Spatial Decay Estimates for the Moore–Gibson–Thompson Heat Equation Based on an Integral Differential Inequality

Abstract

The present work investigates the spatial evolution characteristics of solutions to the Moore–Gibson–Thompson heat equation within a three-dimensional cylindrical geometry. By constructing an integral-differential inequality framework, we establish rigorous estimates demonstrating the exponential spatial decay of the solution as the axial distance from the inlet boundary increases without bound. This finding aligns with a generalized interpretation of the Saint-Venant principle, demonstrating its applicability under the present asymptotic conditions. The integral-differential inequality method proposed in this paper can also be used for the study of the Saint-Venant principle for other equations.

1. Introduction

Over the last fifty years, many authors studied the Saint-Venant principle both in applied mathematics and mechanics. The classical Saint-Venant principle has undergone substantial theoretical extension through extensive investigative efforts. For the systematic examination of advancements in this domain, readers are directed to foundational works by Horgan [1,2] and collaborative studies with Knowles [3]. Established Saint-Venant-type theorems demonstrate that energy experiences exponential decay with increasing axial distance from the proximal end toward infinity along a semi-infinite strip or cylindrical geometries (cf. [4,5,6,7,8,9,10]). A critical prerequisite for deriving these decay properties involves imposing the a priori condition that solutions exhibit algebraic attenuation to zero at asymptotic distances. Recently, some authors have begun to study the Phragmén–Lindelöf alternative principle. They studied the spatial properties of the solutions of the biharmonic equations. In this case, they did not give any decay assumptions for the solutions. They obtained the result of the classical Phragmén–Lindelöf theorem. When the distance tends to infinity, the energy either grows exponentially or decays exponentially. For a comprehensive understanding of the research of Phragmén–Lindelöf in recent years, one can see [11,12,13,14,15].

The following presents a historical context for the central theme of this work—the Moore–Gibson–Thompson (MGT) equation. Early investigations into nonlinear acoustics, particularly those examining second sound phenomena in viscous thermally relaxing fluids (as initially explored in seminal works [16,17,18]), adopted an approximate framework based on the fully compressible Navier–Stokes–Cattaneo system under irrotational flow conditions. Specifically, through the implementation of Lighthill’s perturbation methodology, which systematically retains first- and second-order terms under small deviations from a constant equilibrium state, the governing equations now recognized as the Jordan-Moore–Gibson–Thompson model were formally derived (cf. [19,20] and associated literature):

$$\begin{array}{c}\hfill \tau {\psi }_{ttt}+{\psi }_{tt}-\mathrm{\Delta }\psi -(\delta +\tau )\mathrm{\Delta }{\psi }_t=F({\psi }_t,\nabla \psi ).\end{array}$$

(1)

Here, $\mathrm{\Delta }$ is the harmonic operator, and ∇ is the gradient operator.

The corresponding linearization of the J-M-G-T Equation (1), i.e., the M-G-T Equation (see [21,22,23] and references therein), is given by

$$\begin{array}{c}\hfill \tau {\psi }_{ttt}+{\psi }_{tt}-\mathrm{\Delta }\psi -(\delta +\tau )\mathrm{\Delta }{\psi }_t=0,\end{array}$$

(2)

which has been established in the early works of F.K. Moore, W.E. Gibson [24] in 1960, and P.A. Thompson [25] in 1972. These scalar equations and their related mathematical models have been extensively considered in medical and industrial applications of high-intensity ultra sound, for instance, medical imaging and therapy, ultrasound cleaning, and welding (see [26,27,28] and the references given therein). Nowadays, it has been used in the study of heat transfer or the heat conduction equation for a thermoelastic theory (see [29] for its introduction).

The structural stability analysis of the Moore–Gibson–Thompson (MGT) heat equation has been systematically investigated in [29]. In bounded domains, rigorous findings concerning the continuity and convergence properties for varying coefficient configurations were established. These theoretical results provide quantitative measures for assessing structural stability discrepancies between solutions derived from distinct theoretical frameworks. The mathematical formulation of the aforementioned MGT heat equation is given as follows:

$$\begin{array}{c}\hfill C(\tau \stackrel{⃛}{\alpha }+\ddot{\alpha })=k^{*}\mathrm{\Delta }\alpha +k\mathrm{\Delta }\dot{\alpha },\end{array}$$

(3)

where $\alpha$ represents the thermal displacement variable, while k corresponds to the thermal conductivity coefficient. The parameter k signifies the rate of conductivity, with C and $\tau$, respectively, denoting the thermal capacity and relaxation time—all defined as positive constants. Green and Nagdhi [30,31] posited that the theoretical foundation for adopting Equation (1) as a heat transfer model originates from its incorporation of Type III heat conduction mechanisms. Subsequent developments include the analysis presented in [32], where the Cauchy problem for the Moore–Gibson–Thompson equation was examined under dissipative conditions, extending the understanding of its solution behavior in energy-dissipating scenarios. Particularly, the global (in time) existence of the small data solution to the Cauchy problem for (1) has been proved by [33] when $n=3$ and [34] when $n\ge 1$, whereas carrying the power-type nonlinearities the local (in time) energy solution will blow up in finite time [35,36]. Concerning the linearized Cauchy problem (2), ref. [37] demonstrated the well-posedness and decay estimates of the energy term. Soon after, ref. [32] showed the optimal growth or decay estimates of the solution in the $L^2$ framework with its asymptotic profile as the by product. The recent paper [38] demonstrated global (in time) singular limits as well as a higher-order singular layer as $\tau \to 0$ in (2).

In the current study, the mathematical model governed by Equation (1) is established within a semi-infinite cylindrical domain R featuring boundary $\partial R$. This domain assumes an arbitrary cross-sectional shape D with corresponding boundary $\partial D$, where the cylindrical generators are oriented parallel to the $x_3$ axis. To facilitate subsequent analysis, we introduce the following symbolic conventions:

$$R_z=\left\{(x_1,x_2,x_3)\right|(x_1,x_2)\in D,x_3>z\ge 0\},$$

$$D_z=\left\{(x_1,x_2,x_3)\right|(x_1,x_2)\in D,x_3=z\ge 0\}.$$

In this coordinate system, the variable z is defined as the longitudinal coordinate aligned with the $x_3$ axis. Subsequent analysis will demonstrate that $R_0=R,D_0=D$.

The circumferential surfaces of the cylindrical domain are subject to a homogeneous Dirichlet boundary condition, where the thermal displacement is prescribed to vanish identically. This constraint directly yields the relationship:

$$\begin{array}{c}\hfill \alpha =0on\partial D\times (0,+\infty ).\end{array}$$

(4)

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

$$\begin{array}{c}\hfill \alpha (x_1,x_2,0,t)=f_i(x_1,x_2,t)on(x_1,x_2)\in D,t>0.\end{array}$$

(5)

We give the initial conditions:

$$\begin{array}{c}\hfill \alpha (x_1,x_2,x_3,0)=\dot{\alpha }(x_1,x_2,x_3,0)=\ddot{\alpha }(x_1,x_2,x_3,0)=0.\end{array}$$

(6)

In this paper, we add some a priori asymptotical decay assumptions for the solution at the infinity.

$$\begin{array}{cc}\hfill & \dot{\alpha }(x_1,x_2,x_3,t),{\dot{\alpha }}_{,i}(x_1,x_2,x_3,t),{\alpha }_{,i}(x_1,x_2,x_3,t),{\dot{\alpha }}_{,i}(x_1,x_2,x_3,t),\ddot{\alpha }(x_1,x_2,x_3,t)\to 0\hfill \\ & (uniformlyin{x}_1,x_2)as{x}_3\to \infty .\hfill \end{array}$$

(7)

In this paper, we study the spatial behavior of solutions for Equation (3) under the conditions (4)–(7). We can obtain the spatial decay estimates result for the solution by using an integral-differential inequality. This may be the first instance where the integral-differential inequality method is used to investigate the Saint-Venant principle for the Moore–Gibson–Thompson. We think the method proposed in this paper can also be used for the study of the Saint-Venant principle for other equations. In this work, a comma-based subscript notation is adopted to signify partial differentiation. Specifically, differentiation with respect to the spatial coordinate $x_k$ is denoted as k; thus, ${\alpha }_{,i}$ denotes ${\displaystyle \frac{\partial \alpha }{\partial x_i}}$, and $\dot{\alpha }$ denotes ${\displaystyle \frac{\partial \alpha }{\partial t}}$. The usual summation convection is employed with repeated Latin subscripts i summed from 1 to 3. Hence, ${\alpha }_{,i}{\alpha }_{,i}={{\alpha }_{,1}}^2+{{\alpha }_{,2}}^2+{{\alpha }_{,3}}^2$.

2. The Function Expression $\mathit{F}\mathbf{(}\mathit{z}\mathbf{,}\mathit{t}\mathbf{)}$

To derive the desired decay estimates, we introduce a functional framework defined by $F(z,t)$ characterizing the solution behavior. This auxiliary function will be rigorously constructed through the theoretical propositions presented subsequently in the analysis.

Proposition 1.

Considering a smooth solution α to the governing Equation (3) that complies with the prescribed initial-boundary value conditions (4)–(7), we construct an auxiliary functional $F_1(z,t)$ to characterize the solution’s spatiotemporal behavior. This functional will be systematically developed in the subsequent analytical framework,

$$\begin{array}{cc}\hfill F_1(z,t)& =-c\tau {\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{c\tau {\omega }^2}{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{c\tau \omega }{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\left(\dot{\alpha }\right)}^2\mathrm{d}x+c\tau {\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t)\dot{\alpha }\ddot{\alpha }\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{k^{*}\omega }{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{k^{*}}{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\hfill \\ \hfill & +k{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta .\hfill \end{array}$$

(8)

$F_1(z,t)$ can also be expressed as

$$\begin{array}{c}\hfill F_1(z,t)=-k^{*}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\alpha }_{,3}\mathrm{d}A\mathrm{d}\eta -k{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta .\end{array}$$

(9)

Proof. 

By forming the $\dot{\alpha }$-weighted integral of Equation (5) over the spatial domain, the subsequent operation yields the identity

$$\begin{array}{cc}\hfill 0& ={\int }_0^t{\int }_z^{\infty }{\int }_{L_{\xi }}\exp (-\omega \eta )\dot{\alpha }[c(\tau \stackrel{⃛}{\alpha }+\ddot{\alpha })-k^{*}\mathrm{\Delta }\alpha -k\mathrm{\Delta }\dot{\alpha }]\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & =-c\tau {\int }_0^t{\int }_z^{\infty }{\int }_{L_{\xi }}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta +c\tau \omega {\int }_0^t{\int }_z^{\infty }{\int }_{L_{\xi }}\exp (-\omega \eta )\dot{\alpha }\ddot{\alpha }\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +c\tau {\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t)\dot{\alpha }\ddot{\alpha }\mathrm{d}x+k^{*}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\alpha }_{,i}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +k^{*}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\alpha }_{,3}\mathrm{d}A\mathrm{d}\eta +k{\int }_0^t{\int }_z^{\infty }{\int }_{L_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +k{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & =-c\tau {\int }_0^t{\int }_z^{\infty }{\int }_{L_{\xi }}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{c\tau {\omega }^2}{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{c\tau \omega }{2}}{\int }_z^{\infty }{\int }_{L_{\xi }}\exp (-\omega t){\left(\dot{\alpha }\right)}^2\mathrm{d}x+c\tau {\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t)\dot{\alpha }\ddot{\alpha }\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{k^{*}\omega }{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{L_{\xi }}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{k^{*}}{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\hfill \\ \hfill & +k^{*}{\int }_0^t{\int }_{D_{\xi }}\exp (-\omega \eta )\dot{\alpha }{\alpha }_{,3}\mathrm{d}A\mathrm{d}\eta +k{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +k{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta .\hfill \end{array}$$

If we define $F_1(z,t)$ as (8), we can obtain (9). □

Proposition 2.

Considering a smooth solution α to the governing Equation (3) that complies with the prescribed initial-boundary value conditions (4)–(7), we construct an auxiliary functional $F_2(z,t)$ to characterize the solution’s spatiotemporal behavior. This functional will be systematically developed in the subsequent analytical framework

$$\begin{array}{cc}\hfill F_2(z,t)& ={\displaystyle \frac{c\tau {\omega }^2}{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta +c\tau {\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\hfill \\ \hfill & +\left({\displaystyle \frac{k\omega }{2}}-k^{*}\right){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{k^{*}{\omega }^2}{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{k^{*}\omega }{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x-{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\dot{\alpha }}_{,i}{\alpha }_{,i}\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{k}{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x.\hfill \end{array}$$

(10)

$F_2(z,t)$ can also be expressed as

$$\begin{array}{cc}\hfill F_2(z,t)& =k^{*}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta -k^{*}\omega {\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\alpha }_{,3}\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & -{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\ddot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta .\hfill \end{array}$$

(11)

Proof. 

Multiplying (3) by $\ddot{\alpha }$ and integrating, we have

$$\begin{array}{cc}\hfill 0& ={\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta )\ddot{\alpha }[c(\tau \stackrel{⃛}{\alpha }+\ddot{\alpha })-k^{*}\mathrm{\Delta }\alpha -k\mathrm{\Delta }\dot{\alpha }]\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & ={\displaystyle \frac{c\tau \omega }{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta +{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t)c\tau {\left(\ddot{\alpha }\right)}^2\mathrm{d}x\hfill \\ \hfill & +k^{*}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta )\dot{\alpha }{\dot{\alpha }}_{,ii}\mathrm{d}x\mathrm{d}\eta -k^{*}\omega {\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta )\dot{\alpha }{\alpha }_{,ii}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t)\dot{\alpha }{\alpha }_{,ii}\mathrm{d}x+k{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\ddot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\ddot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & ={\displaystyle \frac{c\tau {\omega }^2}{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta +{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t)c\tau {\left(\ddot{\alpha }\right)}^2\mathrm{d}x\hfill \\ \hfill & -k^{*}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta -k^{*}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{k^{*}{\omega }^2}{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{k^{*}\omega }{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\hfill \\ \hfill & +k^{*}\omega {\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\alpha }_{,3}\mathrm{d}A\mathrm{d}\eta -{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\dot{\alpha }}_{,i}{\alpha }_{,i}\mathrm{d}x\hfill \\ \hfill & -{\int }_{D_z}\exp (-\omega t)\dot{\alpha }{\alpha }_{,3}\mathrm{d}A+{\displaystyle \frac{k\omega }{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{k}{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x+{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\ddot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & -k^{*}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta )\ddot{\alpha }{\alpha }_{,3}\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{k\omega }{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{k}{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t)(\xi -z){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x+{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\ddot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta .\hfill \end{array}$$

(12)

From (12), we obtain the desired results (10) and (11). □

We define a new function

$$\begin{array}{c}\hfill F(z,t)=F_1(z,t)+F_2(z,t).\end{array}$$

(13)

From (8) and (10), we have

$$\begin{array}{cc}\hfill F(z,t)& =\left({\displaystyle \frac{c\tau {\omega }^2}{2}}-c\tau \right){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{c\tau {\omega }^2}{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{c\tau \omega }{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\dot{\alpha }}^2\mathrm{d}x\hfill \\ \hfill & +\left({\displaystyle \frac{k^{*}\omega }{2}}+{\displaystyle \frac{k^{*}{\omega }^2}{2}}\right){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\mathrm{d}\eta +\left({\displaystyle \frac{k^{*}}{2}}+{\displaystyle \frac{k^{*}\omega }{2}}\right){\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\hfill \\ \hfill & +\left(k+{\displaystyle \frac{k\omega }{2}}-k^{*}\right){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{k}{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\hfill \\ \hfill & +c\tau {\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\left(\ddot{\alpha }\right)}^2\mathrm{d}x+c\tau {\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t)\dot{\alpha }\ddot{\alpha }\mathrm{d}x\hfill \\ \hfill & -{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\dot{\alpha }}_{,i}{\alpha }_{,i}\mathrm{d}x.\hfill \end{array}$$

(14)

From (9) and (11), $F(z,t)$ can also be rewritten as

$$\begin{array}{cc}\hfill F(z,t)& =(-k^{*}-k^{*}\omega ){\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\alpha }_{,3}\mathrm{d}A\mathrm{d}\eta -(k-k^{*}){\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & -{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\ddot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta .\hfill \end{array}$$

(15)

From (15), $F(z,t)$ can also be rewritten as

$$\begin{array}{cc}\hfill F(z,t)& ={\displaystyle \frac{\partial }{\partial z}}\left[(-k^{*}-k^{*}\omega ){\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }\alpha \mathrm{d}A\mathrm{d}\eta -{\displaystyle \frac{k-k^{*}}{2}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta \right]\hfill \\ \hfill & +(k^{*}+k^{*}\omega ){\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\dot{\alpha }}_{,3}\alpha \mathrm{d}A\mathrm{d}\eta -{\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\ddot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}A\mathrm{d}\eta .\hfill \end{array}$$

(16)

Equations (14) and (16) will play important roles in deducing our result.

Proposition 3.

Considering a smooth solution α to the governing Equation (3) that complies with the prescribed initial-boundary value conditions (4)–(7), the function $F(z,t)$ defined in (16) satisfies the following estimates:

$$\begin{array}{cc}\hfill {\int }_z^{\infty }F(\xi ,t)\mathrm{d}\xi & \le k_1\left(-{\displaystyle \frac{\partial F(z,t)}{\partial z}}\right)+k_{2}F(z,t),\hfill \end{array}$$

(17)

where $k_1$ and $k_2$ are computable positive constants to be defined later.

Proof. 

By differentiating (14) with respect to z, we have

$$\begin{array}{cc}\hfill -{\displaystyle \frac{\partial F(z,t)}{\partial z}}& =\left({\displaystyle \frac{c\tau {\omega }^2}{2}}-c\tau \right){\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{c\tau {\omega }^2}{2}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta +{\displaystyle \frac{c\tau \omega }{2}}{\int }_{D_z}\exp (-\omega t){\left(\dot{\alpha }\right)}^2\mathrm{d}A\hfill \\ \hfill & +\left({\displaystyle \frac{k^{*}\omega }{2}}+{\displaystyle \frac{k^{*}{\omega }^2}{2}}\right){\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & +\left(k+{\displaystyle \frac{k\omega }{2}}-k^{*}\right){\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}A\mathrm{d}\eta +{\displaystyle \frac{k}{2}}{\int }_{D_z}\exp (-\omega t){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}A\hfill \\ \hfill & +c\tau {\int }_{D_z}\exp (-\omega t){\left(\ddot{\alpha }\right)}^2\mathrm{d}A+c\tau {\int }_{D_z}\exp (-\omega t)\dot{\alpha }\ddot{\alpha }\mathrm{d}A\hfill \\ \hfill & -{\int }_{D_z}\exp (-\omega t){\dot{\alpha }}_{,i}{\alpha }_{,i}\mathrm{d}A+\left({\displaystyle \frac{k^{*}}{2}}+{\displaystyle \frac{k^{*}\omega }{2}}\right){\int }_{D_z}\exp (-\omega t){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}A.\hfill \end{array}$$

(18)

Using the Schwarz inequality for (18), we have

$$\begin{array}{cc}\hfill -{\displaystyle \frac{\partial F(z,t)}{\partial z}}& \ge \left({\displaystyle \frac{c\tau {\omega }^2}{2}}-c\tau \right){\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{c\tau {\omega }^2}{2}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta +\left({\displaystyle \frac{c\tau \omega }{2}}-{\displaystyle \frac{c\tau }{2}}\right){\int }_{D_z}\exp (-\omega t){\left(\dot{\alpha }\right)}^2\mathrm{d}A\hfill \\ \hfill & +\left({\displaystyle \frac{k^{*}\omega }{2}}+{\displaystyle \frac{k^{*}{\omega }^2}{2}}\right){\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}A\mathrm{d}\eta \hfill \\ \hfill & +\left(k+{\displaystyle \frac{k\omega }{2}}-k^{*}\right){\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}A\mathrm{d}\eta +{\displaystyle \frac{c\tau }{2}}{\int }_{D_z}\exp (-\omega t){\left(\ddot{\alpha }\right)}^2\mathrm{d}A\hfill \\ \hfill & +{\displaystyle \frac{k}{4}}{\int }_{D_z}\exp (-\omega t){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}A+\left({\displaystyle \frac{k^{*}}{2}}+{\displaystyle \frac{k^{*}\omega }{2}}-{\displaystyle \frac{1}{k}}\right){\int }_{D_z}\exp (-\omega t){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}A.\hfill \end{array}$$

(19)

If we choose $\omega \ge max\left\{3,{\displaystyle \frac{4k^{*}}{k}},{\displaystyle \frac{4}{k^{*}k}}\right\}$, we have

$$\begin{array}{cc}\hfill -{\displaystyle \frac{\partial F(z,t)}{\partial z}}& \ge 0.\hfill \end{array}$$

Using the similar method for (14), we can obtain

$$\begin{array}{cc}\hfill F(z,t)& \ge \left({\displaystyle \frac{c\tau {\omega }^2}{2}}-c\tau \right){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +{\displaystyle \frac{c\tau {\omega }^2}{2}}{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}x\mathrm{d}\eta +\left({\displaystyle \frac{c\tau \omega }{2}}-{\displaystyle \frac{c\tau }{2}}\right){\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\left(\dot{\alpha }\right)}^2\mathrm{d}x\hfill \\ \hfill & +\left({\displaystyle \frac{k^{*}\omega }{2}}+{\displaystyle \frac{k^{*}{\omega }^2}{2}}\right){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\mathrm{d}\eta \hfill \\ \hfill & +\left(k+{\displaystyle \frac{k\omega }{2}}-k^{*}\right){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x\mathrm{d}\eta +{\displaystyle \frac{c\tau }{2}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\left(\ddot{\alpha }\right)}^2\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{k}{4}}{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\dot{\alpha }}_{,i}{\dot{\alpha }}_{,i}\mathrm{d}x+\left({\displaystyle \frac{k^{*}}{2}}+{\displaystyle \frac{k^{*}\omega }{2}}-{\displaystyle \frac{1}{k}}\right){\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega t){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}x\hfill \\ & =E(z,t).\hfill \end{array}$$

(20)

We can also obtain

$$\begin{array}{cc}\hfill E(z,t)& \ge 0.\hfill \end{array}$$

$E(z,t)$ is the energy function that will be used in deriving our main result. This energy function is formulated based on the quantitative estimates method. We will use its nonnegativity to derive the spatial behavior of the solution.

From (16), we have

$$\begin{array}{cc}\hfill {\int }_z^{\infty }F(\xi ,t)\mathrm{d}\xi & =(-k^{*}-k^{*}\omega ){\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }\alpha \mathrm{d}A\mathrm{d}\eta -{\displaystyle \frac{k-k^{*}}{2}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta \hfill \\ & +(k^{*}+k^{*}\omega ){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,3}\alpha \mathrm{d}x\mathrm{d}\eta -{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta )\ddot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}x\mathrm{d}\eta .\hfill \end{array}$$

(21)

We now begin to bound (21). Using the Schwarz inequality, we have

$$\begin{array}{cc}\hfill & (-k^{*}-k^{*}\omega ){\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }\alpha \mathrm{d}A\mathrm{d}\eta \hfill \\ & \le {\displaystyle \frac{k^{*}+k^{*}\omega }{2}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta +{\displaystyle \frac{k^{*}+k^{*}\omega }{2}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\alpha \right)}^2\mathrm{d}A\mathrm{d}\eta \hfill \\ & \le {\displaystyle \frac{k^{*}+k^{*}\omega }{2}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta +{\displaystyle \frac{k^{*}+k^{*}\omega }{2{\lambda }_1}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}A\mathrm{d}\eta .\hfill \end{array}$$

(22)

In deriving (22), we used the Wirtinger type inequality (see [14])

$$\begin{array}{c}\hfill {\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\alpha \right)}^2\mathrm{d}A\mathrm{d}\eta \le {\displaystyle \frac{1}{{\lambda }_1}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\alpha }_{,i}{\alpha }_{,i}\mathrm{d}A\mathrm{d}\eta ,\end{array}$$

(23)

with ${\lambda }_1$ as a positive constant.

Using (19), we have

$$\begin{array}{c}\hfill (-k^{*}-k^{*}\omega ){\int }_0^t{\int }_{D_z}\exp (-\omega \eta )\dot{\alpha }\alpha \mathrm{d}A\mathrm{d}\eta -{\displaystyle \frac{k-k^{*}}{2}}{\int }_0^t{\int }_{D_z}\exp (-\omega \eta ){\left(\dot{\alpha }\right)}^2\mathrm{d}A\mathrm{d}\eta \le k_1\left(-{\displaystyle \frac{\partial F(z,t)}{\partial z}}\right),\end{array}$$

(24)

with $k_1=max\left\{{\displaystyle \frac{2\left(k^{*}+k+\frac{k^{*}\omega }{2}\right)}{c\tau {\omega }^2}},{\displaystyle \frac{1}{{\lambda }_1\omega }}\right\}$.

By following the same procedure as in deriving (24) and using (14), we have

$$\begin{array}{c}\hfill (k^{*}+k^{*}\omega ){\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta ){\dot{\alpha }}_{,3}\alpha \mathrm{d}x\mathrm{d}\eta -{\int }_0^t{\int }_z^{\infty }{\int }_{D_{\xi }}\exp (-\omega \eta )\ddot{\alpha }{\dot{\alpha }}_{,3}\mathrm{d}x\mathrm{d}\eta \le k_{2}F(z,t),\end{array}$$

(25)

with $k_2=max\left\{{\displaystyle \frac{k^{*}+k^{*}\omega +1}{2k+k\omega -2k^{*}}},{\displaystyle \frac{1}{\omega }},{\displaystyle \frac{1}{c\tau {\omega }^2-2c\tau }}\right\}$.

A combination of (21), (24), and (25) gives the desired result (17). □

3. Spatial Decay Estimates

In this part, we will derive the spatial decay estimates for the energy $E(z,t)$. We can rewrite (17) as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial F(z,t)}{\partial z}}+{\displaystyle \frac{1}{k_1}}{\int }_z^{\infty }F(\xi ,t)\mathrm{d}\xi \le {\displaystyle \frac{k_2}{k_1}}F(z,t).\hfill \end{array}$$

(26)

Next, we define two functions

$$\begin{array}{cc}\hfill & \varphi (z,t)=e^{-{\displaystyle \frac{k_2}{k_1}}z}F(z,t),\hfill \end{array}$$

(27)

and

$$\begin{array}{cc}\hfill & G(z,t)=\varphi (z,t)+r{\int }_z^{\infty }{e}^{{\displaystyle \frac{k_2}{k_1}}(\xi -z)}\varphi (\xi ,t)\mathrm{d}\xi ,\hfill \end{array}$$

(28)

with r as a positive constant, which will be defined later.

Since it is difficult to solve (26), we use the form of $G(z,t)$ to solve it.

Proposition 4.

The function $G(z,t)$ defined in (28) satisfies the following estimates:

$$\begin{array}{cc}\hfill & G(z,t)\le G(0,t)e^{-r_{1}z},\hfill \end{array}$$

(29)

with $r_1$ as a positive constant that will be defined later.

Proof. 

By differentiating (28) with respect to z, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial G(z,t)}{\partial z}}& ={\displaystyle \frac{\partial \varphi (z,t)}{\partial z}}-{\displaystyle \frac{k_2}{k_1}}r{\int }_z^{\infty }{e}^{{\displaystyle \frac{k_2}{k_1}}(\xi -z)}\varphi (\xi ,t)\mathrm{d}\xi -r\varphi (z,t)\hfill \\ \hfill & =-{\displaystyle \frac{k_2}{k_1}}{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(z,t)+e^{-{\displaystyle \frac{k_2}{k_1}}z}{\displaystyle \frac{\partial F(z,t)}{\partial z}}\hfill \\ \hfill & -{\displaystyle \frac{k_2}{k_1}}r{\int }_z^{\infty }{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(\xi ,t)\mathrm{d}\xi -r{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(z,t).\hfill \end{array}$$

(30)

We can easily obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial G(z,t)}{\partial z}}+rG(z,t)& =-{\displaystyle \frac{k_2}{k_1}}{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(z,t)+e^{-{\displaystyle \frac{k_2}{k_1}}z}{\displaystyle \frac{\partial F(z,t)}{\partial z}}\hfill \\ \hfill & -{\displaystyle \frac{k_2}{k_1}}r{\int }_z^{\infty }{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(\xi ,t)\mathrm{d}\xi -r{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(z,t)\hfill \\ \hfill & +r{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(z,t)+r^2{\int }_z^{\infty }{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(\xi ,t)\mathrm{d}\xi .\hfill \end{array}$$

(31)

From (26), we have

$$\begin{array}{cc}\hfill -{\displaystyle \frac{k_2}{k_1}}{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(z,t)+e^{-{\displaystyle \frac{k_2}{k_1}}z}{\displaystyle \frac{\partial F(z,t)}{\partial z}}& \le -{\displaystyle \frac{1}{k_1}}{e}^{-{\displaystyle \frac{k_2}{k_1}}z}{\int }_z^{\infty }F(\xi ,t)\mathrm{d}\xi .\hfill \end{array}$$

(32)

By inserting (32) into (31), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial G(z,t)}{\partial z}}+rG(z,t)\le \left(r^2-{\displaystyle \frac{k_2}{k_1}}r-{\displaystyle \frac{k_2}{k_1}}\right){\int }_z^{\infty }{e}^{-{\displaystyle \frac{k_2}{k_1}}z}F(\xi ,t)\mathrm{d}\xi .\hfill \end{array}$$

(33)

Let $r^2-{\displaystyle \frac{k_2}{k_1}}r-{\displaystyle \frac{k_2}{k_1}}=0$, we choose $r_1={\displaystyle \frac{\frac{k_2}{k_1}+\sqrt{{\left(\frac{k_2}{k_1}\right)}^2+\frac{4k_2}{k_1}}}{2}}>0$. We obtain the result

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial G(z,t)}{\partial z}}+r_{1}G(z,t)\le 0,\hfill \end{array}$$

(34)

By integrating (34), we obtain the desired result (29). □

Proposition 5.

The function $F(z,t)$ defined in (14) satisfies the following estimates:

$$\begin{array}{cc}\hfill & F(z,t)\le {\displaystyle \frac{2r_1{k}_1-k_2}{r_1{k}_1-k_2}}F(0,t)e^{-\left(r_1-{\displaystyle \frac{k_2}{k_1}}\right)z}.\hfill \end{array}$$

(35)

Proof. 

A combination of (28) and (29) gives

$$\begin{array}{cc}\hfill & \varphi (z,t)\le G(0,t)e^{-r_{1}z}.\hfill \end{array}$$

(36)

According to the definition of $\varphi (z,t)$ in (27), we have

$$\begin{array}{cc}\hfill & F(z,t)\le G(0,t)e^{-\left(r_1-{\displaystyle \frac{k_2}{k_1}}\right)z}.\hfill \end{array}$$

(37)

We now want to give a bound for $G(0,t)$ by $F(0,t)$.

Using the (28) and (29), we obtain

$$\begin{array}{cc}\hfill & F(z,t)+r_1{\int }_z^{\infty }F(\xi ,t)\mathrm{d}\xi \le G(0,t)e^{\left({\displaystyle \frac{k_2}{k_1}}-r_1\right)z}.\hfill \end{array}$$

(38)

We rewrite inequality (38) as

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{\partial }{\partial z}}\left[e^{-r_{1}z}{\int }_z^{\infty }F(\xi ,t)\mathrm{d}\xi \right]\le G(0,t)e^{\left({\displaystyle \frac{k_2}{k_1}}-2r_1\right)z}.\hfill \end{array}$$

(39)

Integrating (39) from 0 to ∞, we have

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }F(\xi ,t)\mathrm{d}\xi \le {\displaystyle \frac{G(0,t)}{2r_1-\frac{k_2}{k_1}}}.\hfill \end{array}$$

(40)

Using the definition of $G(0,t)$ in (28), we have

$$\begin{array}{cc}\hfill & G(0,t)=F(0,t)+r_1{\int }_0^{\infty }F(\xi ,t)\mathrm{d}\xi .\hfill \end{array}$$

(41)

By inserting (41) into (40), we have

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }F(\xi ,t)\mathrm{d}\xi \le {\displaystyle \frac{F(0,t)+r_1{\int }_0^{\infty }F(\xi ,t)\mathrm{d}\xi }{2r_1-\frac{k_2}{k_1}}}.\hfill \end{array}$$

(42)

Solving (42), we obtain

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }F(\xi ,t)\mathrm{d}\xi \le {\displaystyle \frac{F(0,t)}{r_1-\frac{k_2}{k_1}}}.\hfill \end{array}$$

(43)

We thus have

$$\begin{array}{cc}\hfill G(0,t)& =F(0,t)+r_1{\int }_0^{\infty }F(0,t)\mathrm{d}\xi \hfill \\ \hfill & \le F(0,t)+{\displaystyle \frac{r_{1}F(0,t)}{r_1-\frac{k_2}{k_1}}}\hfill \\ \hfill & ={\displaystyle \frac{2r_1-\frac{k_2}{k_1}}{r_1-\frac{k_2}{k_1}}}F(0,t).\hfill \end{array}$$

(44)

By inserting (44) into (37), we obtain the desired result (35). □

By combining (20) and (35), we can obtain the following main theorem.

Theorem 1.

Let α be classical solution of Equation (3) and satisfy the initial boundary value problems (4)–(7). We establish the corresponding decay estimates for the energy $E(z,t)$ introduced in (20) as

$$\begin{array}{cc}\hfill & E(z,t)\le {\displaystyle \frac{2r_1{k}_1-k_2}{r_1{k}_1-k_2}}F(0,t)e^{-\left(r_1-{\displaystyle \frac{k_2}{k_1}}\right)z}.\hfill \end{array}$$

(45)

4. Conclusions

Inequality (45) shows the spatial decay estimates result, in which the analytical solution demonstrates exponential decay behavior as the spatial coordinate extends to infinity from the inflow boundary. This asymptotic property aligns with a generalized formulation of the Saint-Venant principle, where far-field disturbances diminish proportionally to the characteristic decay rate. We have never seen such results for the Moore–Gibson–Thompson. Next, we will consider the spatial decay estimates of the solution with a weighted energy function. At this point, the method provided in this article will no longer be applicable, and we will proceed with further research. What is more, the investigation of the structural stability properties for the Moore–Gibson–Thompson heat equation within unbounded spatial domains constitutes a compelling research direction. A comprehensive exploration of this theoretical aspect will be undertaken in a subsequent publication dedicated to the topic.