using (H1) and Young's inequality, we get

for a suitable choice of $\delta_0$ and using the fact that $u^\ell_t$ is bounded in $L^{2}((0, T), L^{2}(\Omega))$, we obtain

Case 2. $G$ is nonlinear. Let $0 < \varepsilon_1 \le \varepsilon$ such that

Then, one can show that

Define the following sets

Then, using (4.17) and (4.18) leads for some $c^{\prime}_{2} > 0$,

Let

and

Using (4.19) and Jensen's inequality, we obtain

Using the convexity of $G$ ($G^\prime$ is increasing), we obtain for $t \in (0, T)$,

Let $G^{*}$ be the convex conjugate of $G$ in the sense of Young (see ^[33], pp. 61–64), then, for $s\in(0, G^{\prime}(\varepsilon)],$

Using the general Young inequality

for

and using the fact that $E^{\ell}(t)\le E^{\ell}(0)$, we get

Integrating (4.24) over $(0, T)$, we obtain

Using (4.21) and the fact that $u^\ell_t$ is bounded in $L^{2}\left((0, T);L^{2}(\Omega)\right)$, we conclude that $g(u^\ell_t)$ is bounded in $L^{2}\left((0, T);L^{2}(\Omega)\right)$. So, we find, up to a subsequence, that

Now, we have to show that $\chi = g(u_t)$. In (4.6), we use $u^{\ell}$ instead of $u^{k}$ and then integrate over $(0, t)$ to get

As $\ell \rightarrow + \infty$, we easily check that

Hence, for $v \in H^2_*(\Omega),$ we have

Since all terms define absolute continuous functions, we get, for $a.e. \; t\in[0, T]$ and for $v \in H^2_*(\Omega),$ the following

This implies that

Using (H1), we see that

So, by using (4.6) and replacing $u^{k}$ by $u^{\ell}$, we get

Taking $\ell \rightarrow +\infty$, we obtain

Replacing $v$ by $u_t$ in (4.30) and integrating over $(0, T)$, we obtain

Adding of (4.34) and (4.35), we get

This gives

Hence,

Let $v = \lambda w + u_t$, where $\lambda > 0$ and $w \in L^2\left(\Omega \times (0, T)\right)$. Then, we get

For $\lambda > 0$, we have

As $\lambda \rightarrow 0$ and using the continuity of $g$ with respect of $\lambda$, we get

Similarly, for $\lambda < 0$, we get

This implies that $\chi = g(u_t)$. Hence, (4.30) becomes

which gives

To handle the initial conditions of problem (4.1), we first note that

Thus, using Lion's Lemma and (4.6), we easily obtain $u^{\ell} \rightarrow u \in C\left([0, T];L^2(\Omega)\right).$ Therefore, $u^{\ell}(x, 0)$ makes sense and $u^{\ell}(x, 0)\rightarrow u(x, 0) \in L^2(\Omega).$ Also, we see that

Hence, $u(x, 0) = u_0(x)$. As in ^[34], let $\phi\in C^{\infty}_{0}(0, T),$ and replacing $u^{k}$ by $u^{\ell}$, we obtain from (4.6) and for any $j \leq \ell$

As $\ell\to +\infty$, we have for any $\phi\in C^{\infty}_{0}((0, T)),$

for all $j\geq1.$ This implies that

for all $v \in H^2_*(\Omega)$. This means that $u_{tt}\in L^{\infty}((0, T);\mathcal{H}(\Omega))$ and $u$ solves the equation

Thus

Consequently, $u_t\in C((0, T);\mathcal{H}(\Omega)).$ So, $u^{\ell}_{t}(x, 0)$ makes sense and follows that

and since

then

This ends the proof of Theorem 4.1.

Now, we proceed to establish the local existence result for problem (1.1).

Theorem 4.2. Let $(u_{0}, u_{1}) \in H^2_*(\Omega)\times L^{2}(\Omega)$ begiven. Then problem (1.1) has a unique local weak solution

Remark 4.3. In this remark, we point out four cases regarding the solution of problem (1.1):

1) If $\beta = 0$, $g$ is linear and $(u_0, u_1)\in \left(H^4(\Omega)\cap H^2_{*}(\Omega)\right)\times H^2_{*}(\Omega)$, then problem (1.1) has a unique classical solution

2) If $\beta = 0$, $g$ is linear and $(u_0, u_1)\in H^2_{*}(\Omega)\times L^2(\Omega)$, then problem (1.1) has a unique weak solution

3) If $\beta > 0$ or $g$ is nonlinear and $(u_0, u_1)\in H^2_{*}(\Omega)\times L^2(\Omega)$, then problem (1.1) has a unique weak solution

4) If $\beta > 0$ or $g$ is nonlinear and $(u_0, u_1)\in \left(H^4(\Omega)\cap H^2_{*}(\Omega)\right)\times H^2_{*}(\Omega)$, then problem (1.1) has a unique strong solution

Proof. To prove Theorem 4.2, we first let $v \in L^{\infty}\left([0, T), H^2_{*}(\Omega)\right)$ and $\widetilde{f}(v) = \vert v \vert^\beta v.$ Then, by the embedding Lemma 3.1, we have

Hence,

Therefore, for each $v \in L^{\infty}([0, T), H^2_{*}(\Omega)),$ there exists a unique solution

satisfying the following nonlinear problem

Now, let

and define the map $K : W_{T} \longrightarrow W_{T}$ by $K(v) = u.$ We note that $W_{T}$ is a Banach space with respect to the following norm

Multiply (4.51) by $u_t$ and integrate over $\Omega \times (0, t)$, we get for all $t \leq T,$

Using Young's inequality and the embedding Lemma 3.1, we have

Thus, (4.52) becomes

where $\lambda_0 = \frac{1}{2} \vert \vert u_1\vert \vert_2^2+\frac{1}{2} \vert \vert \Delta u_0\vert \vert_2^2$ and $C_e$ is the embedding constant. Choosing $\varepsilon$ such that $\frac{\varepsilon T}{2} = \frac{1}{4},$ we get

Suppose that $\vert \vert v \vert\vert_{W_{T}} \leq M$ and for $M^2 > \lambda$ and $T \leq T_0 < \frac{M^2-\lambda}{bM^{2(\beta+1)}}$, we conclude that

Therefore, we deduce that $K:B \longrightarrow B,$ where

Next, we prove, for $T_0$(even smaller), $K$ is a contraction. For this purpose, let $u_1 = K(v_1)$ and $u_2 = K(v_2)$ and set $u = u_1-u_2,$ then $u$ satisfies the following

Multiplying (4.55) by $u_t$ and integrating over $\Omega\times (0, t)$ we get, for all $t \leq T,$

Using (3.5) and (H2), we have

Now, we evaluate

where $v = v_1-v_2$, $\xi = \tau v_1 +(1-\tau)v_2$, $0\leq \tau \leq1$, and $\widetilde{f}'(\xi) = (\beta+1) \vert \xi \vert^\beta$.

Young's inequality implies

Using the embedding Lemma 3.1, we arrive at

Therefore, (4.57) takes the form

Choosing $\delta$ sufficiently small, we see that

Taking $T_0$ small enough so that,

Thus, $K$ is a contraction. The Banach fixed point theorem implies the existence of a unique $u \in B$ satisfying $K(u) = u.$ Thus, $u$ is a local solution of (1.1).

Uniqueness: Suppose that problem (1.1) has two weak solutions $(u, v)$. Taking, $w = u-v,$ that satisfies the following equation, for all $t\in \left(0, T\right),$

Multiplying (4.64) by $w_t$ and integrating over $\Omega \times (0, t)$, we obtain

Using (3.5) and (H2) implies that

By repeating the same above estimates, we obtain

This gives $w \equiv 0.$ The proof of the uniqueness is completed.

5.   Global existence

In this section, we prove that problem (1.1) has a global solution. For this purpose, we introduce the following functionals. The energy functional associated with problem (1.1) is

Direct differentiation of (5.1), using (1.1), leads to

and

Clearly, we have

Lemma 5.1. Suppose that (H1) and (H2) hold and $(u_0, u_1)\in H^2_{*}(\Omega)\times L^2(\Omega)$, such that

then $I(u(t)) > 0, \; \forall t > 0$.

Proof. Since $I(u_0) > 0$, then there exists (by continuity) $T_m < T$ such that $I(u(t)\ge 0$, $\forall t\in [0, T_m]$; which gives

By using (5.2), (5.5) and (5.7), we have

The embedding theorem, (5.6) and (5.8) give, $\forall t\in [0, T_m],$

Therefore,

By repeating this procedure, and using the fact that

$T_m$ is extended to $T.$

Remark 5.2. The restriction (5.6) on the initial data will guarantee the nonnegativeness of $E(t).$

Proposition 5.3. Suppose that (H1) and (H2) hold. Let $(u_0, u_1)\in {H^2_*(\Omega)}\times L^2(\Omega)$ be given, satisfying (5.6). Thenthe solution of (1.1) is global and bounded.

Proof. It suffices to show that $\| u\|_{H^2_*(\Omega)}^{2}+\|u_t\|_2^2$ is bounded independently of $t$. To achieve this, we use (5.2), (5.4) and (5.5) to get

since $I(t)$ is positive. Therefore

where $C$ is a positive constant, which depends only on $\beta$.

6.   Stability result

In this section, we state and prove our stability result. For this purpose, we establish some lemmas.

Lemma 6.1. (Case: $G$ is linear) Let $u$ be the solution of (1.1). Then, for $T > S \geq 0$, the energy functionalsatisfies

Proof. We multiply (1.1) by $\alpha u$ and integrate over $\Omega \times (S, T)$ to get

Adding and subtracting the following terms

to (6.2), and recalling (5.9), we arrive at

Integrating the first term of (6.3) by parts and using (5.1), then (6.3) becomes

Now, we estimate the terms in the right-hand side of (6.4) as follows:

$1)$ Estimate for $-\left[ \alpha \int_{\Omega }uu_{t}dx \right] _{S}^{T}.$

Using Lemma 3.1 and Young's inequality, we obtain

which implies that

$2)$ Estimate for $\int_{S}^{T}\alpha ^{\prime }\int_{\Omega }uu_{t}dxdt$.

The use of (6.5) and (H2) leads to

$3)$ Estimate for $\int_{S}^{T}\alpha \left(\int_{\Omega}u_t^2dx\right)dt.$

Using (H1), (5.2) and recalling that $G$ is linear, we have

$4)$ Estimate for $-\int_{S}^{T}\alpha ^{2}(t) \int_{\Omega }ug(u_t)dxdt.$

Using (H1), Lemma 3.1, Holder's inequality and recalling $G$ is linear, we obtain

Applying Young's inequality to $E^{\frac{1}{2}}(t) (-E'(t))^{\frac{1}{2}}$ with $p = 2$ and $p^* = 2$, to get

which implies that

Combining the above estimates and taking $\varepsilon$ small enough, we get (6.1).

Lemma 6.2. (Case: $G$ is nonlinear) Let $u$ be the solution of (1.1). Then, for $T > S \geq 0$, the energy functionalsatisfies

where $\tilde{\phi}:\mathbb{R}^+ \to \mathbb{R}^+$ is any convex, increasing and of class $C^1[0, \infty)$ function such that $\tilde{\phi}(0) = 0$.

Proof. We multiply (1.1) by $\alpha(t) \frac{\tilde{\phi}(E)}{E}u$ and integrate over $\Omega \times (S, T)$ to get

Adding and subtracting to (6.13) the following terms

we arrive at

Using (5.9), it is easy to deduce that $-\int_{S}^{T}\alpha \frac{\tilde{\phi}(E)}{E} \left(\gamma \| u\|_{H_*^2(\Omega)}^{2}-\|u\|_{\beta+2}^{\beta+2}\right)dt\le 0.$

Integrating by parts in the first term, in the right-hand side of (6.14), we get

Using Cauchy Schwarz' inequality, Lemmas 3.1 and 5.1, we obtain

Using (6.16), the properties of $\alpha(t)$ and the fact that the function $s\to \frac{\tilde{\phi}(s)}{s}$ is non-decreasing and $E$ is non-increasing, we have

Similarly, we get

A combination of (6.15)–(6.18) leads to (6.12).

In order to finalize the proof of our result, we let

where $\varepsilon_0 > 0$ is small enough and $G^*$ and $G_1^*$ denote the dual functions of the convex functions $G$ and $G_1$ respectively in the sense of Young (see, Arnold ^[33], pp. 64).

Lemma 6.3. Suppose $G$ is nonlinear, then the following estimates

and

hold, where $\tilde{\phi}$ is defined earlier in Lemma 6.2.

Proof. Since $G^*$ and $G_1^*$ are the dual functions of the convex functions $G$ and $G_1$ respectively, then

and

Using (6.21) and the definition of $\tilde{\phi}$, we obtain (6.19). For the proof of (6.20), we use (6.22) and the definitions of $G_1$ and $\tilde{\phi}$ to obtain

Now, we state and prove our main decay results.

Theorem 6.4. Let $(u_0, u_1)\in H^2_{*}(\Omega)\times L^2(\Omega)$. Assume that (H1) and (H2) hold. Then there exist positive constants $k$ and $c$ such that, for $t$ large, the solution of (1.1) satisfies

where

and

Proof. To establish (6.24), we use (6.1) and Lemma 3.5 for $\gamma(t) = \int_{0}^{t}\alpha(s)ds$. Consequently the result follows. For the proof of (6.25), we re-estimate the terms of (6.12) as follows: we consider the following partition of $\Omega$:

So,

Using the definition of $\Omega_1$, (3.4) and (5.2), we have

After applying Hölder's and Young's inequalities and Lemma 3.1, we obtain for some $\varepsilon > 0$,

The definition of $\Omega_1$, (3.4), (5.1), (5.2) and (6.27) lead to

Using the definition of $\tilde{\phi}$ and the convexity of $G$, then (6.28) becomes

Combining (6.12), (6.26) and (6.29) and choosing $\varepsilon$ small enough, we obtain

Using Young's inequality and Jensen's inequality (Eq 3.3), (Eq 3.4) and (Eq 5.1), we get

Applying the generalized Young inequality

to the first term of (6.31), with $A = \frac{\tilde{\phi}(E)}{E}$ and $B = G^{-1}\left(\frac{1}{\vert \Omega \vert}\int_{\Omega}u_tg(u_t)dx\right)$, we easily see that

Then we apply it to the second term of (6.31), with $A = \frac{\tilde{\phi}(E)}{E} \sqrt{E}$ and $B = \sqrt{\vert \Omega\vert G^{-1}\left(\frac{1}{\vert \Omega \vert}\int_{\Omega}u_tg(u_t)dx\right)}$ to obtain

Combining (6.31)–(6.33) and using (6.19) and (6.20), we arrive at

Using the definition of $\tilde{\phi}$ and the fact that $s \to (G^{\prime})^{-1}(s)$ is non-decreasing, we deduce that, for $0 < \varepsilon_0\le \frac{1}{2}$,

Combining (6.34) and (6.35) leads to

Then, choosing $\varepsilon_0$ small enough, we deduce from (6.30) and (6.36) that

Using the facts that $E$ is non-increasing and $s \to \frac{\tilde{\phi}(s)}{s}$ is non-decreasing, we obtain

Let $\tilde{E} = E\circ \tilde{\alpha}^{-1}$, where $\tilde{\alpha}(t) = \int_{0}^{t}\alpha(s)ds$. Then we deduce from (6.37) that

Using Lemma 3.6 for $\tilde{E}$ and $\chi(s) = \frac{1}{c} \tilde{\phi}(s)$, we deduce from (3.6) the following estimate

which gives (6.25), by using the definition of $\tilde{E}$ and the change of variables.

Remark 6.5. The stability result (6.25) is a decay result. Indeed,

Hence, $\lim_{t\to\infty}h^{-1}(t) = \infty$, which implies that $\lim_{t \to\infty}h(t) = \infty$. Using the convexity of $G$, we have

Therefore, $\lim_{t\to 0^+}\psi(t) = \infty$ which leads to $\lim_{t\to \infty}\psi^{-1}(t) = 0$.

Examples

$1)$ Let $g(s) = s^m$, where $m\ge 1$. Then the function $G$ is defined in the neighborhood of zero by

which gives, near zero

So, we obtain

and then, in the neighborhood of $\infty$

Using the fact that $h(t) = t$ as $t$ goes to infinity, we obtain from (6.24) and (6.25)

$2)$ Let $g(s) = s^m \sqrt{-\ln{s}}$, where $m\ge 1$. Then the function $G$ is defined in the neighborhood of zero by

which gives, near zero

Therefore, we get

and then, in the neighborhood of $\infty$, we have

Using the fact that $h(t) = t$ as $t$ goes to infinity, we obtain

Acknowledgments

The authors would like to express their profound gratitude to King Fahd University of Petroleum and Minerals (KFUPM) for its continuous support. The authors also thank the referees for their very careful reading and valuable comments. This work was funded by KFUPM under Project #SB201003.

Conflict of interest

The authors declare there is no conflicts of interest.