Existence of asymptotically almost periodic solutions for some second-order hyperbolic integrodifferential equations

Abstract

In this paper we study and obtain the existence of asymptotically almost periodic solutions to some classes of second-order hyperbolic integrodifferential equations of Gurtin–Pipkin type in a separable Hilbert space H. To illustrate our abstract results, the existence of asymptotically almost periodic mild solutions to the well-known Kirchoff plate equation is studied.

1 Introduction

Integrodifferential equations play an important role when it comes to describing various practical problems, see, e.g., [4,5,6, 13, 14, 17, 21,22,23]. In particular, these types of equations are utilized to study practical problems in which some memory effect is taken into account, such as the heat conduction in materials with memory or the sound propagation in viscoelastic media or in homogenization problems in perforated media (Darcy’s Law), see, e.g., [1, 5, 15, 16, 18].

Let $(H, \Vert \cdot \Vert _H, \langle \cdot , \cdot \rangle _H)$ be a separable Hilbert space. The main purpose of this paper consists of establishing the existence of asymptotically almost periodic mild solutions to a class of second-order hyperbolic integrodifferential equations of Gurtin–Pipkin type given by

$$\begin{aligned} \begin{array}{ll} \displaystyle \frac{d^2 u}{dt^2}+A^2u-\displaystyle \int _{-\infty }^tg(t-s)A^2u(s)ds=f(t,u),&t>0 \end{array} \end{aligned}$$

(1)

with initial conditions

$$\begin{aligned} \begin{array}{ll} u(-t)=u_0(t), \ \ t\ge 0 \ \ \text{ and } \ \ u'(0)=u_1, \end{array} \end{aligned}$$

(2)

where $A: {{\mathcal D}}(A)\subset H \mapsto H$ is a positive self-adjoint operator which is bounded below, that is, there exists a constant $\omega > 0$ such that

$$\begin{aligned} \Vert Au\Vert _H \ge \omega \Vert u\Vert _H \ \ \text{ for } \text{ all } \ \ u \in {{\mathcal D}}(A),\end{aligned}$$

(3)

the function $f: [0, \infty ) \times H \mapsto H$ is asymptotically almost periodic in the first variable uniformly in the second one, and the non-increasing differentiable relaxation (kernel) function $g:[0, \infty )\longrightarrow [0, \infty )$ satisfies the following assumptions,

• (A.1) $\displaystyle g(0)>0 \ \ \text{ and }\ \ \beta := 1-\int _0^{\infty }g(s)ds>0$; and

• (A.2) there exists a positive constant $\xi$ such that $g'(t)\le -\xi g(t) \ \ \text{ for } \text{ all } \ \ t\ge 0.$

Our main task in this paper consists of showing that problem (1)–(2), under some suitable assumptions, has an asymptotically almost periodic mild solution. To achieve that, our strategy consists of transforming such a system into a first-order evolution Eq. (7) below. Under assumptions (A.1) and (A.2), it will be shown that the linear operator ${\mathcal {A}}$ appearing in Eq. (7) is the infinitesimal generator of a $C_0$-semigroup of contraction $\left( T(t)\right) _{t\ge 0}$ which actually is exponentially stable (Theorems 2 and 3). Next, one makes use of the appropriate fixed-point tools to obtain the existence of an asymptotically almost periodic mild solution to Eq. (7) which, in turn, yields the existence of an asymptotically almost periodic mild solution to Eqs. (1)–(2).

Recall that the existence of almost periodic and asymptotically almost periodic solutions to integrodifferential equations is an important topic which has numerous applications, see, e.g., [8,9,10]. The novelty in this paper consists of using the semigroup approach to study the existence of solutions to some second-order integrodifferential equations in the case when the forcing term is asymptotically almost periodic. We show that if the forcing term f is asymptotically almost periodic, then, under some additional conditions on the parameters ((A.1) and (A.2)) of the system (1)–(2), there exists a mild solution which converges asymptotically to an almost periodic function. To the best of our knowledge, the existence of asymptotically almost periodic mid solutions to second-order integrodifferential equations of Gurtin–Pipkin type formulated in (1)–(2) is an untreated problem which constitutes the main motivation of this paper. For more information on the system (1)–(2) including its well-posedness or the asymptotic behavior of its solutions, we refer the reader to the work of Vlasov and Rautian [18,19,20].

Let us mention that assumptions (A.1)–(A.2) yield $g(t)\le g(0)e^{-\xi t} \ \ \text{ for } \text{ all } \ \ t\ge 0.$ Consequently, we have

$$\int _0^\infty s^kg(s)ds\le \frac{k!}{\xi ^{k+1}}g(0)\qquad {\mathrm {for}}\quad k=0,1,2,\cdots .$$

Further, it can be shown that the relaxation function K defined by

$$\begin{aligned} K(t) = \sum _{k=1}^\infty a_k e^{-\gamma _k t}, \ \ t \ge 0, \end{aligned}$$

where $a_k > 0$, $\gamma _{k+1}> \gamma _k > 0$ for all $k \in {\mathbb {N}}$, $\gamma _k \rightarrow \infty$ as $k \rightarrow \infty$, and

$$\begin{aligned} \sum _{k=1}^\infty a_k < 1 \end{aligned}$$

satisfies assumptions (A.1)–(A.2) for any $\xi \in (0,\gamma _1]$.

2 Preliminaries

Fix once and for all a separable Hilbert space H whose inner product and norm are given respectively by $\langle \cdot , \cdot \rangle _H$ and $\Vert \cdot \Vert _H$. Let $(X, \Vert \cdot \Vert )$ and $(Y, \Vert \cdot \Vert )$ be two Banach spaces. If A is a linear operator, then the notations ${{\mathcal D}}(A)$ and $\rho (A)$ stand respectively for the domain and resolvent of A.

Let $BC({\mathbb {R}}; X)$ (respectively, $BC({\mathbb {R}}\times X;Y)$) stand for the Banach space of all bounded continuous functions from ${\mathbb {R}}$ into X (respectively, stand for the collection of all jointly continuous bounded functions from ${\mathbb {R}}\times X$ into Y) equipped with the sup-norm defined by $\Vert u\Vert _\infty : = \sup _{t \in {\mathbb {R}}} \left\| u(t)\right\|$ for all $u\in BC({\mathbb {R}}; X)$. Finally, $C_0({\mathbb {R}}_+;X)$ (respectively, $C_0({\mathbb {R}}_+ \times X; Y)$) stands for the collection of all continuous functions $u: {\mathbb {R}}_+ \longrightarrow X$ such that $\lim _{t \rightarrow \infty } \Vert u(t)\Vert = 0$ (respectively, all jointly continuous functions $U: {\mathbb {R}}_+ \times X \longrightarrow Y$ such that $\lim _{t \rightarrow \infty } \Vert U(t, x)\Vert = 0$ for all $x \in X$).

Definition 1

A function $f\in BC({\mathbb {R}};X)$ is almost periodic if for every $\varepsilon >0$ there exists a relatively dense subset of ${\mathbb {R}}$, denoted by ${\mathscr {T}}(\varepsilon ,f,X)$, such that

$$\begin{aligned} \Vert f(t+\tau )-f(t)\Vert <\varepsilon \qquad \forall \,t\in {\mathbb {R}},\quad \forall \,\tau \in {\mathscr {T}}(\varepsilon ,f,X). \end{aligned}$$

Definition 2

A function $f\in BC({\mathbb {R}}_+;X)$ is asymptotically almost periodic if there exist an almost periodic function g and $\phi \in C_0({\mathbb {R}}_+;X)$ such that $f=g+\phi .$

Definition 3

A function $F\in BC({\mathbb {R}}\times X;Y)$ is almost periodic in $t\in {\mathbb {R}}$ uniformly in $y\in Y$ if for each $\varepsilon >0$ and any compact subset K of Y there exists a relatively dense subset of ${\mathbb {R}}$, denoted by ${\mathscr {T}}(\varepsilon ,F,K,X)$, such that

$$\begin{aligned} \Vert F(t+\tau , y)-F(t, y)\Vert <\varepsilon \qquad \forall \,t\in {\mathbb {R}},\quad \forall \,\tau \in {\mathscr {T}}(\varepsilon ,F,K,Y),\quad \forall \,y\in K. \end{aligned}$$

Definition 4

A function $F\in BC({\mathbb {R}}_+\times X;Y)$ is asymptotically almost periodic in $t\in {\mathbb {R}}_+$ uniformly $y\in Y$ if there exist an almost periodic function $G\in AP({\mathbb {R}}\times X;Y)$ and $\varPhi \in C_0({\mathbb {R}}_+\times X;Y)$ such that $F=G+\varPhi .$

Lemma 1

[24] A function $f\in BC({\mathbb {R}}_+;X)$ is asymptotically almost periodic if and only if for every $\varepsilon >0$ there exists $L(\varepsilon ,f,X)>0$ and a relatively dense subset of ${\mathbb {R}}_+$, denoted by ${\mathscr {T}}(\varepsilon ,f,X)$, such that

$$\begin{aligned} \Vert f(t+\tau )-f(t)\Vert <\varepsilon \qquad \forall \,t\ge L(\varepsilon ,f,X),\quad \forall \,\tau \in {\mathscr {T}}(\varepsilon ,f,X). \end{aligned}$$

Lemma 2

[10] A function $F\in BC({\mathbb {R}}\times X;Y)$ is asymptotically almost periodic if for each $\varepsilon >0$ and any compact subset K of Y there exists $L(\varepsilon ,F,K,X)>0$ and a relatively dense subset of ${\mathbb {R}}_+$, denoted by ${\mathscr {T}}(\varepsilon ,F,K,X)$, such that

$$\begin{aligned} \Vert F(t+\tau ,y)-F(t,y)\Vert <\varepsilon \qquad \forall \,t\ge L(\varepsilon ,F,K,X),\quad \forall \,\tau \in {\mathscr {T}}(\varepsilon ,F,K,X),\quad \forall \,y\in K. \end{aligned}$$

3 Preliminary settings

Through out this work we use $c>1$ to represent a generic constant, independent of t and the initial data, and it may also vary from one line to another.

In order to study the system (1)–(2), we rewrite it as a first-order evolution equation which can be easily treated. Indeed, rewrite Eq. (1) as

$$\begin{aligned} \begin{array}{ll} \displaystyle \frac{d^2 u}{dt^2}+A^2u-\displaystyle \int _{0}^\infty g(s)A^2u(t-s)ds=f(t,u),&t>0, \end{array} \end{aligned}$$

(4)

and introduce the following variable as in [2],

$$\begin{aligned} w(t,s)=u(t)-u(t-s),\qquad \forall \,s,t\ge 0, \end{aligned}$$

which in turn satisfies the following system,

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{\partial w}{\partial t} +\frac{\partial w}{\partial s}-\frac{du}{dt}=0, &{}s,t>0,\\ \\ w(0,s)=u_0(0)-u_0(s),&{}s>0. \end{array} \right. \end{aligned}$$

In view of the above, Eqs. (1)–(2) become, under the assumption (A.1),

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{d^2 u}{dt^2}+\beta A^2u+\displaystyle \int _{0}^\infty g(s)A^2w(t,s)ds=f(t,u), &{} t>0,\\ \\ \displaystyle \frac{\partial w}{\partial t} +\frac{\partial w}{\partial s}-\frac{du}{dt}=0, &{} s,t>0,\\ \\ u(0)=u_0,\quad u'(0)=u_1,\quad w(\cdot ,0)=0,\quad w(0,\cdot )=w_0, \end{array}\right. } \end{aligned}$$

(5)

where $u_0=u_0(0)$, $w_0(s)=u_0(0)-u_0(s)$ and $\displaystyle \beta :=1-\int _0^\infty g(s)ds>0$.

It remains to rewrite Eq. (5) as a first-order evolution evolution. For that, let $V :={\mathcal {D}}(A)$ be equipped with the inner product defined by,

$$\langle v,\tilde{v}\rangle _V:=\langle Av,A\tilde{v}\rangle _H,\qquad \forall \,v,\tilde{v}\in V.$$

It follows from the continuous embedding ${\mathcal {D}}(A) \hookrightarrow H$ (see Eq. (3)) that $(V,\Vert \cdot \Vert _V)$ is a Hilbert space.

Let

$$\begin{aligned} W=L^2_g\left( (0,\infty );V\right) :=\left\{ w:(0,\infty )\longrightarrow V: \ \ \int _0^\infty g(s)\Vert w(s)\Vert ^2_Vds<\infty \right\} \end{aligned}$$

be equipped with the inner product

$$\begin{aligned} \langle w,\tilde{w}\rangle _W:=\int _0^\infty g(s)\langle w(s),\tilde{w}(s)\rangle _Vds. \end{aligned}$$

Clearly, the completeness of $(V,\Vert \cdot \Vert _V)$ yields $(W,\Vert \cdot \Vert _W)$ is a Hilbert space.

The state space of our problem is given by ${\mathcal {H}}:=V\times H\times W$ which is equipped with the inner product defined by

$$\begin{aligned} \left\langle U,\tilde{U}\rangle _{\mathcal {H}}:=\beta \langle u,\tilde{u}\rangle _V+\langle v,\tilde{v}\rangle _H+\langle w,\tilde{w}\rangle _W,\right. \end{aligned}$$

for any $U=(u,v,w)^T$ and $\tilde{U}=(\tilde{u},\tilde{v},\tilde{w})^T$ in ${\mathcal {H}}$.

Setting ${\mathcal {A}}:{\mathcal {D}}({\mathcal {A}})\subset {\mathcal {H}}\longrightarrow {\mathcal {H}}$

$$\begin{aligned} {\mathcal {A}}U:=\begin{pmatrix} v\\ \\ -\beta A^2u-\displaystyle \int _0^\infty g(s)A^2w(s)ds\\ \\ \displaystyle v-\frac{dw}{ds} \end{pmatrix} \end{aligned}$$

(6)

for all $U=(u,v,w)^T\in {\mathcal {D}}({\mathcal {A}})$ where

$$\begin{aligned} {\mathcal {D}}({\mathcal {A}})=\left\{ U\in {\mathcal {H}}:\beta u+\displaystyle \int _0^\infty g(s)w(s)ds\in {\mathcal {D}}(A^2),\quad v\in {\mathcal {D}}(A),\quad \frac{dw}{ds}\in W,\quad w(0)=0\right\} \end{aligned}$$

it follows that Eq. (5) can be rewritten as a first-order evolution equation in the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{dU}{dt}={\mathcal {A}}U(t)+F(t,U),&{} t>0,\\ \\ U(0)=U_0, \end{array}\right. } \end{aligned}$$

(7)

where $U=(u,u',w)^T$, $U_0=(u_0,u_1,w_0)^T\in {\mathcal {H}}$ and $F:{\mathbb {R}}_+\times {\mathcal {H}}\longrightarrow {\mathcal {H}}$ is defined by

$$\begin{aligned} F(t,U):=\begin{pmatrix}0\\ f(t,u)\\ 0\end{pmatrix} \ \ \text{ for } \text{ all } \ \ U\in {\mathcal {H}}. \end{aligned}$$

4 Exponential stability of the $C_0$-semigroup associated with ${\mathcal {A}}$

To establish our existence results, we will be using Theorem 1 given below.

Theorem 1

[11] Let $B: {\mathcal {D}}(B) \subset H \mapsto H$ be a densely defined linear operator. If B is dissipative and $0\in \rho (B)$, then B is the infinitesimal generator of a $C_0$-semigroup of contractions on H.

Theorem 2

Under assumptions (A.1) and (A.2), the linear operator ${\mathcal {A}}$ defined in Eq. (6) is the infinitesimal generator of a $C_0$-semigroup of contraction $\left( T(t)\right) _{t\ge 0}$.

Proof

We start by establishing the dissipativeness of the linear operator ${\mathcal {A}}$. Indeed, for any $U=(u,v,w)^T\in {\mathcal {D}}({\mathcal {A}})$, we have

$$\begin{aligned} \langle {\mathcal {A}}U, U \rangle _{\mathcal {H}}= & {} \beta \langle v, u \rangle _V - \left\langle \beta A^2u + \int _0^\infty g(s)A^2w(s) ds, v \right\rangle _H + \langle v-\frac{dw}{ds}, w \rangle _W \\= & {} \beta \langle v, u \rangle _V - \left\langle \beta Au + \int _0^\infty g(s)Aw(s)ds, Av \right\rangle _H\\&+\int _0^\infty g(s)\langle v-\frac{dw}{ds},w(s)\rangle _Vds\\= & {} \beta \langle v,u\rangle _V-\beta \langle Au,Av\rangle _H-\int _0^\infty g(s)\langle Aw(s),Av\rangle _Hds\\&+\int _0^\infty g(s)\langle v,w(s)\rangle _Vds -\int _0^\infty g(s)\langle \frac{dw}{ds},w(s)\rangle _Vds\\= & {} -\frac{1}{2}\int _0^\infty g(s)\frac{d}{ds}\Vert w(s)\Vert ^2_Vds\\= & {} -\frac{1}{2}\left( \lim _{T\rightarrow \infty } \left[ g(s)\Vert w(s)\Vert ^2_V\right] _{s=0}^{s=T}-\int _0^\infty g'(s)\Vert w(s)\Vert ^2_Vds\right) \\= & {} \frac{1}{2}\int _0^\infty g'(s)\Vert w(s)\Vert _V^2ds\le 0 \end{aligned}$$

and hence ${\mathcal {A}}$ is dissipative.

The next step consists of showing that $0\in \rho ({\mathcal {A}})$. Indeed, for any $F=(f_1,f_2,f_3)^T\in {\mathcal {H}}$, consider the solvability of the equation

$$\begin{aligned} {\mathcal {A}}U=F. \end{aligned}$$

Equivalently,

$$\begin{aligned} v=f_1&\qquad {{\mathrm {in}}}\qquad V, \end{aligned}$$

(8a)

$$\begin{aligned} -\beta A^2u-\displaystyle \int _0^\infty g(s)A^2w(s)ds=f_2&\qquad {{\mathrm {in}}}\qquad H, \end{aligned}$$

(8b)

$$\begin{aligned} v-\frac{dw}{ds}=f_3&\qquad {{\mathrm {in}}}\qquad W. \end{aligned}$$

(8c)

From Eq. (8a) and Eq. (8c), we have

$$\begin{aligned} w(s)=sf_1-\int _0^sf_3(\tau )d\tau ,\quad \forall \,s\ge 0. \end{aligned}$$

(9)

Clearly, $w(0)=0$ and $\displaystyle \frac{dw}{ds}=f_1-f_3\in W.$ Using assumption (A.2) and the remarks on the relaxation function g (Introduction) we obtain

$$\begin{aligned} \int _0^\infty g(s)\Vert w(s)\Vert _V^2ds\le & {} 2\int _0^\infty g(s)\Vert sf_1\Vert _V^2ds+2\int _0^\infty \int _0^s g(s)\Vert f_3(\tau )\Vert _V^2d\tau ds\\\le & {} \frac{2}{\xi ^3}g(0)\Vert f_1\Vert _V^2+2\int _0^\infty \int _\tau ^\infty g(s)\Vert f_3(\tau )\Vert _V^2dsd\tau \\\le & {} \frac{2}{\xi ^3}g(0)\Vert f_1\Vert _V^2-\frac{2}{\xi }\int _0^\infty \int _\tau ^\infty g'(s)\Vert f_3(\tau )\Vert _V^2dsd\tau \\= & {} \frac{2}{\xi ^3}g(0)\Vert f_1\Vert _V^2+\frac{2}{\xi }\int _0^\infty g(\tau )\Vert f_3(\tau )\Vert _V^2d\tau \\= & {} \frac{2}{\xi ^3}g(0)\Vert f_1\Vert _V^2+\frac{2}{\xi }\Vert f_3\Vert _W^2<\infty \end{aligned}$$

and hence $w\in W$.

A combination of Eq. (8b) and Eq. (9) yields

$$\begin{aligned} \beta A^2u=-\left( \int _0^\infty sg(s)ds\right) A^2f_1+\int _0^\infty \int _0^sg(s)A^2f_3(\tau )d\tau ds-f_2. \end{aligned}$$

(10)

Consider both the bilinear form $B:V\times V\longrightarrow {\mathbb {R}}$ and the linear form $L:V\longrightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} B(u,\varphi ):=\beta \langle u,\varphi \rangle _V \ \ \text{ for } \text{ all } \ \ (u,\varphi )\in V\times V \end{aligned}$$

and

$$\begin{aligned} L(\varphi ):=-\left( \int _0^\infty g(s)ds\right) \langle f_1,\varphi \rangle )_V+\int _0^\infty \int _0^sg(s)\langle f_3(\tau ),\varphi \rangle _Vd\tau ds -\langle f_2,\varphi \rangle _H \ \ \text{ for } \text{ all }\,\varphi \in V. \end{aligned}$$

Clearly, the bilinear form B is bounded and coercive.

Now

$$\begin{aligned} |L(\varphi )|\le & {} \int _0^\infty sg(s)ds\Vert f_1\Vert _V\Vert \varphi \Vert _V+\int _0^\infty \int _\tau ^\infty g(s)\Vert f_3(\tau )\Vert _V\Vert \varphi \Vert _Vdsd\tau +\Vert f_2\Vert _H\Vert \varphi \Vert _H\\\le & {} \left( \frac{1}{\xi ^2}g(0) \Vert f_1\Vert _V+\frac{\sqrt{1-\beta }}{\xi }\Vert f_3\Vert _W+\frac{1}{\omega }\Vert f_2\Vert _H\right) \Vert \varphi \Vert _V,\qquad \forall \,\varphi \in V \end{aligned}$$

and which yields the linear form L is bounded.

Consequently, the well-known Lax–Milgram Lemma does guarantee the existence of a unique $u\in V$ satisfying

$$\begin{aligned} B(u,\varphi )=L(\varphi ),\qquad \forall \,\varphi \in V. \end{aligned}$$

(11)

The classical regularity argument entails that $u\in {\mathcal {D}}(A^2)$ and satisfies Eq. (10). Combining this with both Eq. (8a) and Eq. (9) we deduce that $0\in \rho ({\mathcal {A}})$.

Arguing as above, it is not hard to show that $I-{\mathcal {A}}$ is surjective, which yields ${\mathcal {A}}$ is densely defined in ${\mathcal {H}}$. Therefore, using Theorem 1, we deduce that the operator ${\mathcal {A}}$ is the infinitesimal generator of a $C_0$-semigroup of contractions which we denote $\left( T(t)\right) _{t\ge 0}$. $\square$

Definition 5

[12] If $G: {\mathbb {R}}_+ \longrightarrow {\mathcal {H}}$ is a continuous function, then the function $U: {\mathbb {R}}_+\longrightarrow {\mathcal {H}}$ is said to be a mild solution to the first-order evolution equation

$$\begin{aligned} \frac{d\varPhi }{dt} = {\mathcal {A}}\varPhi + G(t), \ \ t > 0; \ \ \varPhi (0) = \varPhi _0 \end{aligned}$$

(12)

if it satisfies

$$\begin{aligned} U(t)=T(t)\varPhi _0+\int _0^tT(t-s)G(s)ds,\qquad \forall \,t\ge 0. \end{aligned}$$

From Definition 5, we have

Definition 6

If $F: {\mathbb {R}}_+ \times {\mathcal {H}}\longrightarrow {\mathcal {H}}$ is jointly continuous, then a function $U: {\mathbb {R}}_+\longrightarrow {\mathcal {H}}$ is said to be a mild solution to Eq. (7) if it satisfies

$$\begin{aligned} U(t)=T(t)U_0+\int _0^tT(t-s)F(s,U(s))ds,\qquad \forall \,t\ge 0. \end{aligned}$$

Theorem 3

Under assumptions (A.1)–(A.2), the $C_0$-semigroup $\left( T(t)\right) _{t\ge 0}$ is exponentially stable, that is, there exist two positive constants $M > 0$ and $\delta >0$ such that

$$\begin{aligned} \Vert T(t)\Vert \le Me^{-\delta t},\qquad \forall \,t\ge 0. \end{aligned}$$

Proof

We make extensive use of the multiplier method. Indeed, setting $F\equiv 0$ in Eq. (7) (i.e., $f\equiv 0$), then the energy functional associated with the resulting homogeneous equation can be formulated as follows,

$$\begin{aligned} E(t):=\frac{1}{2}\Vert U(t)\Vert ^2_{\mathcal {H}},\qquad \forall \,t\ge 0, \end{aligned}$$

where U is the mild solution to the corresponding homogeneous equation to Eq. (7), which exists and is given by

$$\begin{aligned} U(t)=T(t)U_0,\qquad \forall \,t\ge 0. \end{aligned}$$

Simple computations show that, the energy functional satisfies

$$\begin{aligned} E'(t)=\frac{1}{2}\int _0^\infty g'(s)\Vert w(t,s)\Vert _V^2ds\le 0,\quad \forall \,t\ge 0. \end{aligned}$$

Indeed, taking the inner product of both sides of Eq. (7) with U(t) in ${\mathcal {H}}$ it follows that

$$\begin{aligned} \langle U'(t),U(t)\rangle _{\mathcal {H}}=\langle {\mathcal {A}}U(t),U(t)\rangle _{\mathcal {H}}=\frac{1}{2}\int _0^\infty g'(s)\Vert w(t,s)\Vert _V^2ds,\quad \forall \,t\ge 0, \end{aligned}$$

which proves our claim.

Define the functionals $I_1$ and $I_2$ by setting

$$\begin{aligned} I_1(t):=\langle u'(t),u(t)\rangle _H\qquad {\mathrm {and}}\qquad I_2(t):=-\left\langle u'(t),\int _0^\infty g(s)w(t,s)ds\right\rangle _H,\qquad \forall \,t\ge 0. \end{aligned}$$

Differentiating $I_1$ and using Eq. (5), we obtain

$$\begin{aligned} I_1'(t)= & {} \Vert u'(t)\Vert ^2_H+\left\langle -\beta A^2u(t)-\int _0^\infty g(s)A^2w(t,s)ds,u(t)\right\rangle _H\nonumber \\= & {} \Vert u'(t)\Vert ^2_H-\beta \Vert Au(t)\Vert ^2_H-\int _0^\infty g(s)\langle Aw(t,s),Au(t)\rangle _H ds. \end{aligned}$$

(13)

Next, we estimate the last term in the above inequality. Indeed, using the Cauchy–Schwarz, Young, and Hölder inequalities, we obtain

$$\begin{aligned} -\int _0^\infty g(s)\langle Aw(t,s),Au(t)\rangle _H ds\le & {} \Vert Au(t)\Vert _H\int _0^\infty g(s)\Vert Aw(t,s)\Vert _Hds\\\le & {} \frac{\beta }{2}\Vert Au(t)\Vert ^2_H+\frac{1}{2\beta }\left( \int _0^\infty g(s)\Vert Aw(t,s)\Vert _Hds\right) ^2\\\le & {} \frac{\beta }{2}\Vert Au(t)\Vert ^2_H+\frac{1-\beta }{2\beta }\int _0^\infty g(s)\Vert Aw(t,s)\Vert ^2_Hds. \end{aligned}$$

Inserting this estimate in Eq. (13), we get

$$\begin{aligned} I_1'(t)\le \Vert u'(t)\Vert _H^2-\frac{\beta }{2}\Vert u(t)\Vert ^2_V+c\int _0^\infty g(s)\Vert w(t,s)\Vert ^2_Vds,\qquad \forall \,t\ge 0. \end{aligned}$$

(14)

Differentiating $I_2$ and exploiting Eq. (5), we get

$$\begin{aligned} I_2'(t)= & {}\, \left\langle \beta A^2u(t)+\int _0^\infty g(s)A^2w(t,s)ds,\int _0^\infty g(s)w(t,s)ds\right\rangle _H\nonumber \\&-\left\langle u'(t),\int _0^\infty g(s)w_t(t,s)ds\right\rangle _H\nonumber \\= & {}\, \left\langle \beta Au(t),\int _0^\infty g(s)Aw(t,s)ds\right\rangle _H+\left\langle \int _0^\infty g(s)Aw(t,s)ds,\int _0^\infty g(s)Aw(t,s)ds\right\rangle _H\nonumber \\&-\left\langle u'(t),\int _0^\infty g(s)(u'(t)-w_s(t,s))ds\right\rangle _H\nonumber \\= & {}\, \beta \left\langle Au(t),\int _0^\infty g(s)Aw(t,s)ds\right\rangle _H+\left\| \int _0^\infty g(s)Aw(t,s)ds\right\| ^2_H\nonumber \\&-(1-\beta )\Vert u'(t)\Vert ^2_H+\left\langle u'(t),\int _0^\infty g(s)w_s(t,s)ds\right\rangle _H. \end{aligned}$$

(15)

Next, we estimate the terms in the above inequality. Using the Cauchy–Schwarz, Young and Hölder inequalities, it follows that, for any $\eta >0$,

$$\begin{aligned} \beta \left\langle Au(t),\int _0^\infty g(s)Aw(t,s)ds\right\rangle _H\le & {} \,\eta |Au(t)\Vert ^2_H+\frac{\beta ^2}{4\eta }\left( \int _0^\infty g(s)\Vert Aw(t,s)\Vert _Hds\right) ^2\\\le & {} \,\eta \Vert u(t)\Vert ^2_V+\frac{c}{\eta }\int _0^\infty g(s)\Vert w(t,s)\Vert ^2_Vds. \end{aligned}$$

Using Hölder’s inequality, we have

$$\begin{aligned} \left\| \int _0^\infty g(s)Aw(t,s)ds\right\| ^2_H\le & {} \left( \int _0^\infty g(s)ds\right) \int _0^\infty g(s)\Vert Aw(t,s)\Vert ^2_Hds\\= & {} (1-\beta )\int _0^\infty g(s)\Vert w(t,s)\Vert ^2_Vds. \end{aligned}$$

Lastly, using integration by parts together with both Young’s and Hölder’s inequalities, it follows that, for any $\eta >0$,

$$\begin{aligned} \left\langle u'(t),\int _0^\infty g(s)w_s(t,s)ds\right\rangle _H= & {} \,\left\langle u'(t),g(s)w(t,s)\Big |_{s=0}^\infty -\int _0^\infty g'(s)w(t,s)ds\right\rangle _H\\= & {}\, -\left\langle u'(t),\int _0^\infty g'(s)w(t,s)ds\right\rangle _H\\\le & {} \eta \Vert u'(t)\Vert ^2_H+\frac{1}{4\eta }\left( \int _0^\infty g'(s)\Vert w(t,s)\Vert _Hds\right) ^2\\\le & {}\, \eta \Vert u'(t)\Vert ^2_H-\frac{c}{\eta }\int _0^\infty g'(s)\Vert w(t,s)\Vert _Vds. \end{aligned}$$

Plugging the above estimates in Eq. (15), we obtain

$$\begin{aligned} I_2'(t)\le & {} -((1-\beta )-\eta )\Vert u'(t)\Vert ^2_H+\eta \Vert u(t)\Vert ^2_V\nonumber \\&+\left( (1-\beta )+\frac{c}{\eta }\right) \int _0^\infty g(s)\Vert w(t,s)\Vert ^2_Vds\nonumber \\&-\frac{c}{\eta }\int _0^\infty g'(s)\Vert w(t,s)\Vert ^2_Vds. \end{aligned}$$

(16)

Define the functional ${\mathscr {L}}$ by

$$\begin{aligned} {\mathscr {L}}(t):=NE(t)+\varepsilon _1I_1(t)+\varepsilon _2I_2(t),\qquad \forall \,t\ge 0, \end{aligned}$$

where $N,\,\varepsilon _1,\,\varepsilon _2$ are positive constants to be specified later.

From Eq. (14) and Eq. (16) we have

$$\begin{aligned} {\mathscr {L}}'(t)\le & {} -\left[ ((1-\beta )-\eta )\varepsilon _2-\varepsilon _1\right] \Vert u'(t)\Vert ^2_H-\left( \frac{1}{2}\beta \varepsilon _1-\eta \varepsilon _2\right) \Vert u(t)\Vert ^2_V\nonumber \\&+\left[ c\varepsilon _1+\left( (1-\beta )+\frac{c}{\eta }\right) \varepsilon _2\right] \int _0^\infty g(s)\Vert w(t,s)\Vert ^2_Vds\nonumber \\&+\left( \frac{N}{2}-\frac{c}{\eta }\varepsilon _2\right) \int _0^\infty g'(s)\Vert w(t,s)\Vert ^2_Vds. \end{aligned}$$

(17)

Now choose $\eta >0$ small enough so that

$$\begin{aligned} \eta <\min \left\{ \frac{1}{2}(1-\beta ),\,\frac{1}{8}\beta (1-\beta )\right\} . \end{aligned}$$

Consequently, for any fixed $\varepsilon _2>0$, we pick $\varepsilon _1>0$ satisfying

$$\begin{aligned} \frac{1}{4}(1-\beta )\varepsilon _2<\varepsilon _1<\frac{1}{2}(1-\beta )\varepsilon _2. \end{aligned}$$

Then,

$$\begin{aligned} c_1:=((1-\beta )-\eta )\varepsilon _2-\varepsilon _1>\frac{1}{2}(1-\beta )\varepsilon _2-\varepsilon _1>0 \end{aligned}$$

and

$$\begin{aligned} c_2:=\frac{1}{2}\beta \varepsilon _1-\eta \varepsilon _2>\frac{1}{2}\beta \varepsilon _1-\frac{1}{8}\beta (1-\beta )=\frac{1}{2}\beta \left( \varepsilon _1-\frac{1}{4}(1-\beta )\varepsilon _2\right) >0. \end{aligned}$$

Finally, we choose N large enough so that $\displaystyle N>\frac{2c}{\eta }\varepsilon _2$ and ${\mathscr {L}}\sim E$. Thus Eq. (17) becomes, for some fixed $\alpha >0$,

$$\begin{aligned} {\mathscr {L}}'(t)\le & {} -c_1\Vert u'(t)\Vert ^2_H-c_2\Vert u(t)\Vert ^2_V+c\int _0^\infty g(s)\Vert w(t,s)\Vert ^2_Vds\\\le & {} -\alpha E(t)+c\int _0^\infty g(s)\Vert w(t,s)\Vert ^2_Vds\\\le & {} -\alpha E(t)+\frac{c}{\xi }\int _0^\infty \xi g(s)\Vert w(t,s)\Vert ^2_Vds\\\le & {} -\alpha E(t)-\frac{c}{\xi }\int _0^\infty g'(s)\Vert w(t,s)\Vert ^2_Vds\\\le & {} -\alpha E(t)-cE'(t),\qquad \forall \,t\ge 0. \end{aligned}$$

Set ${\mathcal {F}}:={\mathscr {L}}+cE$, then ${\mathcal {F}}\sim E$ and thus we deduce from the above estimate that

$$\begin{aligned} {\mathcal {F}}'(t)\le -\lambda {\mathcal {F}}(t),\qquad \forall \,t\ge 0, \end{aligned}$$

for some fixed $\lambda >0$. A simple integration over (0, t) yields

$$\begin{aligned} {\mathcal {F}}(t)\le {\mathcal {F}}(0)e^{-\lambda t},\qquad \forall \,t\ge 0 \end{aligned}$$

which in turn yields

$$\begin{aligned} E(t)\le CE(0)e^{-\lambda t},\qquad \forall \,t\ge 0. \end{aligned}$$

Thus

$$\begin{aligned} \Vert T(t)U_0\Vert =\Vert U(t)\Vert \le \sqrt{C}e^{-\frac{\lambda }{2}t}\Vert U_0\Vert ,\qquad \forall \,t\ge 0. \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert T(t)\Vert \le Me^{-\delta t}\, \, \text{ for } \text{ all } \, \, t\ge 0, \end{aligned}$$

where $M=\sqrt{C}$ and $\delta =\frac{\lambda }{2}.$ $\square$

5 Existence of asymptotically almost periodic mild solutions

The next lemma can be proved using similar arguments as in [10]. However for the sake of completeness, we provide the proof.

Lemma 3

Let $g\in AAP(X)$. Then the function v defined by

$$\begin{aligned} v(t)=\int _0^tT(t-s)g(s)ds \end{aligned}$$

is asymptotically almost periodic.

Proof

Let $\varepsilon >0$, since $g\in AAP(X)$, there exists $L=L(\frac{\delta \varepsilon }{2M},g,X)$ and a relatively dense subset ${\mathscr {T}}(\frac{\delta \varepsilon }{2M},g,X)$ of ${\mathbb {R}}_+$ such that

$$\begin{aligned} \Vert g(t+\tau )-g(t)\Vert <\frac{\delta \varepsilon }{2M}, \quad \forall \,t\ge L,\quad \forall \,\tau \in {\mathscr {T}}\left( \frac{\delta \varepsilon }{2M},g,X\right) . \end{aligned}$$

Let $L_1$ be a positive constant such that $L_1>\frac{1}{\delta }\ln (\frac{6M^2\Vert g\Vert _\infty }{\delta \varepsilon })$, then any $t\ge L+L_1$ and $\tau \in {\mathscr {T}}\left( \frac{\delta \varepsilon }{2M},g,X\right)$ we have

$$\begin{aligned} \Vert v(t+\tau )-v(t)\Vert= & {} \left\| \int _0^{t+\tau }T(t+\tau -s)g(s)ds-\int _0^tT(t-s)g(s)ds\right\| \\\le & {} \underbrace{\int _0^\tau \Vert T(t+\tau -s)g(s)\Vert ds}_{:=I_3}\\&+\underbrace{\int _0^t\Vert T(t-s)(g(s+\tau )-g(s))\Vert ds}_{:=I_4}. \end{aligned}$$

Next, we estimate each term on the right-hand side of the above inequality.

$$\begin{aligned} I_3\le & {} \int _0^\tau \Vert T(t)\Vert \Vert T(\tau -s)g(s)\Vert ds\\\le &\, {} Me^{-\delta t}\int _0^\tau Me^{-\delta (\tau -s)}\Vert g(s)\Vert ds\\\le &\, {} M^2\Vert g\Vert _\infty e^{-\delta t}\int _0^\tau e^{-\delta s}ds=\frac{1}{\delta }M^2\Vert g\Vert _\infty e^{-\delta t}.\\ I_4= & {} \left( \int _0^L+\int _L^t\right) \Vert T(t-s)(g(s+\tau )-g(s))\Vert ds\\= & {} \int _0^L\Vert T(t-L)T(L-s)(g(s+\tau )-g(s))\Vert ds\\&+\int _L^t\Vert T(t-s)(g(s+\tau )-g(s))\Vert ds\\\le & {}\, M^2e^{-\delta (t-L)}\int _0^Le^{-\delta (L-s)}\Vert g(s+\tau )-g(s)\Vert ds\\&+M\int _L^te^{-\delta (t-s)}\Vert g(s+\tau )-g(s)\Vert ds\\\le & {}\, 2M^2\Vert g\Vert _\infty e^{-\delta (t-L)}\int _0^Le^{-\delta (L-s)}ds+M\int _L^te^{-\delta (t-s)}\frac{\delta \varepsilon }{2M}ds\\\le & {}\, \frac{2}{\delta }M^2\Vert g\Vert _\infty e^{-\delta (t-L)}+\frac{\varepsilon }{2}. \end{aligned}$$

Combining the above estimates, we obtain,

$$\begin{aligned} \Vert v(t+\tau )-v(t)\Vert\le & {} \,\frac{1}{\delta }M^2\Vert g\Vert _\infty e^{-\delta t}+\frac{2}{\delta }M^2\Vert g\Vert _\infty e^{-\delta (t-L)}+\frac{\varepsilon }{2}\\\le & {}\, \frac{3}{\delta }M^2\Vert g\Vert _\infty e^{-\delta (t-L)}+\frac{\varepsilon }{2}\le \frac{3}{\delta }M^2\Vert g\Vert _\infty e^{-\delta L_1}+\frac{\varepsilon }{2}\\< & {}\, \frac{\varepsilon }{2}+\frac{\varepsilon }{2}=\varepsilon \qquad \forall \,t\ge L\quad {{\mathrm {and}}}\quad \forall \,\tau \in {\mathscr {T}}\left( \frac{\delta \varepsilon }{2M},g,X\right) , \end{aligned}$$

which yields $v\in AAP(X)$. $\square$

Theorem 4

Under assumptions (A.1)–(A.2), suppose that the initial data $U_0 \in {\mathcal {H}}$ and that the function f belongs to $AAP({\mathbb {R}}_+\times H,H)$. Further, suppose that there exists $K>0$ such that

$$\begin{aligned} \Vert f(t,u)-f(t,v)\Vert _H\le K\Vert u-v\Vert _H\qquad \forall \,u,v\in H\quad {{\mathrm {and}}}\quad \forall \,t\in {\mathbb {R}}_+. \end{aligned}$$

(18)

Then, Eq. (7) has a unique asymptotically almost periodic mild solution whenever K is small enough.

Proof

First of all, note that using the assumptions on f and the composition of asymptotically almost periodic functions (see [3]) it follows that $F \in AAP({\mathbb {R}}_+ \times {\mathcal {H}}, {\mathcal {H}})$. Furthermore, F is Lipschitzian in the second variable uniformly in the first one with K as a Lipschitz constant.

Consider the nonlinear integral operator $\varGamma$ given on $AAP({\mathcal {H}})$ by

$$\begin{aligned} \varGamma U(t)=T(t)U_0+\int _0^tT(t-s)F(s,U(s))ds, \end{aligned}$$

for any $t\in {\mathbb {R}}_+$.

Using Lemmas 1 and 3 it follows that $\varGamma$ is well-defined and maps $AAP({\mathcal {H}})$ into itself. Now, for any $U,\tilde{U}\in AAP({\mathcal {H}})$ we have

$$\begin{aligned} \Vert \varGamma U(t)-\varGamma \tilde{U}(t)\Vert _{{\mathcal {H}}}\le & {} \int _0^t\Vert T(t-s)\left( F(s,U(s))-F(s,\tilde{U}(s))\right) \Vert _{{\mathcal {H}}}ds\\\le & {} \int _0^tMe^{-\delta (t-s)}\Vert F(s,U(s))-F(s,\tilde{U}(s))\Vert _{{\mathcal {H}}}ds\\\le & {} \,MK\Vert U-V\Vert _\infty \int _0^\infty e^{-\delta s}ds\\= & {}\, \delta ^{-1} MK\Vert U-V\Vert _\infty \qquad \forall \,t\ge 0. \end{aligned}$$

Thus $\varGamma$ is a strict contraction whenever K is small enough, that is, $\delta ^{-1} MK <1$. Therefore, using the Banach contraction mapping theorem, we deduce that $\varGamma$ has a unique fixed point which obviously is the only asymptotically mild solution to Eq. (7). $\square$

Corollary 1

Under assumptions (A.1)–(A.2), suppose that the initial data $u_0 \in {\mathcal {D}}(A)$ and that the function f belongs to $AAP({\mathbb {R}}_+\times H,H)$. Further, suppose that f satisties Eq. (18). Then, the system (1)–(2) has a unique asymptotically almost periodic mild solution whenever $K < \delta M^{-1}$.

6 Example

In order to illustrate our previous abstract results, we consider the so-called Kirchhoff plate equation with infinite memory. For that, let $\varOmega \subset {\mathbb {R}}^n$ be an open bounded subset with smooth boundary $\partial \varOmega$ and let $H=L^2(\varOmega )$ be equipped with its usual $L^2$-norm given, for all $u \in L^2(\varOmega )$, by

$$\begin{aligned} \Vert u\Vert _{L^2(\varOmega )} = \left( \int _{\varOmega }|u(x)|^2 dx\right) ^{\frac{1}{2}}. \end{aligned}$$

Consider the so-called Kirchhoff plate with infinite memory given by

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}(x,t)+\varDelta ^2u(x,t)-\displaystyle \int _{-\infty }^tg(t-s)\varDelta ^2u(x,s)ds=f(t,u(x,t)),&{}{\mathrm {in}}\quad \varOmega \times (0,\infty ),\\ \\ u(x,t)=\varDelta u(x,t)=0,&{}{\mathrm {on}}\quad \partial \varOmega \times [0,\infty ),\\ \\ u(x,-t)=u_0(x,t),\qquad u_t(x,0)=u_1(x)&{}{\mathrm {in}}\quad \varOmega \times (0,\infty ), \end{array}\right. } \end{aligned}$$

(19)

where $u_0$ $u_1$ are given initial data and f belongs to $AAP({\mathbb {R}}_+\times L^2(\varOmega ),L^2(\varOmega ))$ and satisfies Eq. (18).

Consequently, letting $A=-\varDelta$ with domain ${\mathcal {D}}(A)=H^2(\varOmega )\cap H^1_0(\varOmega )$, one can easily see that Eq. (19) is a particular case of our system (1)–(2). Therefore, under assumptions of Corollary 1, we conclude that Eq. (19) has a unique asymptotically almost periodic solution if the Lipschitz constant K is small enough. That is, Eq. (19) has a unique mild solution which converges asymptotically to an almost periodic function.