New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory

1.   Introduction

The Moore-Gibson-Thompson (MGT) equation is one of the equations of nonlinear acoustics describing acoustic wave propagation in gases and liquids [13,15,30] and arising from modeling high frequency ultrasound waves [9,18] accounting for viscosity and heat conductivity as well as effect of the radiation of heat on the propagation of sound. This research field is highly active due to a wide range of applications such as the medical and industrial use of high intensity ultrasound in lithotripsy, thermotherapy, ultraound cleaning, etc. The classical nonlinear acoustics models include the Kuznetson's equation, the Westervelt equation and the Kokhlov-Zabolotskaya-Kuznetsov equation.

In order to gain a better understanding of the nonlinear MGT equation, we shall begin with the linearized model. In [15], Kaltenbacher, Lasiecka and Marchand investigated the following linearized MGT equation

For equation (1.1), they disclosed a critical parameter $\gamma = \alpha-\frac{c^{2}\tau}{b}$ and showed that when $\gamma>0$, namely in the subcritical case, the problem is well-posed and its solution is exponentially stable; while $\gamma = 0$, the energy is conserved. Since its appearance, an increasing interest has been developed to study the MGT equation, see [4,5,8,10,14]. Caixeta, Lasiecka and Cavalcanti [4] considered the following nonlinear equation

They proved that the underlying PDE generates a well-posed dynamical system which admits a global and finite dimensional attractor. They also overcomed the difficulty of lacking the Lyapunov function and the lack of compactness of the trajectory.

Now, we concentrate on the stabilization of MGT equation with memory which has received a considerable attention recently. For instance, Lasiecka and Wang [17] studied the following equation:

where $\alpha-\frac{c^{2}\tau}{b}\geq0$ and the form of $w$ classifies the memory into three types. By imposing the assumption on the relaxation function $g$, for a positive constant $c_{0}$, as

they discussed the effect of memory described by three types on decay rates of the energy when $\alpha-\frac{c^{2}\tau}{b}>0$. Moreover, in the critical case $\alpha-\frac{c^{2}\tau}{b} = 0$, they proved an exponential rate of decay for the solution of (1.3) under "the right mixture" of memory. Lasiecka and Wang [18] showed the general decay result of the equation (1.3) when $w = u$, and established their result under weaker condition on $g$. In [9], Filippo et al. investigated the critical case of equation (1.3) (that is $\alpha b-c^{2}\tau = 0$) for $w = u$ and $g$ satisfies (1.4), and obtained an exponential decay result if and only if $\mathcal{A}$ is a bounded operator. When $\int_{0}^{t}$ is replaced by $\int_{0}^{\infty}$, (1.3) turns to

Alves et al. [1] investigated the uniform stability of equation (1.5) encompassing three different types of memory in a history space set by the linear semigroup theory. Moreover, we refer the reader to [3,6,7,12,24,25,26,28] for other works of the equation(s) with memory.

More recently, Filippo and Vittorino [11] considered the fourth-order MGT equation

They investigated the stability properties of the related solution semigroup. And, according to the values of certain stability numbers depending on the strictly positive parameters $\alpha$, $\beta$, $\gamma$, $\delta$, $\varrho$, they established the necessary and sufficient condition for exponential stability. For other related results on the higher-order equations, please see [20,27,34,35,36,37] and the references therein.

Motivated by the above results, we intend to study the following abstract version of the fourth-order Moore-Gibson-Thompson (MGT) equation with a memory term

where $\alpha$, $\beta$, $\gamma$, $\delta$, $\varrho$ are strictly positive constants, $\mathcal{A}$ is a strictly positive self-adjoint linear operator defined in a real Hilbert space $H$ where the (dense) embedding $\mathcal{D}(\mathcal{A})\subset H$ need not to be compact. And we consider the following initial conditions

A natural question that arised in dealing with the general decay of fourth-order MGT equation with memory:

● Can we get a general decay result for a class of relaxation functions satisfying $g'(t)\leq-\xi(t)M(g(t))$ for $M$ to be increasing and convex function near the origin and $\xi(t)$ to be a nonincreasing function?

Mustafa answered this question for viscoelastic wave equations in [31,32]. Messaoudi and Hassan [29] considered the similar question for memory-type Timoshenko system in the cases of equal and non-equal speeds of wave propagation. Moreover, they extended the range of polynomial decay rate optimality from $p\in\left[1,\frac{3}{2}\right)$ to $p\in[1,2)$ when $g$ satisfies $g'(t)\leq-\xi(t)g^{p}(t)$. We refer to [19] for the non-equal wave speeds case. And, Liu et al. [22,23] also concerned with the similar question for third-order MGT equations with memory term.

The aim of this paper is to establish the well-posedness and answer the above mention question for fourth-order MGT equation with memory (1.7). We first use the Faedo-Galerkin method to prove the well-posedness result. We then use the idea developed by Mustafa in [31,32], taking into consideration the nature of fourth-order MGT equation, to prove new general decay results for the case $\gamma-\frac{\delta}{\alpha}>0$ and $\beta-\frac{\alpha\varrho}{\delta}>0$, based on the perturbed energy method and on some properties of convex functions. Our result substantially improves and generalizes the earlier related results in previous literature.

The rest of our paper is organized as follows. In Section $2$, we give some assumptions and state our main results. In Section $3$, we give the proof of well-posedness. In Section $4$, we state and prove some technical lemmas that are relevant in the entire work. In Section $5$, we prove the general decay result.

2.   Preliminaries and main results

In this section, we consider the following assumptions and state our main results. We use $c>0$ to denote a positive constant which does not depend on the initial data.

First, we consider the following assumptions as in [11] for $(A1)$, in [18] for $(A3)$, $(A5)$ and in [31] for $(A2)$, $(A4)$:

${\rm (A1)}$ $\gamma-\frac{\delta}{\alpha}>0$ and $\beta-\frac{\alpha\varrho}{\delta}>0$.

${\rm (A2)}\ g : \mathbb{R}^+\rightarrow\mathbb{R}^+\ $ is a non-increasing differentiable function such that

${\rm (A3)}$ $g''(t)\geq 0$ almost everywhere.

${\rm (A4)}$ There exists a non-increasing differentiable function $\xi$ : $\mathbb{R}^+\rightarrow\mathbb{R}^+$ and a $C^{1}$ function $M:$ $[0,\infty)\rightarrow[0,\infty)$ which is either linear or strictly increasing and strictly convex $C^{2}$ function on $(0,r]$, $r\leq g(0)$, with $M(0) = M^{'}(0) = 0$, such that

${\rm (A5)}$ There exists $\lambda_{0}>0$ such that $\mathcal{A}$ satisfies $\|u\|^{2}\leq \lambda_{0}\left\|\mathcal{A}^{\frac{1}{2}}u\right\|^{2}$ for all $u\in H$.

Remark 1. ([31,Remark 2.8]) (1) From assumption $(A2)$, we deduce that

Furthermore, from the assumption $(A4)$, we obtain that there exists $t_{0}\geq0$ large enough such that

The non-increasing property of $g(t)$ and $\xi(t)$ gives

A combination of these with the continuity of $H$, for two constants $a,d>0$, yields

Consequently, for any $t\in[0,t_{0}]$, we get

and, hence,

(2) If $M$ is a strictly increasing and strictly convex $C^{2}$ function on $(0,r]$, with $M(0) = M'(0) = 0$, then it has an extension $\overline{M}$, which is strictly increasing and strictly convex $C^{2}$ function on $(0,\infty)$. For example, if we set $M(r) = A$, $M'(r) = B$, $M''(r) = C$, we can define $\overline{M}$, for any $t>r$, by

Then, inspired by the notations in [11], we define the Hilbert spaces

In order to simplify the notation, we denote the usual space $H_{0}$ by $H$. The phase space of our problem is

Moreover, we will denote the inner product of $H$ by $(\cdot,\cdot)$ and its norm by $\|\cdot\|$.

After that, we introduce the following energy functional

where $G(t) = \int_{0}^{t}g(s)ds$ and for any $v\in L_{loc}^{2}\left(\mathbb{R}^{+};L^{2}(\Omega)\right)$,

As in [31], we set, for any $0<\nu<1$,

The following lemmas play an important role in the proof of our main results.

Lemma 2.1. ([31] ) Assume that condition $(A2)$ holds. Then for any $u\in L_{loc}^{2}\left(\mathbb{R}^{+};\right.$ $\left.L^{2}(\Omega)\right)$, we have

Lemma 2.2. (Jensen's inequality) Let $P$: $[b,c]\rightarrow \mathbb{R}$ be a convex function. Assume that functions $f : \Omega\rightarrow [b,c]$ and $h: \Omega\rightarrow \mathbb{R}$ are integrable such that $h(x)\geq 0$, for any $x\in\Omega$ and $\int_{\Omega} h(x)dx = k>0$. Then

Lemma 2.3. ([2])(The generalized Young inequality) If $f$ is a convex function defined on a real vector space $X$ and ite convex conjugate is denoted by $f^{*}$, then

where

We are now in a position to state the well-posedness and the general decay result for problem (1.7)-(1.8).

Theorem 2.4. (Well-posedness) Assume that $(A1)-(A5)$ hold. Then, for given $(u_{0}, u_{1}, u_{2}, u_{3})\in \mathcal{H}$ and $T>0$, there exists a unique weak solution $u$ of problem (1.7)-(1.8) such that

Theorem 2.5. (General decay) Let $(u_{0},u_{1},u_{2},u_{3})\in\mathcal{H}$. Assume that $(A1)$-$(A5)$ hold. Then there exist positive constants $k_{1}$ and $k_{2}$ such that, along the solution of problem (1.7)-(1.8), the energy functional satisfies

where $M_{1}(t) = \int_{t}^{r}\frac{1}{sM'(s)}ds$ and $M_{1}$ is strictly decreasing and convex on $(0,r]$, with ${\lim\limits_{t\rightarrow 0}}M_{1}(t) = +\infty$.

Remark 2. Assume that $M(s) = s^{p}$, $1\leq p<2$ in $(A4)$, then by simple calculations, we see that the decay rate of $E(t)$ is given by, for constants $\overline{k}$, $\widetilde{k}$ and $C$,

3.   Proof of the well-posedness

In this section, we will prove the global existence and uniqueness of the solution of problem (1.7)-(1.8). Firstly, we give the following lemmas.

Lemma 3.1. If $0<g(0)<\frac{2\alpha\varrho}{\delta}(\alpha\gamma-\delta)$, then there is $\sigma>0$ such that $\alpha\left(\gamma-\frac{\delta}{\alpha}\right)-\frac{\delta g(0)}{2(\alpha-\sigma)\varrho}>0$.

Proof. Since $g(0)<\frac{2\alpha\varrho}{\delta}(\alpha\gamma-\delta)$, all we need is to show

which is trivially true.

Lemma 3.2. Assume that $(A1)$-$(A5)$ hold. Then, the energy functional $E(t)$ satisfies, for all $t\geq 0$,

Proof. From the definition of $E(t)$, we have

Then, we estimate the sixth term of the above equality

A combination of the above results, we complete the proof of lemma.

Now, we prove the well-posedness result of problem (1.7)-(1.8).

Proof of Theorem 2.1. The proof is given by Faedo-Galerkin method and combines arguments from [16,39,38]. We present only the main steps.

Step $1$. Approximate problem

We construct approximations of the solution $u$ by the Faedo-Galerkin method as follows. For every $m\geq 1$, let $W_{m} = {\rm span}\{w_{1},\cdots, w_{m}\}$ be a Hilbertian basis of the space $H^{1}_{0}(\Omega)$. We choose four sequences $(u_{0}^{m})$, $(u_{1}^{m})$, $(u_{2}^{m})$ and $(u_{3}^{m})$ in $W_{m}$ such that $u_{0}^{m}\rightarrow u_{0}$ strongly in $\mathcal{D}\left(\mathcal{A}^{\frac{1}{2}}\right)$, $u_{1}^{m}\rightarrow u_{1}$ strongly in $\mathcal{D}\left(\mathcal{A}^{\frac{1}{2}}\right)$, $u_{2}^{m}\rightarrow u_{2}$ strongly in $\mathcal{D}\left(\mathcal{A}^{\frac{1}{2}}\right)$ and $u_{3}^{m}\rightarrow u_{3}$ strongly in $H$. We define now the approximations:

where $u^{m}(t)$ are solutions to the finite dimensional Cauchy problem (written in normal form):

with initial conditions

According to the standard theory of ordinary differential equation, the finite dimensional problem (3.2)-(3.3) has a local solution $\left(u^{m}(t), u^{m}_{t}(t),u^{m}_{tt}(t),u^{m}_{ttt}(t)\right)$ in some interval $[0,T_{m})$ with $0<T_{m}\leq T$, for every $m\in \mathbb{N}$. Next, we present some estimates that allow us to extend the local solutions to the interval $[0,T]$, for any given $T>0$.

Step $2$. Weak solutions

Multiplying equation (3.2) by $a_{jttt}^{m}+\alpha a_{jtt}^{m}+\frac{\alpha\varrho}{\delta}a_{jt}^{m}$ and integrating over $\Omega$, we have

where

From assumptions $(A1)-(A3)$ and Lemma 3.1, we get, for $\varepsilon\in(0,\alpha)$,

and so

Therefore, we have

Integrating (3.6) from $0$ to $t\leq T_{m}$, one has

Now, since the sequences $(u_{0}^{m})_{m\in N}$, $(u_{1}^{m})_{m\in N}$, $(u_{2}^{m})_{m\in N}$ and $(u_{3}^{m})_{m\in N}$ converge and using $(A1)-(A3)$, we can find a positive constant $C$ independent of $m$ such that

Therefore, using the fact $\varrho-\int_{0}^{t}g(s)ds\geq l$, the last estimate (3.8) together with (3.4) give us, for all $m\in N, T_{m} = T$, we deduce that

Consequently, we may conclude that

From (3.9), we get that $(u^{m})_{m\in N}$ is bounded in $L^{\infty}\left(0,T; \mathcal{D}\left(\mathcal{A}^{\frac{1}{2}}\right)\right)$. Then, $(u^{m})_{m\in N}$ is bounded in $L^{2}\left(0,T; \mathcal{D}\left(\mathcal{A}^{\frac{1}{2}}\right)\right)$. Since $(u_{t}^{m})_{m\in N}$ is bounded in $L^{\infty}\left(0,T; \mathcal{D}\left(\mathcal{A}^{\frac{1}{2}}\right)\right)$, $(u_{t}^{m})_{m\in N}$ is bounded in $L^{2}\left(0,T; \mathcal{D}\left(\mathcal{A}^{\frac{1}{2}}\right)\right)$. Consequently, $(u_{tt}^{m})_{m\in N}$ is bounded in $L^{2}\left(0,T; \mathcal{D}\left(\mathcal{A}^{\frac{1}{2}}\right)\right)$ and $(u_{ttt}^{m})_{m\in N}$ is bounded in $L^{2}\left(0,T; H\right)$. Moreover, $(u^{m})_{m\in N}$ is bounded in $H^{3}(0,T; H^{1}(\Omega))$.

Since the embedding $H^{3}(0,T; H^{1}(\Omega))\hookrightarrow L^{2}(0,T; H(\Omega))$ is compact, using Aubin-Lions theorem [21], we can extract a subsequence $(u^{n})_{n\in N}$ of $(u^{m})_{m\in N}$ such that

Therefore,

The proof now can be completed arguing as in [21].

Step $3$. Uniqueness

It is sufficient to show that the only weak solution of (1.7)-(1.8) with $u_{0} = u_{1} = u_{2} = u_{3} = 0$ is

According to the energy estimate (3.8) and noting that $E(u(0)) = 0$, we obtain

So, we have

And this implies (3.10). Thus, we conclude that problem (1.7)-(1.8) has at most one solution.

4.   Technical lemmas

In this section, we state and prove some lemmas needed to establish our general decay result.

Lemma 4.1. Let $(u,u_{t},u_{tt}, u_{ttt})$ be the solution of (1.7). Assume that $(A1)$-$(A3)$ hold. Then, we have

Proof. Multiplying (1.7) by $u_{ttt}+\alpha u_{tt}+\frac{\alpha\varrho}{\delta}u_{t}$ and integrating over $\Omega$ yield

We proceed to show that, for a constant $\varepsilon\in(0,\alpha)$,

Then, combining (4.1) and (4.2), we can obtain

According to $(A1)$-$(A3)$ and Lemma 3.1, we complete the proof of lemma.

Lemma 4.2. Assume that $(A1)$-$(A5)$ hold. Then, the functional $F_{1}(t)$ defined by

satisfies the estimate

Proof. Taking the derivative of $F_{1}(t)$ with respect to $t$, exploiting (1.7) and integrating by parts, we get

Using Young's inequality, Lemma 2.1, $(A5)$ and the fact $\gamma-\frac{\delta}{\alpha}>0$ and $\beta-\frac{\alpha\varrho}{\delta}>0$, we have

and

Also, we have

Then, combining the above inequalities, we complete the proof of (4.3).

Lemma 4.3. Assume that $(A1)$-$(A5)$ hold. Then the functional $F_{2}(t)$ defined by

satisfies the estimate

where $0<\varepsilon_{1}<1$.

Proof. By differentiating $F_{2}(t)$ with respect to $t$, using (1.7) and integrating by parts, we obtain

Now, we estimate the terms in the right-hand side of the above identity.

Using Young's inequality, we obtain, for $0<\varepsilon_{1}<1$,

and

Also, we have

and

Exploiting Young's inequality and $(A5)$, we get

A combination of all the above estimates gives the desired result.

As in [11], we introduce the following auxiliary functional

Lemma 4.4. Assume that $(A1)$-$(A5)$ hold. Then the functional $F_{3}(t)$ satisfies the estimate

where $0<\varepsilon_{2}<1$.

Proof. Using the equation (1.7), a direct computation leads to the following identity

Now, the first and third terms in the right-hand side of (4.6) can be estimated as follows:

and

where $0<\varepsilon_{2}<1$.

Using Young's inequality and Lemma 2.1, we get

and

Then, combining the above inequalities, we obtain the desired result.

Lemma 4.5. Assume that $(A1)$-$(A5)$ hold. Then the functional $F_{4}(t)$ defined by

satisfies the estimate

where $f(t) = \int_{t}^{\infty}g(s)ds$.

Proof. Noting that $f'(t) = -g(t)$, we see that

Exploiting Young's inequality and the fact $\int_{0}^{t}g(s)ds\leq \varrho-l$, we obtain

Moreover, taking account of $f(t)\leq f(0) = \varrho-l$, we have

Combining the above estimates, we arrive at the desired result.

Lemma 4.6. Assume that $(A1)-(A5)$ hold. The functional $\mathcal{L}(t)$ defined by

satisfies, for a suitable choice of $N, N_{2}, N_{3}$,

and the estimate, for all $t\geq t_{0}$,

where $t_{0}$ has been introduced in Remark $2.1$.

Proof. Combining Lemmas 4.1-4.4 and recalling that $g' = \nu g-h$, we obtain, for all $t\geq t_{0}$,

At this point, we need to choose our constants very carefully. First, we choose

The above choice yields

Then, we choose $N_{2}$ large enough so that

Next, we choose $N_{3}$ large enough so that

Now, as $\frac{\nu^{2}g(s)}{\nu g(s)-g'(s)}<g(s)$, it is easy to show, using the Lebesgue dominated convergence theorem, that

Hence, there is $0<\nu_{0}<1$ such that if $\nu<\nu_{0}$, then

Now, let us choose $N$ large enough and choose $\nu$ satisfying

which means

and

So we arrive at, for positive constant $c$,

On the other hand, from Lemma 3.2, we find that

Therefore, we can choose $N$ even large (if needed) so that (4.8) is satisfied.

5.   Proof of the general decay result

In this section, we will give an estimate to the decay rate for the problem (1.7)-(1.8).

Proof of Theorem 2.2. Our proof starts with the observation that, for any $t\geq t_{0}$,

which are derived from (2.2) and Lemma 4.1 and can be used in (4.8).

Taking $\mathcal{F}(t) = \mathcal{L}(t)+cE(t)$, which is obviously equivalent to $E(t)$, we get, for all $t\geq t_{0}$,

where $m$ is a positive constant. Then, we obtain that

We consider the following two cases relying on the ideas presented in [31].

(ⅰ) $M(t)$ is linear.

We multiply (5.1) by $\xi(t)$, then on account of $(A1)$-$(A4)$ and Lemma 4.1, we obtain, for all $t\geq t_{0}$,

Therefore,

As $\xi(t)$ is non-increasing and $\mathcal{F}(t)\sim E(t)$, we have

and

It follows immediately that

We may now integrate over $(t_{0},t)$ to conclude that, for two positive constants $k_{1}$ and $k_{2}$

By the continuity of $E(t)$, we have

(ⅱ) $M$ is nonlinear.

First, we define the functional

Obviously, $L(t)$ is nonnegative. And, by Lemma 4.5 and Lemma 4.6, there exists $b>0$ such that

Therefore, integrating the above inequality over $(t_{0},t)$, we see at once that

It is sufficient to show that

and

Now, we define a functional $\lambda(t)$ by

Clearly, we have

After that, we define another functional $I(t)$ by

Now, the following inequality holds under Lemma 4.1 and (5.2) that

Then (5.4) allows for a constant $0<q<1$ chosen so that, for all $t\geq t_{0}$,

otherwise we get an exponential decay from (5.1). Moreover, recalling that $M$ is strict convex on $(0,r]$ and $M(0) = 0$, then

From assumptions $(A2)$ and $(A4)$, (5.5) and Lemma 2.2, it follows that

According to $\overline{M}$ is an extension of $M$ (see Remark $2.1 (2)$), we also have

In this way,

and (5.1) becomes

Let $0<\varepsilon_{0}<r$, we define the functional $\mathcal{F}_{1}(t)$ by

Then, recalling that $E'(t)\leq 0$, $\overline{M}'>0$ and $\overline{M}''>0$ as well as making use of estimate (5.6), we deduce that $\mathcal{F}_{1}(t)\sim E(t)$ and also, for any $t\geq t_{0}$, we have

Taking account of Lemma 2.3, we obtain

where

So, combining (5.7), (5.8) and (5.9), we obtain

From this, we multiply the above inequality by $\xi(t)$ to get

Then, using the fact that, as $\varepsilon_{0}\frac{E(t)}{E(0)}<r$, $\overline{M}'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right) = M'\left(\varepsilon_{0}\frac{E(t)}{E(0)}\right)$ and (5.3), we get

Consequently, defining $\mathcal{F}_{2}(t) = \xi(t)\mathcal{F}_{1}(t)+cE(t)$, then since $\mathcal{F}_{1}(t)\sim E(t)$, we arrive at

and with a suitable choice of $\varepsilon_{0}$, we get, for some positive constant $k$ and for any $t\geq t_{0}$,

Define

Moreover, it suffices to show that $M'_{2}(t), M_{2}(t)>0$ on $(0,1]$ by the strict convexity of $M$ on $(0,r]$. And, it is easily seen that

According to (5.10) and (5.12), there exist $\lambda_{2}, \lambda_{3}>0$ such that

Then, it follows that there exists $k_{1}>0$ such that

Next, we define

And based on the properties of $M$, we know that $M_{1}$ is strictly decreasing function on $(0,r]$ and ${\lim_{t\rightarrow 0}}M_{1}(t) = +\infty$.

Now, we integrate (5.14) over $(t_{0},t)$ to obtain

so

which implies that

It is easy to obtain that

A combining of (5.13) and (5.15) gives the proof.

Acknowledgments

The authors are grateful to the anonymous referees and the editor for their useful remarks and comments.