Spatial Behaviour of Solutions of the Moore-Gibson-Thompson Equation

Abstract

In this note we study the spatial behaviour of the Moore-Gibson-Thompson equation. As it is a hyperbolic equation, we prove that the solutions do not grow along certain spatial-time lines. Given the presence of dissipation, we show that the solutions also decay exponentially in certain directions.

Appendix

In the development of the paper we have assumed the existence, uniqueness and regularity of the solutions to the proposed problem (3-6). To be precise we have used that

$$\begin{aligned} \hat{u}\in W^{2,2}(R);\, \, \, \tilde{u}, \, u, \, \, \dot{u}\in W_0^{1,2}(R). \end{aligned}$$

(51)

In this appendix we prove that these properties hold whenever we assume that

$$\begin{aligned} g\in C^4\left( 0,T,W^{2,2}(D) \cap W_0^{1.2}(D )\right) , \end{aligned}$$

(52)

as well as

$$\begin{aligned} g\left( x_1,x_2,0,0\right) =u^0\left( x_1,x_2,0\right) ,\, \, \, \dot{g}\left( x_1,x_2,0,0\right) = \vartheta ^0\left( x_1,x_2,0\right) ,\, \, \, \ddot{g}\left( x_1,x_2,0,0\right) =\phi ^0\left( x_1,x_2,0\right) ,\, \, \, \end{aligned}$$

(53)

and

$$\begin{aligned}&k^* u^0+k\vartheta ^0 \in W^{2,2}(R), \, \, \, \, u^0-{g} \left( x_1,x_2,0\right) \exp \left( -{ \gamma } x_3\right) \in W_0^{1,2}(R),\nonumber \\&\quad \vartheta ^0 -\dot{g} (x_1,x_2,0) \exp (-{ \gamma } x_3),\, \, \phi ^0-\ddot{g} (x_1,x_2,0) \exp (-{ \gamma } x_3)\in W_0^{1,2}(R), \end{aligned}$$

(54)

where ${ \gamma }$ is an arbitrary positive constant.

First, we will need the following result:

Theorem 6.1

Let us consider the problem determined by the equation

$$\begin{aligned} \tau \dddot{w}+\ddot{w}-k\triangle \dot{w}-k^{*}\triangle w=f(\mathbf {x} ,t), \end{aligned}$$

(55)

the initial conditions

$$\begin{aligned} w(\mathbf {x},0)=a^{0}(\mathbf {x}),\,\,\dot{w}(\mathbf {x},0)=b^{0}(\mathbf {x} ),\,\,\ddot{w}(\mathbf {x},0)=c^{0}(\mathbf {x}), \end{aligned}$$

(56)

and the boundary conditions

$$\begin{aligned} w(\mathbf {x},t) =0,\,\,\,\mathbf {x}\in \partial R \times [0,\infty ), \end{aligned}$$

(57)

where $a^0 (\mathbf {x})\in W_0^{1,2} (R),\, \, \, b^0(\mathbf {x})\in W_0^{1,2}(R),\, \, \, c^0(\mathbf {x}) \in W_0^{1,2} (R) \text{ and } k^*a^0 (\mathbf {x})+k b^0(\mathbf {x})\in L^2(R).$ We also assume that $f(\mathbf {x} ,t)\in C^1(0,T, L^2(R))$. Then, there exists a unique solution $w(\mathbf {x} ,t) \in W_0^{1,2}(R)$ such that $\dot{w}(\mathbf {x} ,t) \in W_0^{1,2}(R)$ , $\ddot{w}(\mathbf {x} ,t) \in L^{2}(R)$, $\hat{w}(\mathbf {x} ,t) \in W^{2,2}(R)$, $\tilde{w}(\mathbf {x} ,t) \in W_0^{1,2}(R)$

Proof

We consider the Hilbert space ${\mathcal {H}} =W_{0}^{1,2}(R)\times W_{0}^{1,2}(R)\times L^{2}(R)$ and we can write our problem as

$$\begin{aligned} \frac{d}{dt} \begin{pmatrix} w \\ \theta \\ \varphi \end{pmatrix} ={\mathcal {A}} \begin{pmatrix} w \\ \theta \\ \varphi \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ f \end{pmatrix} ,\,\,\,w(0)=a^{0}(\mathbf {x}),\,\,\theta (0)=b^{0}(\mathbf {x}),\,\,\varphi (0)=c^{0}(\mathbf {x}), \end{aligned}$$

(58)

where

$$\begin{aligned} {\mathcal {A}} \begin{pmatrix} w \\ \theta \\ \varphi \end{pmatrix} = \begin{pmatrix} \theta \\ \varphi \\ \tau ^{-1}[k^{*}\Delta w+k\Delta \theta -\varphi ] \end{pmatrix} . \end{aligned}$$

(59)

The domain of this operator is the subspace of ${\mathcal {H}}$ such that $k^{*}\Delta w+k\Delta \theta \in L^{2}(R)$ and $\varphi \in W_{0}^{1,2}(R)$. Following the arguments proposed in ([25], p. 4025), we can prove that ${\mathcal {A}}$ generates a $C^{0}$-semigroup of contractions. As $f(\mathbf {x},t)\in C^{1}(0,T,L^{2}(R))$ we can conclude the existence and uniqueness of solutions (see [2], p.117). We also know that the solutions belong to the domain of ${\mathcal {A}}$ and then the required regularity conditions are satisfied.

Theorem 6.2

Let us assume that the boundary and initial datas satisfy the conditions proposed at the begin of the appendix (52-54). Then, there exists a unique solution to the problem (3-6). Moreover, this solution satisfies the regularity conditions.

Proof

We define the function $H(\mathbf {x},t)=g(x_1,x_2,t) \exp (- { \gamma } x_3)$ where ${ \gamma }$ is an arbitrary positive number. We denote by w(t) the solution of the problem (55-57) when

$$\begin{aligned} f(\mathbf {x},t)=\tau \dddot{H}+\ddot{H}-k\Delta \dot{ H}-k^*\Delta H, \end{aligned}$$

and

$$\begin{aligned} a^0(\mathbf {x})=u^0(\mathbf {x})-H(\mathbf {x},0),\, \, \, b^0(\mathbf {x} )=\vartheta ^0(\mathbf {x})-\dot{H}(\mathbf {x},0),\, \, \, c^0(\mathbf {x} )=\phi ^0(\mathbf {x})-\ddot{H}(\mathbf {x},0). \end{aligned}$$

We note that $f(\mathbf {x},t)$ and $a^0(\mathbf {x}),b^0(\mathbf {x}),c^0( \mathbf {x})$ satisfy the conditions of the previous theorem.

The solution to the problem (3-6) can be obtained by taking

$$\begin{aligned} u(\mathbf {x},t)=w(\mathbf {x},t)+H(\mathbf {x},t). \end{aligned}$$

(60)

Existence, uniqueness and regularity of the solution are clearly checked in view of the previous theorem.