Dissipative structure and diffusion phenomena for doubly dissipative elastic waves in two space dimensions

In this paper we consider the following Cauchy problem for doubly dissipative elastic waves in two space dimensions:

$$\left\{\begin{aligned} &u_{tt}-a^{2}\Delta u-\left(b^{2}-a^{2}\right)\nabla\operatorname{div}u+(-\Delta)^{\rho}u_{t}+(-\Delta)^{\theta}u_{t}=0,&&x\in\mathbb{R}^{2},\,\,t>0,\\ &(u,u_{t})(0,x)=(u_{0},u_{1})(x),&&x\in\mathbb{R}^{2},\end{aligned}\right.$$

(1.1)

where the unknown $u=u(t,x)\in\mathbb{R}^{2}$ denotes the elastic displacement. The positive constants $a$ and $b$ in (1.1) are related to the Lamé constants and fulfill $b>a>0$. Moreover, the parameters $\rho$ and $\theta$ in (1.1) satisfy $0\leq\rho<1/2<\theta\leq 1$.

Let us recall some related works to our problem (1.1). Taking $a=b=1$, $\rho=0$ and $\theta=1$ in (1.1), then we immediately turn to doubly dissipative wave equation, where the damping terms consist of friction $u_{t}$ as well as viscoelastic damping $-\Delta u_{t}$

$$\left\{\begin{aligned} &u_{tt}-\Delta u+u_{t}-\Delta u_{t}=0,&&x\in\mathbb{R}^{n},\,\,t>0,\\ &(u,u_{t})(0,x)=(u_{0},u_{1})(x),&&x\in\mathbb{R}^{n},\end{aligned}\right.$$

(1.2)

with $n\geq 1$. The recent paper [15] derived asymptotic profiles of solutions to (1.2) in a framework of weighted $L^{1}$ data. Precisely, the authors found that from asymptotic profiles of solutions point of view, friction $u_{t}$ is more dominant than viscoelastic damping $-\Delta u_{t}$ as $t\rightarrow\infty$. Later in [13], the authors obtained higher-order asymptotic expansions of solutions to (1.2) and gave some lower bounds estimates to show the optimality of these expansions. For the other related works on (1.2), we refer the reader to the recent papers [16, 6, 18]. However, asymptotic profiles of solutions to general doubly dissipative wave equation, where the damping terms consist of friction or structural damping (i.e., taking $a=b=1$ in (1.1)), are still open. This open problem is proposed in [15]. The main difficulty is to answer what is the dominant profile of solutions, due to the fact that the asymptotic profiles for wave equation with damping term $(-\Delta)^{\rho}u_{t}$ for $0\leq\rho<1/2$, or with damping term $(-\Delta)^{\theta}u_{t}$ for $1/2<\theta\leq 1$, are quite different. One may see, for example, [24, 22, 23, 9, 27, 28, 34, 11, 7, 14, 25, 26, 17, 33, 29, 8].

Let us come back to dissipative elastic waves. In recent years the Cauchy problem for dissipative elastic waves have aroused wide concern, which can be modeled by

$$\left\{\begin{aligned} &u_{tt}-a^{2}\Delta u-\left(b^{2}-a^{2}\right)\nabla\operatorname{div}u+\mathcal{A}u_{t}=0,&&x\in\mathbb{R}^{n},\,\,t>0,\\ &(u,u_{t})(0,x)=(u_{0},u_{1})(x),&&x\in\mathbb{R}^{n},\end{aligned}\right.$$

(1.3)

where $b>a>0$ and the term $\mathcal{A}u_{t}$ describes several kinds of damping mechanisms.
In the case when

$\displaystyle\mathcal{A}u_{t}=u_{t},\,\,\,\,\text{i.e., \emph{friction} or \emph{external damping}},$

the authors of [12] proved almost sharp energy estimates for $n\geq 2$ by using energy methods in the Fourier space and the Haraux-Komornik inequality, and then the recent paper [5] investigated propagation of singularities, sharp energy estimates and diffusion phenomenon for $n=3$.
Furthermore, in the case when

$\displaystyle\mathcal{A}u_{t}=(-\Delta)^{\theta}u_{t}\,\,\,\,\text{with}\,\,\,\,\theta\in(0,1],\,\,\,\,\text{i.e., \emph{structural damping}},$

energy estimates are derived with different data spaces in [12] for $n\geq 2$, and in [30] for $n=2$. Moreover, some qualitative properties of solutions, including smoothing effect, sharp energy estimate and diffusion phenomena (especially, double diffusion phenomena when $\theta\in(0,1/2)$) are obtained for $n=3$.
Finally, in the case when

$\displaystyle\mathcal{A}u_{t}=(-a^{2}\Delta-(b^{2}-a^{2})\nabla\operatorname{div})u_{t},\,\,\,\,\text{i.e., \emph{Kelvin-Voigt damping}},$

by applying energy methods in the Fourier space, almost sharp energy estimates for $n\geq 2$ have been obtained in [35]. Then, sharp energy estimates, $L^{p}-L^{q}$ estimates as well as asymptotic profiles of solutions are derived for $n=2$ in [3]. Other studies on dissipative elastic waves can be found in literatures [1, 2]. Nevertheless, concerning about decay properties and diffusion phenomena for the Cauchy problem for doubly dissipative elastic waves it seems that we still do not have any previous research manuscripts. Moreover, this problem is strongly related to the open problem proposed in [15]. In this paper we give the answer to the two-dimensional case.

Let us point out that the study of the Cauchy problem (1.1) is not simply a generalization of elastic waves with friction or structural damping in [30, 5]. On one hand, because there exists two different damping terms $(-\Delta)^{\rho}u_{t}$ and $(-\Delta)^{\theta}u_{t}$ with $0\leq\rho<1/2<\theta\leq 1$ in our problem (1.1), it is not clear which damping term has a dominant influence on dissipative structure. On the other hand, from the paper [5], the authors derived diffusion phenomena to elastic waves with the damping term $(-\Delta)^{\theta}u_{t}$ where $\theta\in[0,1/2)\cup(1/2,1]$, which are described by the following so-called reference system.

• •

  In the case when $\theta=0$, the reference system consist of heat-type system with mass term as follows:

  $\displaystyle\widetilde{V}_{t}-\mathcal{D}_{1}\Delta\widetilde{V}+\mathcal{D}_{2}\widetilde{V}=0,$

  with real diagonal matrices $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$.

• •

  In the case when $\theta\in(0,1/2)$, the reference system consist of two different parabolic systems as follows:

  $\displaystyle\widetilde{V}_{t}+\mathcal{D}_{3}(-\Delta)^{1-\theta}\widetilde{V}+\mathcal{D}_{4}(-\Delta)^{\theta}\widetilde{V}=0,$

  with real diagonal matrices $\mathcal{D}_{3}$ and $\mathcal{D}_{4}$.

• •

  In the case when $\theta\in(1/2,1]$, the reference system consist of parabolic system and half-wave system as follows:

  $\displaystyle\widetilde{V}_{t}+\mathcal{D}_{5}(-\Delta)^{\theta}\widetilde{V}+i\mathcal{D}_{6}(-\Delta)^{\frac{1}{2}}\widetilde{V}=0,$

  with real diagonal matrices $\mathcal{D}_{5}$ and $\mathcal{D}_{6}$.

Hence, for different choices of damping terms, which mainly depend on the value of the parameter $\theta$ in the damping term, the diffusion phenomena are quite different. In the Cauchy problem (1.1), the damping terms consist of $(-\Delta)^{\rho}u_{t}$ with $\rho\in[0,1/2)$, and $(-\Delta)^{\theta}u_{t}$ with $\theta\in(1/2,1]$. Thus, it is not clear that the reference system is make up of what kind of evolution systems, and how do two different damping terms influence on diffusion structure. Furthermore, from [5] we know the threshold of diffusion structure is $\theta=1/2$ for elastic waves with structural damping. In other words, the structure of reference system will be changed from $\theta\in(0,1/2)$ to $\theta\in(1/2,1]$. Then, the natural question is what is the threshold of diffusion structure for doubly dissipative elastic waves. Again, we give the answers for these questions in two dimensions.

Our main purpose of the present paper is to investigate dissipative structure and diffusion phenomena for doubly dissipative elastic waves with different assumptions on initial data. We find that the damping term $(-\Delta)^{\rho}u_{t}$ with $0\leq\rho<1/2$ has the dominant influence on energy estimates (see Theorems 3.1 and 3.2). Furthermore, in the case when $\rho+\theta<1$, the damping terms $(-\Delta)^{\rho}u_{t}$ and $(-\Delta)^{\theta}u_{t}$ with $0\leq\rho<1/2<\theta\leq 1$ have the influence on diffusion structure at the same time. However, in the case when $\rho+\theta\geq 1$, the diffusion structure is determined by the damping term $(-\Delta)^{\rho}u_{t}$ with $0\leq\rho<1/2$ only. Hence, one of our novelties is to derive a threshold $\rho+\theta=1$ of diffusion structure for doubly dissipative elastic waves.

This paper is organized as follows. In Section 2 we derive representation of solutions by applying WKB analysis and multistep diagonalization procedure. In Section 3 we obtain pointwise estimate in the Fourier space and energy estimates by using this representation. In Section 4 we derive diffusion phenomena with different assumptions on initial data. Finally, in Section 5 some concluding remarks complete the paper.

In this section we will derive asymptotic behavior of solutions and representation of solutions in the Fourier space. Let us apply the partial Fourier transform with respect to spatial variable such that $\hat{u}(t,\xi)=\mathcal{F}_{x\rightarrow\xi}(u(t,x))$ to obtain

$$\left\{\begin{aligned} &\hat{u}_{tt}+|\xi|^{2}A(\eta)\hat{u}+\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)\hat{u}_{t}=0,&&\xi\in\mathbb{R}^{2},\,\,t>0,\\ &(\hat{u},\hat{u}_{t})(0,\xi)=(\hat{u}_{0},\hat{u}_{1})(\xi),&&\xi\in\mathbb{R}^{2},\end{aligned}\right.$$

(2.4)

where

$\displaystyle A(\eta)=\left({\begin{array}[]{*{20}c}a^{2}+\left(b^{2}-a^{2}\right)\eta_{1}^{2}&\left(b^{2}-a^{2}\right)\eta_{1}\eta_{2}\\ \left(b^{2}-a^{2}\right)\eta_{1}\eta_{2}&a^{2}+\left(b^{2}-a^{2}\right)\eta_{2}^{2}\\ \end{array}}\right)$

with $\eta=\xi/|\xi|\in\mathbb{S}^{1}$. Similar as [30, 3], we introduce the matrix

$$\begin{split}M(\eta):=\left({\begin{array}[]{*{20}c}\eta_{1}&\eta_{2}\\ \eta_{2}&-\eta_{1}\\ \end{array}}\right),\end{split}$$

and define a new variable $W=W(t,\xi)$ such that

$$W:=\left(\begin{aligned} &v_{t}+i|\xi|\text{diag}(b,a)v\\ &v_{t}-i|\xi|\text{diag}(b,a)v\end{aligned}\right),$$

where $v:={M}^{-1}(\eta)\hat{u}$. Moreover, we have $U(t,x)=\mathcal{F}^{-1}_{\xi\rightarrow x}(W(t,\xi))$. Next, the following first-order system can be derived:

$$\left\{\begin{aligned} &W_{t}+\left(\frac{1}{2}{B}_{0}|\xi|^{2\rho}+i{B}_{1}|\xi|+\frac{1}{2}{B}_{0}|\xi|^{2\theta}\right)W=0,&&\xi\in\mathbb{R}^{2},\,\,t>0,\\ &W(0,\xi)=W_{0}(\xi),&&\xi\in\mathbb{R}^{2},\end{aligned}\right.$$

(2.5)

where the coefficient matrices ${B}_{0}$ and ${B}_{1}$ are respectively given by

$$\begin{split}{B}_{0}=\left({\begin{array}[]{*{20}c}1&0&1&0\\ 0&1&0&1\\ 1&0&1&0\\ 0&1&0&1\\ \end{array}}\right)\,\,\,\,\text{and}\,\,\,\,{B}_{1}=\left({\begin{array}[]{*{20}c}-b&0&0&0\\ 0&-a&0&0\\ 0&0&b&0\\ 0&0&0&a\\ \end{array}}\right).\end{split}$$

(2.6)

Let us point out that throughout this section, we will study representation of solutions $U=U(t,x)$ to the following Cauchy problem by deriving representation of its partial Fourier transform $W(t,\xi)=\mathcal{F}_{x\rightarrow\xi}(U(t,x))$:

$$\left\{\begin{aligned} &U_{t}+\frac{1}{2}{B}_{0}(-\Delta)^{\rho}U+i{B}_{1}(-\Delta)^{\frac{1}{2}}U+\frac{1}{2}{B}_{0}(-\Delta)^{\theta}U=0,&&x\in\mathbb{R}^{2},\,\,t>0,\\ &U(0,x)=U_{0}(x),&&x\in\mathbb{R}^{2},\end{aligned}\right.$$

(2.7)

where the coefficient matrices $B_{0}$ and $B_{1}$ are given in (2.6). Moreover, to derive qualitative properties of solutions to (1.1), we only need to study the solutions to (2.7).

With the aim of deriving representation of solutions, we may apply WKB analysis and multistep diagonalization procedure (see for example [31, 36, 20, 19, 21, 30, 4]). Before doing these, we should understand the influence of the parameter $|\xi|$ on the asymptotic behavior of solutions to (2.5). Due to our assumption $0\leq 2\rho<1<2\theta\leq 2$, we now discuss the influence of $|\xi|$ by three parts. Specifically, we will apply diagonalization procedure for small frequencies $\xi\in\mathcal{Z}_{\operatorname{int}}(\varepsilon)$ and large frequencies $\xi\in\mathcal{Z}_{\operatorname{ext}}(N)$ in Subsections 2.1 and 2.2, respectively. Then, the contradiction argument will be applied to prove an exponential stability of solutions for bounded frequencies $\xi\in\mathcal{Z}_{\text{bdd}}(\varepsilon,N)$ in Subsection 2.3.