The effect of time-varying delay damping on the stability of porous elastic system

1. Introduction

Problems in porous media are essential in the field of petroleum engineering, soil mechanics, power technology, biology, material science, etc. Thus, this has attracted the attention of scientists and mathematicians in particular, see for instance the results in [1,2,3,4,5,6,7,8] and the references cited therein for related theory of porous-elastic materials. The basic equations describing the motion of a classical porous system are given by where \(\varphi = \varphi(x,t)\) and \(\psi = \psi(x,t)\) are the displacements of solid elastic material and the volume fraction, respectively. The physical parameters \(\rho\) and \(J\) are respectively, mass density and product of the equilibrated inertia by the mass density. The constitutive laws \(S, G\) and \(Q\) are: stress tensor, equilibrated stress vector and equilibrated body force, respectively. Time delays occur in systems modeling different types of phenomena in areas such as: biosciences, medicine, physics, chemical and structural engineering. These phenomena depend naturally on the present state and past history of the system. It is a well known fact that the presence of a delay term in a system, which is a priori a stable system, might cause an instability in the system, see for instance, the result of Nicaise and Pignotti [9]. In past decades, a great number of researchers have investigated the effect of delay on the stability of various systems or wave equations (with or without memory), see for example [10,11,12,13,14,15] and references therein. Back to system (1), we should mention that, there are very few results in literature that studied the effect of delay on this system. With memory and time varying delay dampings, the constitutive laws in (1) are given by where the constitutive physical parameters, \(k,b,\delta,a\) satisfies \(\mu_1, \mu_2\) are real constants, \(\tau(t)>0\) is the time-dependent delay and \(g\) is a given function to be specified later. For simplicity, we set \(L=1\), then substituting (2) into (1), we arrive at the following porous-viscoelastic system with varying time dependent delay; When \(g = \mu_1 =\mu_2 =0,\) Quintanilla [16] investigated \begin{equation} \nonumber \begin{cases} \rho \varphi_{tt} -k\varphi_{xx}-b \psi_x=0,& \text{ in}\ (0,1)\times (0,+\infty), \\ J\psi_{tt} -\delta \psi_{xx} + b\varphi_x + a \psi-\gamma \psi_t =0,&\text{ in}\ (0,1)\times (0,+\infty), \end{cases} \end{equation} where \(\mu_1=\gamma>0\) and showed the lack of exponential stability. However, he established a slow non-exponential decay result. Casas and Quintanilla [17] studied and improved the result in [16] (exponential stability). Soufyane et al., [18] considered (5) with viscoelastic damping on the boundaries and proved a general decay estimate. Recently, Apalara [19] looked at \begin{equation} \nonumber \begin{cases} \rho \varphi_{tt} -k\varphi_{xx}-b \psi_x=0,& \text{ in}\ (0,1)\times (0,+\infty), \\ J\psi_{tt} -\delta \psi_{xx} + b\varphi_x + a \psi +\int_0^t g(t-s)\psi_{xx}(x,s)ds=0, &\text{ in}\ (0,1)\times (0,+\infty), \end{cases} \end{equation} where the memory \(g\) satisfies \(g'(t)\leq-\xi(t)g(t)\) and established a general decay estimate. Feng and Apalara [20] improved the result in [19] when the relaxation function \(g\) satisfies \(g'(t)\leq-\xi(t)H(g(t)).\)

For results in porous systems with delay damping, not much has been done in this direction. we refer the reader to the result of Khochemane et al., [21], where they considered a porous elastic system with weak internal damping and constant delay damping. Precisely, they studied

and proved a general decay result provided \(g\) satisfies \begin{eqnarray*} \begin{cases} c_1|s|\leq |g(s)|\leq c_2|s|,& \text{if}\ |s|\geq\epsilon,\\ s^2 + g^2(s)\leq G^{-1}(sg(s)),& \text{if} \ |s|\leq \epsilon. \end{cases} \end{eqnarray*} Recently, Borges Filho and Santos [22] considered \begin{equation*} \begin{cases} \rho \varphi_{tt} -k\varphi_{xx}-b \psi_x=0,& \text{ in}\ (0,1)\times (0,+\infty), \\ J\psi_{tt} -\delta \psi_{xx} + b\varphi_x + a \psi + \mu_1\psi_t +\mu_2\psi_t(.,t-\tau(t)) =0,& \text{ in}\ (0,1)\times (0,+\infty), \end{cases} \end{equation*} and showed that the system is exponentially stable. More related results can be found in [23,24,25,26,27] and references therein.

The novelty of this work is to study the stability of system (7). In fact, we show that the solution energy has an optimal decay estimate even in the presence of time varying delay term, from which the results in [21,22] are particular cases. To the best of our knowledge, system (7) has not been considered before in the literature. The rest of work is organized as follows: In Section 2, we recall some preliminaries and assumptions on the memory term. In Section 3, we state and prove several lemmas needed for establishing our main results. In Section 4, we establish the uniform stability result and in Section 5, we give some examples in support of our results.

2. Problem setting and assumptions

In this work, we consider the following system: In addition to (3), we need the following:

Assumptions

• (A1)   The relaxation function \(g: [0,+\infty)\rightarrow (0,+\infty)\) is a \({\cal C}^1\) non-increasing function and satisfies
  \begin{equation} \label{ass1} g(0)>0, \hspace{2cm} \delta-\displaystyle\int_0^{+\infty} g(s)ds=l>0. \end{equation}

  (8)
• (A2)   There exists a \({\cal C}^1-\)function \(M: [0,+\infty)\rightarrow [0,+\infty)\), which is either linear or is a strictly increasing and strictly convex \({\cal C}^2\) function on \([0,\alpha],\ \alpha>0,\) \(\alpha\leq g(0)\), with \(M(0)= M'(0)=0\), such that
  \begin{equation} \label{ass2} g'(t)\leq -\xi(t)M(g(t)), \hspace{0.5cm} \forall\ t\geq 0, \end{equation}

  (9)
  where \(\xi\) is a positive non-increasing differentiable function.
• (A3)   There exist \(\tau_0,\tau_1>0\) such that
  \begin{equation} \label{ass3} 0< \tau_0\leq \tau(t)\leq \tau_1, \ \forall\ t>0. \end{equation}

  (10)
• (A4)
  \begin{equation} \label{ass4} \tau(t)\in W^{2,+\infty}(0,T) \ \text{ and}\ \tau'(t)\leq d< 1, \ \forall \ t,T>0. \end{equation}

  (11)
• (A5)   The real constants \(\mu_1\) and \(\mu_2\) satisfies \(|\mu_2|< \mu_1\sqrt{1-d}.\)
From \((A1)\) and \((A2)\), we can deduce the following:
• (I)   It follows from \((A1)\) that \(\displaystyle\lim_{t\rightarrow\infty}g(t)=0.\) Thus, there exists \(t_0\geq 0\) large enough, such that
  \begin{equation} \label{imp1} g(t_0)=\alpha \hspace{0.3cm} \text{ and} \hspace{0.3cm} g(t)\leq \alpha, \hspace{0.3cm} \forall\ t\geq t_0. \end{equation}

  (12)
• (II)   Since \(g\) and \(\xi\) are positive, non-increasing and continuous functions, in addition to \(M\) being a positive continuous function, it follows that, for all \(t\in[0,t_0]\), \begin{equation*} \left. \begin{aligned} & 0< g(t_0)\leq g(t)\leq g(0)\\ & 0< \xi(t_0)\leq \xi(t)\leq \xi(0) \end{aligned} \right\} \Rightarrow \beta_1\leq\xi(t)M(g(t))\leq \beta_2 \end{equation*} for some positive constants \(\beta_1\) and \(\beta_2\). Hence,
  \begin{equation} \label{imp2} g'(t)\leq -\xi(t)M(g(t))\leq -\frac{\beta_1}{g(0)}g(0)\leq -\frac{\beta_1}{g(0)}g(t), \ \forall\ t\in [0,t_0]. \end{equation}

  (13)
• (III)   \(M\) has an extension \(\overline M\), which is a strictly increasing and strictly convex \({\cal C}^2\) function on \((0,\infty)\). As an example, given that \(M(\alpha)=a_1\), \(M'(\alpha)=a_2\) and \(M''(\alpha)=a_3\), then we can define \(\overline M\) by
  \begin{equation} \label{imp3} \overline M(t)= \frac{a_3}{2}t^2+(a_2-a_3\alpha)t+\bigg( a_1+\frac{a_3}{2}\alpha^2-a_2\alpha\bigg), \ \forall\ t>\alpha. \end{equation}

  (14)

From now on, \(C\) denotes a positive constant that may change within lines or from line to line. We denote by \(\|.\|_2\) the usual norm in \(L^2(0,1)\) and define the following spaces: \begin{equation} \nonumber L^2_{*}(0,1)=\left\lbrace w\in L^2(0,1): \int_0^1w(x)dx=0\right\rbrace ,\ H^1_{*}(0,1)= H^1(0,1)\cap L^2_{*}(0,1), \end{equation} and \begin{equation} \nonumber H^2_{*}(0,1)=H^2(0,1)\cap H^1_{*}(0,1). \end{equation} Let \( W=(\varphi,\varphi_t,\psi,\psi_t), \ W_0 =(\varphi_0,\varphi_1,\psi_0,\psi_1) \) and set \(\ \ \ \ \mathcal{H}= H^1_{*}(0,1)\times L^2_{*}(0,1)\times H^1_{0}(0,1)\times L^2(0,1), \) \( \ \ \mathcal{H}_1=H^2_{*}(0,1)\times H^1_{*}(0,1)\times H^2(0,1)\cap H^1_{0}(0,1)\times H^1_0(0,1). \) We have the following well-possedness result, which is obtained by using the Classical Faedo-Galerkin method.

Theorem 1. Suppose assumptions \((A1)-(A5)\) hold. Let \(W_0\in \mathcal{H}\) and \(f_0\in H^1((0,1)\times (-\tau(0),0)),\) then (7) possesses a unique weak solution \(W\in \mathcal{C}([0,+\infty),\mathcal{H}).\) Moreover, if \(W_0\in \mathcal{H}_1\) and \(f_0\in H^2((0,1)\times (-\tau(0),0))\), then the solution is more regular in the class \( W\in \mathcal{C}([0,+\infty),\mathcal{H}_1)\cap \mathcal{C}^1([0,+\infty),\mathcal{H}).\)

We recall the following useful lemmas that will be applied repeatedly throughout this article.

Lemma 1. Let \(w\in L^2_{loc}([0,+\infty),L^2(0,1))\), we have

and where \((g\circ w)(t)=\int_0^t g(t-s)\|w(t)-w(s)\|_2^2 ds.\)

Lemma 2. Let \( w\in H_{0}^1(0,1)\), then

where \( C_p >0\) is the Poincaré's constant and \((g\circ w)(t)=\int_0^t g(t-s)\|w(t)-w(s)\|_2^2 ds.\)

Lemma 3. Let \( (\varphi,\psi)\) be the solution of (7). Then, for any \(0< \alpha< 1\), we have

where \( h(t)=\alpha g(t)-g'(t)\ \ \ \text{ and }\ \ \ A_{\alpha}=\int_0^{+\infty}\frac{g^2(s)}{\alpha g(s)-g'(s)}ds. \)

Proof. Cauchy-Schwarz inequality gives

Lemma 4.(Jensen's inequality) Given that \(G\) is a convex function on \([a,b]\), \(f:\, \Omega\rightarrow\,[m,n]\) and \(h\) are integrable functions on \(\Omega\), \(q(x)\geq 0\), and \(\int_{\Omega}q(x)dx=\varrho>0\), then \begin{equation*} G\left[\frac{1}{\varrho}\int_{\Omega}f(x)q(x)dx \right]\leq \frac{1}{\varrho}\int_{\Omega}G[f(x)]q(x)dx. \end{equation*}

3. Strategic lemmas

For convenience, we will denote the norm \(\|\cdot\|_{L^2(0,1)}\) and the inner product \( \langle .,. \rangle_{L^2(0,1)}\) of the Lebesgue space \(L^2(0,1)\) by \(\|\cdot\|\) and \( \langle .,. \rangle\) respectively. The constants \(c>0\) and \(C>0\) are generic constants which may change in value from one line to the other or within the same line. We define the energy functional of problem (7) as where \(\zeta>0\) is a constant to be specified later, see [28] and \(\lambda>0\) satisfies and \((g\circ \psi_x)(t)=\int_0^t g(t-s)\|\psi_x(t)-\psi_x(s)\|^2ds.\)

Lemma 5. Assume the conditions \((A1)-(A5)\) hold. Then, the energy functional (20) satisfies

Proof. Differentiation of (20) gives

Also, multiplying \((7)_1\) by \(\varphi_t\), \((7)_2\) by \(\psi_t\), integrating over \((0,1)\) and adding the two equations, we get Young's inequality yields Substituting (24) into (23) and taking into account (25), assumptions (A3) and (A4), we get From condition (21), we can select \(\zeta>0\) so that Hence, (22) follows from (26) by virtue of \((A1)-(A2)\) and (27). This completes the proof.

Lemma 6. Let \(t_0>0\). Then, the functional \( F_1(t)=-J\int_0^1\psi_t \int_0^t g(t-s)\left(\psi(t)-\psi(s)\right)dsdx\) along the solution of \((7)\) for any \( \epsilon_1,\epsilon_2>0\) and \(0< \alpha< 1\) satisfies the estimate

Proof. Differentiating \(F_1\), using \((7)_2\) and integration by parts, we get

Using Cauchy-Schwarz, Young's and Poincaré's inequalities, Lemmas 1-3 and similar computations as in (19), we estimate \(I_1-I_7\) as follows: Substituting the estimates in (30) into (29), we arrive at From \((A1)\), we have that \(g(0)>0\) and \(g\) is continuous. Therefore, for \(t\geq t_0>0,\) we obtain Thus, we select \(\sigma_1=\dfrac{Jg_0}{2}\) to get (28). This completes the proof.

Lemma 7. Let \((\varphi,\psi)\) be the solution of Problem \((7)\). Then, the functional \( F_2(t)=-\rho\int_0^1 \varphi_t\varphi dx\) satisfies the estimate

Proof. Differentiation of \(F_2\), using \((7)_1\) and integration by parts, we obtain \begin{equation} \nonumber F_2'(t)=-\rho\|\varphi_t\|^2 +k \|\varphi_x\|^2+ b\int_0^1 \varphi_x\psi dx. \end{equation} Applying Young's and Poincaré's inequalities, we obtain (33). This completes the proof.

Lemma 8. The functional \( F_3(t)=J\int_0^1\psi_t\psi dx+\frac{b\rho}{k}\int_0^1\psi\int_0^x \varphi_t (y)dy dx\) along the solution of problem \((7)\) for any \(\epsilon_4>0\) and \(0< \alpha< 1\) satisfies the estimate

Proof. Differentiation of \(F_3\), using \((7)\) and integration by parts leads to

Using Cauchy-Schwarz, Young's and Poincaré's inequalities together with Lemmas 1-3, we have and Substituting (36)-(39) into (35), we arrive at We choose \(\sigma_2=\dfrac{l}{2}\) to obtain (34). This completes the proof.

Lemma 9. Assume \(\frac{k}{\rho}=\frac{\delta}{J}\). Then, the functional \[ F_4(t)=\frac{|b|\delta\rho}{bk}\int_0^1 \varphi_t\psi_xdx+\frac{|b|J}{b}\int_0^1 \psi_t\varphi_xdx -\frac{|b|\rho}{bk}\int_0^1 \varphi_t\int_0^tg(t-s)\psi_x(s)dsdx,\] along the solution of \((7),\) for any \(\epsilon_5>0\) and \(0< \alpha< 1\) satisfies the estimate

Proof. Differentiating \(F_4\), we get \begin{equation} \nonumber \begin{aligned} F'_4(t)=&\frac{|b|\delta\rho}{bk}\int_0^1 \varphi_{tt}\psi_xdx + \frac{|b|\delta\rho}{bk}\int_0^1 \varphi_t\psi_{xt}dx+ \frac{|b|J}{b}\int_0^1 \psi_{tt}\varphi_xdx\\ &+\frac{|b|J}{b}\int_0^1 \psi_t\varphi_{xt}dx-\frac{|b|\rho}{bk}\int_0^1 \varphi_{tt}\int_0^tg(t-s)\psi_x(s)dsdx\\ & -\frac{|b|\rho}{bk}\int_0^1 \varphi_t\int_0^tg'(t-s)\psi_x(s)dsdx -\frac{|b|\rho}{bk}g(0)\int_0^1 \varphi_t\psi_xdx. \end{aligned} \end{equation} Using these equations in (7) and integration by parts, we arrive at \begin{equation} \nonumber \begin{aligned} F'_4(t)=&\frac{|b|\delta\rho}{k}\|\psi_x\|^2 + \frac{|b|}{b}\left(\frac{\delta\rho}{k}-J \right)\int_0^1 \varphi_{x}\psi_{xt}dx -|b|\|\varphi_x\|^2\\ &-\frac{|b|\mu_1}{b}\int_0^1 \psi_t\varphi_{x}dx -\frac{|b|\mu_2}{b}\int_0^1 \psi_t(t-\tau(t))\varphi_{x}dx\\ & - \frac{|b|a}{b}\int_0^1\psi\varphi_xdx-\frac{|b|}{k}\int_0^1 \psi_{x}\int_0^tg(t-s)\psi_x(s)dsdx\\ & -\frac{|b|\rho}{bk}\int_0^1 \varphi_t\int_0^tg'(t-s)\psi_x(s)dsdx -\frac{|b|\rho}{bk}g(0)\int_0^1 \varphi_t\psi_xdx. \end{aligned} \end{equation} Using the fact that \(\frac{k}{\rho}=\frac{\delta}{J}\), we get

Applying Cauchy-Schwarz, Young's and Poincaré's inequalities, taking into account Lemmas 1-3, \(h=\alpha g-g'\), we have for any \(\sigma_3, \epsilon_5>0\) and Substitution of (43)-(48) into (42) yields Finally, we choose \(\sigma_3=\dfrac{|b|}{2}\) to obtain (41). This completes the proof.

Lemma 10. The functional \(F_5(t)= \int_0^t f(t-s)\|\psi_x(s)\|^2ds,\) where \(f(t)=\displaystyle \int_t^{+\infty}g(s)ds \), along the solution of \((7)\) satisfies the estimate

Proof. Differentiation of \(F_5\) and using the fact that \(f'(t)=-g(t)\) lead to \begin{equation*} \begin{aligned} F'_5(t)=&\int_0^t f'(t-s)\|\psi_x(s)\|^2ds + f(0)\|\psi_x\|^2\\ =& -(g\circ \psi_x)(t) + f(t)\|\psi_x\|^2-2\int_0^1 \psi_x\int_0^t g(t-s)\left(\psi_x(s)-\psi_x(t) \right)dsdx. \end{aligned} \end{equation*} Cauchy-Schwarz inequality and condition \((A1)\) give \begin{equation*} -2\int_0^1 \psi_x\int_0^t g(t-s)\left(\psi_x(s)-\psi_x(t) \right)dsdx \leq 2(\delta-l)\|\psi_x\|^2 +\frac{\int_0^t g(s)ds}{2(\delta-l)}(g\circ \psi_x)(t) \leq 2(\delta-l)\|\psi_x\|^2 +\frac{1}{2}(g\circ\psi_x)(t). \end{equation*} Therefore, \begin{equation*} F'_5(t)\le 2(\delta-l)\|\psi_x\|^2- \frac{1}{2}(g\circ \psi_x)(t) + f(t)\|\psi_x\|^2. \end{equation*} Since \(f'(t)=-g(t)\leq 0\), it follows that \(f(t)\leq f(0)=\delta-l\). Thus, we have \begin{equation} \nonumber F_5'(t)\leq 3(\delta-l)\|\psi_x\|^2- \frac{1}{2} (g\circ \psi_x)(t), \ \forall\ t\geq 0. \end{equation} For the next lemma, we consider the Lyapunov functional \(K\) defined by

where \(N,\ N_j,\ j=1,2,3,4\) are positive constants to be specified later.

Lemma 11. Assume \(\frac{k}{\rho}=\frac{\delta}{J}\). Then, for suitable choice of \(N,\ N_j,\ j=1,2,3,4\), the Lyapunov functional \(K\) along the solution of \((7)\) satisfies the estimate

for some \(\beta>0\) and \(K\sim E\), that is holds for some \(\ \alpha_1,\ \alpha_2>0.\)

Proof. Using (51) and recalling that \(h(t)=\alpha g(t)-g'(t)\), then Lemmas 6-10 yields, for all \(t\geq t_0,\)

where \(\gamma_1=\left( a-\frac{b^2}{k}\right)>0,\ \gamma_2=\left(\frac{\mu_1}{2}-\frac{\zeta}{2} \right)>0,\ \gamma_3=\left(\frac{\zeta}{2}e^{-\lambda\tau_1}(1-d)-\frac{\mu_2^2}{2\mu_1} \right)>0\) by virtue of (3) and (27). Next, we choose and (54) becomes \begin{align*} K'(t)\leq & -\frac{\rho}{2}\|\varphi_t\|^2-\left[\frac{N_4|b|}{4}-C\right]\|\varphi_x\|^2 -\left[N\gamma_2 + \frac{N_1Jg_0 }{2}- N_3\left(1+\frac{4N_3}{\rho} \right) -N_4C\right] \|\psi_t\|^2 \\ &-\left[\frac{N_3l}{4}-N_4C\left( 1+\frac{4N_4}{\rho}\right)-C \right]\|\psi_x\|^2-\frac{N_3\gamma_1}{2}\| \psi\|^2 \\ &-\left[N\gamma_3 -\left( N_1+N_3+N_4\right)C \right]\|\psi_t(t-\tau(t))\|^2\\ &+\frac{N\alpha}{2}\left( g\circ\psi_x\right)(t) -\frac{N\lambda\zeta}{2}\int_{t-\tau(t)}^te^{-\lambda(t-s)}\|\psi_t(s)\|^2ds \\ & -\left[\frac{N}{2}-CA_{\alpha}\left(N_1\left(1+\frac{4 N_1}{N_3l}+\frac{4 N_1}{N_4|b|}+\frac{2 N_1}{N_3\gamma_1}\right)+N_3+\frac{N_4^2}{\rho}\right)\right] \left( h\circ\psi_x\right)(t). \end{align*} Now, we choose the remaining constants: First, we select \(N_4\) so that Then, we choose \(N_3\) large enough such that Hence \(N_3\) and \(N_4\) are fixed, we choose \(N_1\) large so that We have that \(\frac{\alpha g^2(s)}{h(s)}=\frac{\alpha g^2(s)}{\alpha g(s)-g'(s)}< g(s).\) Thus, using the dominated convergence theorem, we get Thus, there exist \(0< \alpha_0< 1\) such that for all \(0< \alpha\leq \alpha_0,\) we have Finally, we choose \(N\) so large and take \(\alpha=\frac{1}{2N}\) Such that The analysis from (55)-(61) yields (52). Applying Young's, Cauchy-Schwarz, and Poincaré's inequalities, we obtain (53) easily. This completes the proof.

4. Main stability result

The main stability result of the work is the following:

Theorem 2. Assume \(\frac{k}{\rho}=\frac{\delta}{J}\) and \((A1)- (A5)\) hold. Then, there exist \(\lambda_1>0,\lambda_2>0\) such that the solution energy (20) satisfies

where \( M_1(t)=\int_t^r \frac{1}{s M'(s)}ds\) and \( M_1\) is a strictly decreasing and strictly convex function on \((0,r]\) with \( \lim _{t\rightarrow 0} M_1(t)=+\infty.\)

Proof. From (13) and (22), we have \(\forall\ t\geq t_0\)

Thus, (52) and (63) implies \begin{eqnarray*} K'(t)\leq -\eta E(t) + \frac{1}{2}(g\circ \psi_x)(t) \leq -\eta E(t)-C E'(t)+\frac{1}{2}\int_{t_0}^{t}g(s)\|\psi_x(t)-\psi_x(t-s)\|^2ds, \end{eqnarray*} for some \(\eta>0.\) Let \(K_1= K+ C E\sim E,\) by virtue of (53), it follows that To complete the proof, we divide it into two parts:

Case 1: when \(M(t)\) is linear. Multiplying (64) by \(\xi(t)\), keeping in mind (22) and \((A2),\) we get

From (A2), \(\xi\) is non-increasing, thus, we get and Since \(K\sim E,\) thus, setting \(K_2(t)=\xi(t)K_1(t)+C E(t)\), there exists \(\eta_1>0\) so such that Integration of (68) over \((t_0,t)\) and recalling (67), we obtain \begin{equation} \nonumber E(t)\leq \displaystyle \lambda_1 e^{-\lambda_2\displaystyle\int_{t_0}^t\xi(s)ds} = \lambda_1 M_1^{-1}\left( \lambda_2\int_{t_0}^t\xi(s)ds\right). \end{equation}

Case 2: when \(M(t)\) is nonlinear. In this case, we consider \(\mathcal{K}(t)=K(t)+ F_5(t).\) Then, Lemmas 10 and (52) yield for some \(d>0\)

It follows that \( d\int_{t_0}^t E(s)ds\leq \mathcal{K}(t_0)-\mathcal{K}(t)\leq \mathcal{K}(t_0), \) from which we get From (70), we can define \(p(t)\) by \( p(t):= \varpi\int_{t_0}^t\|\psi_x(t)-\psi_x(t-s)\|^2ds, \) where \(0< \varpi< 1\) so that Furthermore, we assume \(p(t)>0\) for all \(t\geq t_0\); otherwise, it follows from (64) that the solution energy is exponentially stable. Also, we define the functional \(q(t)\) by \( q(t)=-\int_{t_0}^t g'(s)\|\psi_x(t)-\psi_x(t-s)\|^2ds \) and it's easy to see that \(q(t)\leq -CE'(t),\ \forall\ t\geq t_0.\) From \((A_2),\) \(M\) is strictly convex on \((0,r]\) (where \(r=g(t_0)\)) and \(M(0)=0\), this implies Using (71),(72), assumption \((A2)\) and Jensen's inequality, we have where \(\bar{M}\) is the convex extension of \(M\) on \((0,+\infty)\), see (14). From (73), we have \begin{equation} \nonumber \int_{t_0}^t g(s)\|\psi_x(t)-\psi_x(t-s)\|^2ds\leq \frac{1}{\varpi}\bar{M}^{-1}\left(\frac{\varpi q(t)}{\xi(t)} \right). \end{equation} Thus, (64) gives Let \(r_0< r\) to be specified later and define the functional \(K_3\) by \( K_3(t):=\bar{M}'\left(r_0\frac{E(t)}{E(0)}\right)K_1(t)+ E(t)\sim E(t). \) Since \(K_1\sim E\). Using (74) and recalling that \(E'(t)\leq 0,\ \bar{M}'(t)>0,\ \bar{M}''(t)>0,\) we have for all \( t\geq t_0\) Now, we consider conjugate of \(\bar{M}\) denoted by \(\bar{M}^{\ast}\) define in the sense of Young, see Arnold [29] page 61-64, define by and \(\bar{M}^{\ast}\) satisfies the Young's inequality Setting \( X=\bar{M}'\left(r_0\frac{E(t)}{E(0)}\right) \) and \(Y=\bar{M}^{-1}\left(\varpi\frac{q(t)}{\xi(t)}\right),\) then (22) and (75)-(77) yield for all \(t\geq t_0\) Now, multiplying (78) by \(\xi(t)\) keeping in mind that \(r_0\frac{E(t)}{E(0)}< r\) and \(\bar{M}'\left(r_0\frac{E(t)}{E(0)}\right)=M'\left(r_0\frac{E(t)}{E(0)}\right),\) we arrive at We set \( K_4(t)=\xi(t) K_3(t) + CE(t)\sim E(t)\) since \(K_3\sim E\). Thus there exist \(n_0,n_1\) positive such that Therefore, estimate (79) yields \begin{equation} \nonumber K_4'(t)\leq -(\eta E(0)-Cr_0)\xi(t)\frac{E(t)}{E(0)}\xi(t)M'\left(r_0\frac{E(t)}{E(0)}\right), \ \forall\ t\geq t_0. \end{equation} We choose \(r_0< r\) small enough so that \(\eta E(0)-Cr_0>0\) to arrive at for some constant \(\eta_2>0\) and \(M_2(s)=sM'(r_0 s).\) We note that \(M_2'(s)=M'(r_0 s)+r_0 sM''(r_0s),\) hence, the strict convexity of \(M\) on \((0,r],\) yields \(M_2(s)>0,\ M_2'(s)>0\) on \((0,r].\) Let \(R(t)=n_0\frac{ K_4(t)}{E(0)}. \) Using (80) and (81), we obtain and for some \(\eta_3>0.\) Integration of (83) over \((t_0,t),\) gives from which we get Using properties of \(M\), we easily see that \(M_1\) is strictly decreasing function on \((0,r]\) and \(\displaystyle \lim _{t\longrightarrow 0} M_1(t)=+\infty.\) Therefore, (62) follows from (82) and (85). This completes the proof.

Remark 1. The stability result in Theorem 2 is general and optimal in the sense that it agrees with the decay rate of \(g\), see [30], Remark 2.3.

Corollary 1. Suppose \(\frac{k}{\rho}=\frac{\delta}{J},\) and \((A1)-(A5)\) hold. Assume the function \(M\) in \((A_2)\) be defined by \( H(s)=s^p, \ \ p\geq 1, \) then there exist \(\lambda_1,\lambda_2, C>0\) such that (20) satisfies

5. Examples

• (1).   Let \(g(t)=\nu_1e^{-\nu_2t},\ t\geq 0, \ \ \nu_1,\ \nu_2>0\) and \(\nu_2\) is chosen so that (A\(_1\)) holds. Then, \( g'(t)=- \nu_1\nu_2e^{-\nu_2t}=-\nu_2M(g(t))\) with \(M(t)=t. \) Thus, from (62), the solution energy (20) satisfies \( E(t)\leq Ce^{-\lambda t}, \ \forall \ t\geq 0,\) where \(\lambda=\nu_2\lambda_1. \)
• (2).   Let \(g(t)=ue^{-(1+t)^{v}},\ t\geq 0, \ \ u>0, \ 0< v< 1\) are constants and \(u\) is chosen such that (A\(_1)\) holds. Then, \( g'(t)=-uv(1+t)^{v-1}e^{-(1+t)^{v}}=-\xi(t) M(g(t)), \) where \(\xi(t)=v(1+t)^{v-1}\) and \(M(t)=t.\) Thus, we get from (62) that \( E(t)\leq \lambda_2e^{-\lambda_1(1+t)^v}, \ \forall \ t\geq 0. \)
• (3).   Let \(g(t)=\frac{u}{(1+t)^{v}},\ t\geq 0, \ \ u>0, \ v>1\) are constants and \(u\) is chosen such that (A\(_1)\) holds. We have \( g'(t)=\frac{-uv}{(1+t)^{v+1}}=-\xi\left(\frac{u}{(1+t)^v} \right)^{\frac{v+1}{v}}=-\xi g^p(t)=-\xi M(g(t)), \) where \( M(t)=t^p,\ \ p=\frac{v+1}{v}\) satisfying \(1< p< 2\) and \(\xi=\frac{v}{u^\frac{1}{v}}>0. \) Hence, we deduce from (86) that \( E(t)\leq \frac{C}{(1+t)^v}, \ \forall \ t\geq 0. \)

Acknowledgments

The author appreciate the continuous support of University of Hafr Al Batin.

Conflicts of Interest

The author declares no conflict of interest.