Spatial Decay Estimates for the Moore-Gibson-Thompson Heat Equation

Jincheng SHI

doi:10.1051/wujns/2024295397

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Volume 29 / No 5 (October 2024)

Wuhan Univ. J. Nat. Sci., 29 5 (2024) 397-402

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Issue		Wuhan Univ. J. Nat. Sci. Volume 29, Number 5, October 2024


Page(s)		397 - 402
DOI		https://doi.org/10.1051/wujns/2024295397
Published online		20 November 2024

Wuhan University Journal of Natural Sciences, 2024, Vol.29 No.5, 397-402

Mathematics

CLC number: O175.29

Spatial Decay Estimates for the Moore-Gibson-Thompson Heat Equation

Moore-Gibson-Thompson热方程的空间衰减估计

Jincheng SHI (石金诚)

School of Artificial Intelligence, Guangzhou Huashang College, Guangzhou 511300, Guangdong, China

Received: 26 December 2023

Abstract

In this article, the Moore-Gibson-Thompson heat equation in three-dimensional cylindrical domain are studied. Using a second order differential inequality, we obtain that the solution can decay exponentially as the distance from the entry section tends to infinity. Our result can be seen as a version of Saint-Venant principle.

摘要

本文研究三维柱形区域中的Moore-Gibson-Thompson热方程, 利用二阶微分不等式, 得到当离入口端的距离趋于无穷大时, 方程的解呈指数衰减。该结果可以看成是Saint-Venant原理的一个具体应用。

Key words: decay estimates / Moore-Gibson-Thompson heat equation / Saint-Venant principle

关键字 : 衰减估计 / Moore-Gibson-Thompson热方程 / Saint-Venant原理

Cite this article: SHI Jincheng. Spatial Decay Estimates for the Moore-Gibson-Thompson Heat Equation[J]. Wuhan Univ J of Nat Sci, 2024, 29(5): 397-402.

Biography: SHI Jincheng, male, Associate professor, research direction: partial differential equation. E-mail: hning0818@163.com

Fundation item: Supported by the National Natural Science Foundation of China (11371175), the Research Team of Guangzhou Huashang College (2021HSKT01), and Guangzhou Huashang College Mentorship Program(2020HSDS16)

© Wuhan University 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

In Ref.[1], the authors studied the structural stability for the Moore-Gibson-Thompson (M-G-T) heat equation. They obtained the results of the continuous and convergence for different coefficients in a bounded domain. These results gave measures of the structural stability between the solutions for different theories. The so-called M-G-T heat equation has the form:

$C (τ \overset{⃛}{α} + \ddot{α}) = k^{*} Δ α + k Δ \dot{α},$ (1)

where $α$ denotes the thermal displacement, $k$ is the thermal conductivity, $k^{*}$ denotes the conductivity rate, $C$ and $τ$ denote the thermal capacity and the relaxation parameter, respectively. In Refs.[2] and [3], Green and Nagdhi proposed that the motivation for considering (1) as a heat equation came from the heat conduction of type III.

The M-G-T heat equation and similar equations have deserved much interest in the last years, see Refs. [4-8]. In the present paper, equation (1) is defined in a semi-infinite cylindrical pipe $R$ with boundary $\partial R$ . The pipe has arbitrary cross section denotes by $D$ and the boundary $\partial D$ and the generators of the pope are parallel to the $x_{3}$ axis. We introduce the notations:

$R_{z} = {(x_{1}, x_{2}, x_{3}) | (x_{1}, x_{2}) \in D, x_{3} > z \geq 0}, D_{z} = {(x_{1}, x_{2}, x_{3}) | (x_{1}, x_{2}) \in D, x_{3} = z \geq 0},$

where $z$ is a running variable along the $x_{3}$ axis. Clearly, $R_{0} = R$ , $D_{0} = D$ .

The lateral sides of the cylinder are constrained to have zero thermal displacement. Thus, we have

$α = 0 o n \partial D \times (0, + \infty) = D .$ (2)

To have a well-determined problem, we impose boundary conditions on the finite end of the cylinder. Thus, we take as assumption

$α (x_{1}, x_{2}, 0, t) = f_{i} (x_{1}, x_{2}, t) o n (x_{1}, x_{2}) \in D, t > 0 .$ (3)

We give the initial conditions:

$α (x_{1}, x_{2}, x_{3}, 0) = \dot{α} (x_{1}, x_{2}, x_{3}, 0) = \ddot{α} (x_{1}, x_{2}, x_{3}, 0) = 0 .$ (4)

The classical Phragmén-Lindelöf theorem, without some a priori asymptotical decay assumptions for solution, states that harmonic function which vanishes on the cylindrical surface must either grow or decay exponentially with the distance from the finite end of the cylinder. In this paper, we add some a priori asymptotical decay assumptions for solution at the infinity. We can get the spatial decay estimates result. It can be seen as a version of Saint-Venant principle. For a review of results of the Saint-Venant principle have been published recently (in several situations), one could see Refs.[9-14]. In the present article, the comma is used to indicate partial differentiation and the differentiation with respect to the direction $x_{i}$ is denoted as " $, i$ ", thus $α_{, i}$ denotes $\frac{\partial α}{\partial x_{i}}$ , and $\dot{α}$ denotes $\frac{\partial α}{\partial t}$ . The usual summation convection is employed with repeated Latin subscripts $i$ summed from 1 to 3. Hence, $α_{, i} α_{, i} = α_{, 1}^{2} + α_{, 2}^{2} + α_{, 3}^{2} .$

1 The Function Expression $φ (z, t)$

In order to obtain the decay estimates result, we must define a function $φ (z, t)$ for the solutions. The function will be constructed by the following Lemmas.

Lemma 1

Let $α$ be classical solution of equation (1) and satisfy the initial boundary value problems (2)-(4), we define a function

$φ_{1} (z, t) = k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \dot{α} α_{, 3} d x d η + k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \dot{α} {\dot{α}}_{, 3} d x d η .$ (5)

$φ_{1} (z, t)$ can also be expressed as

$\begin{array}{l} φ_{1} (z, t) = - C τ \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + \frac{C τ ω^{2}}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\dot{α})}^{2} d x d η \\ + C τ \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) \dot{α} \ddot{α} d x + \frac{C ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\dot{α})}^{2} d x d η \\ + \frac{C}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\dot{α})}^{2} d x + \frac{k^{*} ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) α_{, i} α_{, i} d x d η \\ + \frac{k^{*}}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) α_{, i} α_{, i} d x + \frac{k^{*} ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η . \end{array}$ (6)

Proof

Multiplying (1) by $\dot{α}$ and integrating, we have

$\begin{array}{l} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) \dot{α} [C (τ \overset{⃛}{α} + \ddot{α}) - k^{*} Δ α + k Δ \dot{α}] d x d η \\ = - C τ \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + C τ ω \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) \dot{α} \ddot{α} d x d η + C τ \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) \dot{α} \ddot{α} d x \\ + \frac{C ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\dot{α})}^{2} d x d η + \frac{C}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\dot{α})}^{2} d x + k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} α_{, i} d x d η \\ - k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \dot{α} α_{, 3} d x d η + k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η - k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \dot{α} {\dot{α}}_{, 3} d x d η \\ = - C τ \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + \frac{C τ ω^{2}}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\dot{α})}^{2} d x d η \\ + \frac{C τ ω}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\dot{α})}^{2} d x + C τ \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) \dot{α} \ddot{α} d x + \frac{C ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\dot{α})}^{2} d x d η \end{array}$

$\begin{array}{l} + \frac{C}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\dot{α})}^{2} d x + \frac{k^{*} ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) α_{, i} α_{, i} d x d η + \frac{k^{*}}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) α_{, i} α_{, i} d x \\ - k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \dot{α} α_{, 3} d x d η + k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η - k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \dot{α} {\dot{α}}_{, 3} d x d η = 0 . \end{array}$ (7)

From (5) and (7), we can obtain (6).

Lemma 2

Let $α$ be classical solution of equation (1) and satisfy the initial boundary value problems (2)-(4), we define a function

$φ_{2} (z, t) = k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} α_{, 3} d x d η + k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} {\dot{α}}_{, 3} d x d η .$ (8)

$φ_{2} (z, t)$ can also be expressed as

$\begin{array}{l} φ_{2} (z, t) = (\frac{C τ ω}{2} + C) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + \frac{C τ}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\dot{α})}^{2} d x \\ + (\frac{k ω}{2} - k^{*}) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η + \frac{k^{*} ω^{2}}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) α_{, i} α_{, i} d x d η \\ + \frac{k^{*} ω}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) α_{, i} α_{, i} d x + k^{*} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} α_{, i} d x + \frac{k}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x . \end{array}$ (9)

Proof

Multiplying (1) by $\ddot{α}$ and integrating, we have

$\begin{array}{l} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) \ddot{α} [C (τ \overset{⃛}{α} + \ddot{α}) - k^{*} Δ α + k Δ \dot{α}] d x d η \\ = \frac{C τ ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + C \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η \\ + \frac{C τ}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\ddot{α})}^{2} d x + k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\ddot{α}}_{, i} α_{, i} d x d η - k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} α_{, 3} d x d η \\ + k \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\ddot{α}}_{, i} α_{, i} d x - k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} α_{, 3} d x d η \\ = (\frac{C τ ω}{2} + C) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + \frac{C τ}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\ddot{α})}^{2} d x - k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η \\ + k^{*} ω \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} α_{, i} d x d η + k^{*} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} α_{, i} d x - k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} α_{, 3} d x d η \\ + \frac{k ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η + \frac{k}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x - k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} α_{, 3} d x d η \\ = (\frac{C τ ω}{2} + C) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + \frac{C τ}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\ddot{α})}^{2} d x \\ - k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η + \frac{k^{*} ω^{2}}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) α_{, i} α_{, i} d x d η \\ + \frac{k^{*} ω}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) α_{, i} α_{, i} d x + k^{*} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} α_{, i} d x - k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} α_{, 3} d x d η \\ + \frac{k ω}{2} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η + \frac{k}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x - k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} α_{, 3} d x d η = 0 . \end{array}$ (10)

From (8) and (10), we get the desired results (9).

We define a new function $φ (z, t)$ :

$φ (z, t) = \tilde{k} φ_{1} (z, t) + φ_{2} (z, t),$ (11)

where $\tilde{k}$ is an arbitrary positive constant which will be defined later.

From (5) and (8), we have

$\begin{array}{l} φ (z, t) = k^{*} \tilde{k} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \dot{α} α_{, 3} d x d η - \frac{k \tilde{k}}{2} \int_{0}^{t} \int_{D_{z}}^{} e x p (- ω η) {(\dot{α})}^{2} d A d η \\ + k^{*} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} α_{, 3} d x d η + k \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) \ddot{α} {\dot{α}}_{, 3} d x d η . \end{array}$ (12)

From (6) and (9), $φ (z, t)$ can also be rewritten as

$\begin{array}{l} φ (z, t) = (\frac{C τ ω}{2} + C - C τ \tilde{k}) \int_{}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + \frac{C τ}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\ddot{α})}^{2} d x \\ + (\frac{k ω}{2} - k^{*} + k \tilde{k}) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η + (\frac{k^{*} ω^{2}}{2} + \frac{k^{*} ω \tilde{k}}{2}) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) α_{, i} α_{, i} d x d η \\ + (\frac{k^{*} ω}{2} + \frac{k^{*} \tilde{k}}{2}) \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) α_{, i} α_{, i} d x + \frac{k}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x \\ + (\frac{C τ ω^{2}}{2} + \frac{C ω}{2}) \tilde{k} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\dot{α})}^{2} d x d η + C τ \tilde{k} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) \dot{α} \ddot{α} d x \\ + \frac{C \tilde{k}}{2} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\dot{α})}^{2} d x + k^{*} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} α_{, i} d x \end{array}$ (13)

We will use the function $φ (z, t)$ to obtain our main result in the next section.

2 Spatial Decay Estimates

Differentiating (13) and using the Schwarz's inequality, we have

$\begin{array}{l} - \frac{\partial φ (z, t)}{\partial z} \geq (\frac{C τ ω}{2} + C - C τ \tilde{k}) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) {(\ddot{α})}^{2} d x d η + (\frac{C τ}{2} - C τ^{2} \tilde{k}) \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) {(\ddot{α})}^{2} d x \\ + (\frac{k ω}{2} - k^{*} + k \tilde{k}) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η + (\frac{k^{*} ω^{2}}{2} + \frac{k^{*} ω \tilde{k}}{2}) \int_{0}^{t} \int_{D_{z}}^{} e x p (- ω η) α_{, i} α_{, i} d A d η \\ + [\frac{k^{*} ω}{2} + \frac{k^{*} \tilde{k}}{2} - \frac{{(k^{*})}^{2}}{k}] \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) α_{, i} α_{, i} d x + \frac{k}{4} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) {\dot{α}}_{, i} {\dot{α}}_{, i} d x \\ + \frac{C \tilde{k}}{4} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) {(\dot{α})}^{2} d x + (\frac{C τ ω^{2}}{2} + \frac{C ω}{2}) \tilde{k} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) {(\dot{α})}^{2} d x d η . \end{array}$ (14)

If we choose $\tilde{k} = \frac{1}{4 τ}$ , $ω = \frac{4 k^{*}}{k}$ , we have

$(\frac{C τ ω}{2} + C - C τ \tilde{k}) > 0, (\frac{C τ}{2} - C τ^{2} \tilde{k}) > 0, \frac{k ω}{2} - k^{*} + k \tilde{k} > 0, \frac{k^{*} ω}{2} + \frac{k^{*} \tilde{k}}{2} - \frac{{(k^{*})}^{2}}{k} > 0 .$

Using the same method, we have

$\begin{array}{l} \frac{\partial^{2} φ (z, t)}{\partial z^{2}} \geq \frac{C τ ω}{4} \int_{0}^{t} \int_{D_{z}}^{} e x p (- ω η) {(\ddot{α})}^{2} d A d η + \frac{C τ}{4} \int_{D_{z}}^{} e x p (- ω t) {(\ddot{α})}^{2} d A + \frac{k ω}{4} \int_{0}^{t} \int_{D z}^{} e x p (- ω η) {\dot{α}}_{, i} {\dot{α}}_{, i} d A d η \\ + (\frac{k^{*} ω^{2}}{2} + \frac{k^{*} ω \tilde{k}}{2}) \int_{0}^{t} \int_{D_{z}}^{} e x p (- ω η) α_{, i} α_{, i} d A d η + \frac{k^{*} ω}{4} \int_{D_{z}}^{} e x p (- ω t) α_{, i} α_{, i} d A + \frac{k}{4} \int_{D_{z}}^{} e x p (- ω t) {\dot{α}}_{, i} {\dot{α}}_{, i} d A \\ + \frac{C \tilde{k}}{4} \int_{D_{z}}^{} e x p (- ω t) {(\dot{α})}^{2} d A + (\frac{C τ ω^{2}}{2} + \frac{C ω}{2}) \tilde{k} \int_{0}^{t} \int_{D_{z}}^{} e x p (- ω η) {(\dot{α})}^{2} d A d η . \end{array}$ (15)

We can easily get

$\begin{array}{l} φ (z, t) \geq \frac{C τ ω}{4} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\ddot{α})}^{2} d x d η + \frac{C τ}{4} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\ddot{α})}^{2} d x \\ + \frac{k ω}{4} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x d η + (\frac{k^{*} ω^{2}}{2} + \frac{k^{*} ω \tilde{k}}{2}) \int_{0}^{t} \int_{z}^{\infty} \int_{D_{z}}^{} e x p (- ω η) (ξ - z) α_{, i} α_{, i} d x d η \\ + \frac{k^{*} ω}{4} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) α_{, i} α_{, i} d x + \frac{k}{4} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {\dot{α}}_{, i} {\dot{α}}_{, i} d x \\ + \frac{C \tilde{k}}{4} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω t) (ξ - z) {(\dot{α})}^{2} d x + (\frac{C τ ω^{2}}{2} + \frac{C ω}{2}) \tilde{k} \int_{0}^{t} \int_{z}^{\infty} \int_{D_{ξ}}^{} e x p (- ω η) (ξ - z) {(\dot{α})}^{2} d x d η = E (z, t) . \end{array}$ (16)

Combining (12), (14) and (15), we have

$φ (z, t) \leq k_{1} (- \frac{\partial φ}{\partial z}) + k_{2} (\frac{\partial^{2} φ}{\partial z^{2}}),$ (17)

where $k_{1}$ , $k_{2}$ are positive computable constants.

We can rewrite (17) as

$\frac{\partial^{2} φ}{\partial z^{2}} - \frac{k_{1}}{k_{2}} \frac{\partial φ}{\partial z} - \frac{1}{k_{2}} \geq 0 .$ (18)

We want to obtain the following result

$(\frac{\partial}{\partial z} - a) (\frac{\partial φ}{\partial z} + b) \geq 0 .$ (19)

Combining (18), (19) and (15), we obtain

$b - a = - \frac{k_{1}}{k_{2}}, a b = \frac{1}{k_{2}} .$

We obtain

$a = \frac{\sqrt[]{{(\frac{k_{1}}{k_{2}})}^{2} + \frac{4}{k_{2}}} + \frac{k_{1}}{k_{2}}}{2}, b = \frac{\sqrt[]{{(\frac{k_{1}}{k_{2}})}^{2} + \frac{4}{k_{2}}} - \frac{k_{1}}{k_{2}}}{2} .$

From (19), we have

$\frac{\partial}{\partial z} {e^{- a z} (\frac{\partial φ}{\partial z} + b φ)} \geq 0 .$ (20)

Integrating (20) with respect to $z$ from $z$ to $\infty$ , we have

$\frac{\partial φ (z, t)}{\partial z} + b φ (z, t) \leq 0 .$ (21)

Solving inequality (21), we easily get

$φ (z, t) \leq φ (0, t) e^{- b z} .$ (22)

Inserting (16) into (22), we obtain

$E (z, t) \leq φ (0, t) e^{- b z} .$ (23)

Inequality (23) implies that the energy can decay as $z \to \infty$ .We thus have proved the following theorem:

Theorem 1 Let $α$ be classical solution of equation (1) and satisfy the initial boundary value problems (2)-(4), we can get the decay estimates $E (z, t) \leq φ (0, t) e^{- b z} .$

References

Pellicer M, Quintanilla R. Continuous dependence and convergence for Moore-Gibson-Thompson heat equation[J]. Acta Mechanica, 2023, 234(8): 3241-3257. [CrossRef] [MathSciNet] [Google Scholar]
Green A E, Naghdi P M. On undamped heat waves in an elastic solid[J]. Journal of Thermal Stresses, 1992, 15(2): 253-264. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Green A E, Naghdi P M. Thermoelasticity without energy dissipation[J]. Journal of Elasticity, 1993, 31(3): 189-208. [CrossRef] [MathSciNet] [Google Scholar]
Chen W H, Ikehata R. The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case[J]. Journal of Differential Equations, 2021, 292: 176-219. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Chen W H, Dao T A. The Cauchy problem for the nonlinear viscous Boussinesq equation in the L framework[J]. Journal of Differential Equations, 2022, 320: 558-597. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Chen W H. Cauchy problem for thermoelastic plate equations with different damping mechanisms[J]. Communications in Mathematical Sciences, 2020, 18(2): 429-457. [CrossRef] [MathSciNet] [Google Scholar]
Liu Y, Qin X L, Zhang S H. Global existence and estimates for Blackstock's model of thermoviscous flow with second sound phenomena[J]. Journal of Differential Equations, 2022, 324(2): 76-101. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Liu Y, Shi J C. Coupled plate equations with indirect damping: Smoothing effect, decay properties and approximation[J]. Zeitschrift Für Angewandte Mathematik und Physik, 2021, 73(1): 11. [Google Scholar]
Liu Y, Qin X L, Shi J C, et al. Structural stability of the Boussinesq fluid interfacing with a Darcy fluid in a bounded region in R²[J]. Appl Math Comput, 2021, 411: 126488. [Google Scholar]
Liu Y, Xiao S Z. Structural stability for the Brinkman fluid interfacing with a Darcy fluid in an unbounded domain[J]. Nonlinear Analysis: Real World Applications, 2018, 42: 308-333. [CrossRef] [MathSciNet] [Google Scholar]
Liu Y. Continuous dependence for a thermal convection model with temperature-dependent solubility[J]. Applied Mathematics and Computation, 2017, 308: 18-30. [CrossRef] [MathSciNet] [Google Scholar]
Li Y F, Xiao S Z, Zeng P. The applications of some basic mathematical inequalities on the convergence of the primitive equations of moist atmosphere[J]. Journal of Mathematical Inequalities, 2021, 15(1): 293-304. [CrossRef] [MathSciNet] [Google Scholar]
Li Y F, Chen X J, Shi J C. Structural stability in resonant penetrative convection in a Brinkman-Forchheimer fluid interfacing with a Darcy fluid[J]. Applied Mathematics & Optimization, 2021, 84(1): 979-999. [CrossRef] [MathSciNet] [Google Scholar]
Li Y F, Chen X J. Phragmén-Lindelöf alternative results in time-dependent double-diffusive Darcy plane flow[J]. Mathematical Methods in the Applied Sciences, 2022, 45(11): 6982-6997. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

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[1] Pellicer M, Quintanilla R. Continuous dependence and convergence for Moore-Gibson-Thompson heat equation[J]. Acta Mechanica, 2023, 234(8): 3241-3257. [CrossRef] [MathSciNet] [Google Scholar]

[2] Green A E, Naghdi P M. On undamped heat waves in an elastic solid[J]. Journal of Thermal Stresses, 1992, 15(2): 253-264. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[3] Green A E, Naghdi P M. Thermoelasticity without energy dissipation[J]. Journal of Elasticity, 1993, 31(3): 189-208. [CrossRef] [MathSciNet] [Google Scholar]

[4] Chen W H, Ikehata R. The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case[J]. Journal of Differential Equations, 2021, 292: 176-219. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[5] Chen W H, Dao T A. The Cauchy problem for the nonlinear viscous Boussinesq equation in the L framework[J]. Journal of Differential Equations, 2022, 320: 558-597. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[6] Chen W H. Cauchy problem for thermoelastic plate equations with different damping mechanisms[J]. Communications in Mathematical Sciences, 2020, 18(2): 429-457. [CrossRef] [MathSciNet] [Google Scholar]

[7] Liu Y, Qin X L, Zhang S H. Global existence and estimates for Blackstock's model of thermoviscous flow with second sound phenomena[J]. Journal of Differential Equations, 2022, 324(2): 76-101. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[8] Liu Y, Shi J C. Coupled plate equations with indirect damping: Smoothing effect, decay properties and approximation[J]. Zeitschrift Für Angewandte Mathematik und Physik, 2021, 73(1): 11. [Google Scholar]

[9] Liu Y, Qin X L, Shi J C, et al. Structural stability of the Boussinesq fluid interfacing with a Darcy fluid in a bounded region in R²[J]. Appl Math Comput, 2021, 411: 126488. [Google Scholar]

[10] Liu Y, Xiao S Z. Structural stability for the Brinkman fluid interfacing with a Darcy fluid in an unbounded domain[J]. Nonlinear Analysis: Real World Applications, 2018, 42: 308-333. [CrossRef] [MathSciNet] [Google Scholar]

[11] Liu Y. Continuous dependence for a thermal convection model with temperature-dependent solubility[J]. Applied Mathematics and Computation, 2017, 308: 18-30. [CrossRef] [MathSciNet] [Google Scholar]

[12] Li Y F, Xiao S Z, Zeng P. The applications of some basic mathematical inequalities on the convergence of the primitive equations of moist atmosphere[J]. Journal of Mathematical Inequalities, 2021, 15(1): 293-304. [CrossRef] [MathSciNet] [Google Scholar]

[13] Li Y F, Chen X J, Shi J C. Structural stability in resonant penetrative convection in a Brinkman-Forchheimer fluid interfacing with a Darcy fluid[J]. Applied Mathematics & Optimization, 2021, 84(1): 979-999. [CrossRef] [MathSciNet] [Google Scholar]

[14] Li Y F, Chen X J. Phragmén-Lindelöf alternative results in time-dependent double-diffusive Darcy plane flow[J]. Mathematical Methods in the Applied Sciences, 2022, 45(11): 6982-6997. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

Spatial Decay Estimates for the Moore-Gibson-Thompson Heat Equation

Moore-Gibson-Thompson热方程的空间衰减估计

0 Introduction

1 The Function Expression φ(z,t)

2 Spatial Decay Estimates

References

1 The Function Expression $φ (z, t)$