Spatial estimates for a class of hyperbolic equations with nonlinear dissipative boundary conditions

Abstract

This paper is concerned with investigating the spatial behavior of solutions for a class of hyperbolic equations in semi-infinite cylindrical domains, where nonlinear dissipative boundary conditions imposed on the lateral surface of the cylinder. The main tool used is the weighted energy method.

Mathematics Subject Classification (2010) 35B40, 35L05, 35L35

1 Introduction

The aim of this paper is to study the spatial asymptotic behavior of solutions of the problem determined by the equation

$$u_{tt}=\Delta u_t-a{u}_t-{\Delta }^{2}u,\left(x,t\right)\in \Omega \times \left(0,\infty \right),$$

(1.1)

where a is a positive constant and

$$\Omega =\left\{x\in R^n:x_n\in R^{+},x^{\prime }=\left(x_1,\dots ,x_{n-1}\right)\in {\Gamma }_{x_n}\subset R^{n-1}\right\},$$

where

$${\Gamma }_{\tau }=\left\{\left(x^{\prime },x_n\right)\in \Omega :x_n=\tau \right\}.$$

When we consider equation (1.1), we impose the initial and boundary conditions

$$u\left(x,0\right)=u_t\left(x,0\right)=0,x\in \Omega ,$$

(1.2)

$$u\left(x^{\prime },0,t\right)=h_1\left(x^{\prime },t\right),\frac{\partial u}{\partial v}\left(x^{\prime },0,t\right)=h_2\left(x^{\prime },t\right),\left(x^{\prime },t\right)\in {\Gamma }_0\times \left(0,\infty \right),$$

(1.3)

$$u=0,\Delta u=-f\left(\frac{\partial u}{\partial \nu }\right),\left(x,t\right)\in {\Sigma }_0\times \left(0,\infty \right),$$

(1.4)

where ν is the outward normal to the boundary and

$${\Sigma }_{\tau }=\left\{x\in R^n:x^{\prime }\in \partial {\Gamma }_{x_n},\tau \le x_n<\infty \right\},$$

where $\tau \to {\bar{\Gamma }}_{\tau }$ is a map from R⁺ into family of bounded domains in R^n-1with sufficiently smooth boundary ∂ Γ _τ such that

$$0<m_0\le \underset{\tau }{inf}\left|{\Gamma }_{\tau }\right|\le \underset{\tau }{sup}\left|{\Gamma }_{\tau }\right|\le m_1<\infty .$$

In the sequel, we are using

$${\Omega }_{\tau }=\Omega \cap \left\{x\in R^n:0<x_n<\tau \right\},$$

$$R_{\tau }=\Omega \cap \left\{x\in R^n:\tau <x_n<\infty \right\},$$

and assume f satisfies

$$F\left(v\right)=\underset{0}{\overset{v}{\int }}f\left(\xi \right)\mathsf{\text{d}}\xi \ge \alpha vf\left(v\right)>0,\alpha >0,\forall v\in R,$$

(1.5)

$$vf\left(v\right)\ge \gamma {\left|v\right|}^{2p},p>\frac{1}{2},\gamma >0,\forall v\in R.$$

(1.6)

In recent years, much attention has been directed to the study of spatial behavior of solutions of partial differential equations and systems. The history and development of this question is explained in the work of Horgan and Knowles [1]. The interested reader is referred to the papers [2–9] and the reviews by Horgan and Knowles [1, 10, 11]. The energy method is widely used to study such results.

Spatial growth or decay estimates for nontrivial solutions of initial -boundary value problems in semi-infinite domains with nonlinearities on the boundary have been studied by many authors. Since 1908, when Edvard Phragmén and Ernst Lindelöf published their idea [12], many authors have obtained spatial growth or decay results by Phragmén-Lindelöf theorems. In [13], Horgan and Payne proved some these types of theorems and showed the asymptotic behavior of harmonic functions defined on a three-dimensional semi-infinite cylinder when homogeneous nonlinear boundary conditions are imposed on the lateral surface of the cylinder. Payne and Schaefer [14] proved such results for some classes of heat conduction problems. In [15], Quintanilla investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions, (see also [16, 17]).

Under nonlinear dissipative feedbacks on the boundary, Nouria [18] proved a polynomial stability for regular initial data and exponential stability for some analytic initial data of a square Euler-Bernoulli plate. For the used methodology, one can see [19, 20] where the stabilities are investigated in the cases bounded and unbounded feedbacks for some evolution equations. Recently, Celebi and Kalantarov [21] established a Phragmén-Lindelöf type theorems for a linear wave equation under nonlinear boundary conditions. In our study, we establish Phragmén-Lindelöf type theorems for equation (1.1) with nonlinear dissipative feedback terms on the boundary. Our study is inspired by the results of [21].

For the proof of our results, we will use the following Lemma.

Lemma [22] Let ψ be a monotone increasing function with ψ(0) = 0 and lim_z→∞ψ(z) = ∞. Then φ(z) > 0 satisfying φ(z) < ψ(φ'(z)), z > 0, tends to +∞ when z → +∞.

• (i) If ψ(z) ≤ cz^m for some c and m > 1 for z ≥ z₁, then

  $$\underset{z\to +\infty }{liminf}{z}^{-\frac{m}{m-1}}\phi \left(z\right)>0.$$

• (ii) If ψ(z) ≤ cz for some c and z ≥ z₁, then

  $$\underset{z\to +\infty }{liminf}\phi \left(z\right)exp\left(-\frac{z}{c}\right)>0.$$

2 Spatial estimates

With the solutions of (1.1-1.4) with h _i (x', t) = 0, i = 1, 2 is naturally associated an energy function

$$E\left(\tau \right)=\underset{0}{\overset{T}{\int }}\left[\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2+||\nabla u_t|{|}_{{\Omega }_{\tau }}^2+\left|\right|\Delta u|{|}_{{\Omega }_{\tau }}^2+\underset{0}{\overset{\tau }{\int }}\underset{\partial {\Gamma }_{\eta }}{\int }\nabla uf\left(\nabla u\right)\mathsf{\text{d}}s\mathsf{\text{d}}\eta \right]\mathsf{\text{d}}t,$$

(2.1)

where ||.||_Ω denotes the usual norm in L²(Ω).

A multiplication of equation (1.1) by u _t , integrating over Ω _τ and using (1.3-1.5):

$$\begin{array}{c}\frac{\mathsf{\text{d}}}{\mathsf{\text{d}}t}\left[\frac{1}{2}\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2+\frac{1}{2}|\left|\Delta u\right|{|}_{{\Omega }_{\tau }}^2+\underset{0}{\overset{\tau }{\int }}\underset{\partial {\Gamma }_{\eta }}{\int }F\left(\nabla u\right)\mathsf{\text{d}}s\mathsf{\text{d}}\eta \right]+a\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2\\ +\left|\right|\nabla u_t|{|}_{{\Omega }_{\tau }}^2=-{\left(u_t,u_{x_n{x}_n{x}_n}\right)}_{{\Gamma }_{\tau }}+{\left(u_{t{x}_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}+{\left(u_t,u_{t{x}_n}\right)}_{{\Gamma }_{\tau }}.\end{array}$$

Since

$${\left(u_t,u_{x_n{x}_n{x}_n}\right)}_{{\Gamma }_{\tau }}=-{\left(u_{t{x}_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }},$$

we obtain

$$\begin{array}{c}\frac{\mathsf{\text{d}}}{\mathsf{\text{d}}t}\left[\frac{1}{2}\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2+\frac{1}{2}|\left|\Delta u\right|{|}_{{\Omega }_{\tau }}^2+\underset{0}{\overset{\tau }{\int }}\underset{\partial {\Gamma }_{\eta }}{\int }F\left(\nabla u\right)\mathsf{\text{d}}s\mathsf{\text{d}}\eta \right]+a\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2\\ +\left|\right|\nabla u_t|{|}_{{\Omega }_{\tau }}^2=2{\left(u_{t{x}_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}+{\left(u_t,u_{t{x}_n}\right)}_{{\Gamma }_{\tau }}.\end{array}$$

(2.2)

Let δ > 0. Multiplying (1.1) by δu, integrating over Ω _τ , and adding to (2.2), we obtain

$$\begin{array}{c}\frac{\mathsf{\text{d}}}{\mathsf{\text{d}}t}\left\{\frac{1}{2}\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2+\frac{1}{2}|\left|\Delta u\right|{|}_{{\Omega }_{\tau }}^2+\delta {\left(u,u_t\right)}_{{\Omega }_{\tau }}\right.\\ \left.+\frac{a\delta }{2}\left|\right|u|{|}_{{\Omega }_{\tau }}^2+\frac{\delta }{2}||\nabla u|{|}_{{\Omega }_{\tau }}^2+\underset{0}{\overset{\tau }{\int }}\underset{\partial {\Gamma }_{\eta }}{\int }F\left(\nabla u\right)\mathsf{\text{d}}s\mathsf{\text{d}}\eta \right\}\\ +\left(a-\delta \right)\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2+||\nabla u_t|{|}_{{\Omega }_{\tau }}^2+\delta \left|\right|\Delta u|{|}_{{\Omega }_{\tau }}^2+\delta \underset{0}{\overset{\tau }{\int }}\underset{\partial {\Gamma }_{\eta }}{\int }\nabla uf\left(\nabla u\right)\mathsf{\text{d}}s\mathsf{\text{d}}\eta \\ =2{\left(u_{t{x}_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}+{\left(u_t,u_{t{x}_n}\right)}_{{\Gamma }_{\tau }}+2\delta {\left(u_{x_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}+\delta {\left(u,u_{t{x}_n}\right)}_{{\Gamma }_{\tau }}.\end{array}$$

(2.3)

Integrating (2.3) with respect to t over (0, T) and using (1.5), one can find

$$\begin{array}{l}\frac{1}{2}\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2+\frac{1}{2}|\left|\Delta u\right|{|}_{{\Omega }_{\tau }}^2+\frac{\delta }{2}\left|\right|\nabla u|{|}_{{\Omega }_{\tau }}^2+\frac{a\delta }{2}|\left|u\right|{|}_{{\Omega }_{\tau }}^2\\ +\delta {(u,u_t)}_{{\Omega }_{\tau }}+\alpha {\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{\partial \Gamma \eta }{\int }\nabla uf(\nabla u)\text{d}s\text{d}\eta }}\\ +(a-\delta ){\displaystyle \underset{0}{\overset{T}{\int }}\left|\right|u_t|{|}_{{\Omega }_{\tau }}^2}\text{d}t+\delta {\displaystyle \underset{0}{\overset{T}{\int }}\left|\right|\Delta u|{|}_{{\Omega }_{\tau }}^2}\text{d}t+{\displaystyle \underset{0}{\overset{T}{\int }}\left|\right|\nabla u_t|{|}_{{\Omega }_{\tau }}^2}\text{d}t\\ +\delta {\displaystyle \underset{0}{\overset{T}{\int }}{\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{\partial \Gamma \eta }{\int }\nabla uf(\nabla u)\text{d}s\text{d}\eta dt}}}\le {\displaystyle \underset{0}{\overset{T}{\int }}[\text{2}({\text{u}}_{{\text{tx}}_{\text{n}}},{\text{u}}_{{\text{x}}_{\text{n}}{\text{x}}_{\text{n}}}){\Gamma }_{\tau }\text{+}({\text{u}}_{\text{t}},{\text{u}}_{{\text{tx}}_{\text{n}}}){\Gamma }_{\tau }]\text{dt}}\\ +{\displaystyle \underset{0}{\overset{T}{\int }}[2\delta (u_{x_n},u_{x_n{x}_n}){\Gamma }_{\tau }+\delta (u,u_{t{x}_n}){\Gamma }_{\tau }]\text{dt}.}\end{array}$$

(2.4)

On exploiting (2.1) and the inequality $-\left(\frac{1}{4}\right){∥u_t∥}_{{\Omega }_{\tau }}^2-{\delta }^2{∥u∥}_{{\Omega }_{\tau }}^2\le \delta {\left(u,u_t\right)}_{{\Omega }_{\tau }}$, the estimate (2.4) takes the form

$$\begin{array}{cc}\hfill {\sigma }^{-1}E\left(\tau \right)\hfill & \hfill \le \underset{0}{\overset{T}{\int }}\mathsf{\text{[}}\left(u_{t{x}_n},u_{x_n{x}_n}\right){\Gamma }_{\tau }+\left(u_t,u_{t{x}_n}\right){\Gamma }_{\tau }\mathsf{\text{]}}\mathsf{\text{d}}t\hfill \\ \hfill +\underset{0}{\overset{T}{\int }}\mathsf{\text{[}}\left(u_{x_n},u_{x_n{x}_n}\right){\Gamma }_{\tau }+\left(u,u_{t{x}_n}\right){\Gamma }_{\tau }\mathsf{\text{]}}\mathsf{\text{d}}t,\hfill \end{array}$$

(2.5)

by choosing $\delta =\frac{a}{2}$, ${\delta }_1=min\left\{1,\frac{a}{2}\right\}$, $\sigma =max\left\{\frac{a}{{\delta }_1},\frac{2}{{\delta }_1}\right\}$. Now we find upper bounds for the right hand side of (2.5). Using the Young's and Schwartz inequalities, we have

$$\underset{0}{\overset{T}{\int }}{\left(u_{t{x}_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t\le \frac{1}{2}\underset{0}{\overset{T}{\int }}\left|\right|\nabla u_t|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t+\frac{1}{2}\underset{0}{\overset{T}{\int }}|\left|\Delta u\right|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t,$$

(2.6)

$$\underset{0}{\overset{T}{\int }}{\left(u_t,u_{t{x}_n}\right)}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t\le \frac{1}{2}\underset{0}{\overset{T}{\int }}\left|\right|u_t|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t+\frac{1}{2}\underset{0}{\overset{T}{\int }}||\nabla u_t|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t,$$

(2.7)

$$\underset{0}{\overset{T}{\int }}{\left(u_{x_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t\le \underset{0}{\overset{T}{\int }}\left|\right|u_{x_n}\left|{|}_{{\Gamma }_{\tau }}\right|\left|u_{x_n{x}_n}\right|{|}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t.$$

(2.8)

By the Poincaré inequality, it is not difficult to see

$${∥v∥}_D^2\le {\lambda }^{-1}{∥\nabla v∥}_D^2+{\left|D\right|}^{-1}{\left(\underset{D}{\int }v\mathsf{\text{d}}A\right)}^2.$$

(2.9)

Inserting (2.9) into (2.8), we get

$$\begin{array}{c}\underset{0}{\overset{T}{\int }}{\left(u_{x_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t\\ \le \underset{0}{\overset{T}{\int }}\left\{{\lambda }_{\tau }^{-\frac{1}{2}}\left|\right|{\Delta }^{\prime }u|{|}_{{\Gamma }_{\tau }}+{\left|{\Gamma }_{\tau }\right|}^{-\frac{1}{2}}\left|\underset{{\Gamma }_{\tau }}{\int }{\nabla }^{\prime }u\mathsf{\text{d}}A\right|\right\}\left|\right|u_{x_n{x}_n}|{|}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t,\end{array}$$

(2.10)

where Δ' and ∇' are Laplacian and gradient operators in R^n-1, respectively, |Γ _τ | is the area of Γ _τ and λ _τ is the Poincaré constant. Now, we recall the inequality

$$\underset{D}{\int }v\mathsf{\text{d}}A\le \frac{r_0}{2}\underset{\partial D}{\int }\left|v\right|\mathsf{\text{d}}s+\frac{I_0^{\frac{1}{2}}}{2}{\left(\underset{D}{\int }{\left|\nabla v\right|}^2\mathsf{\text{d}}A\right)}^{\frac{1}{2}},$$

(2.11)

from [13] where $r_0^2=\underset{D}{sup}|x^{\prime }{|}^2$ and $I_0={\int }_D|x^{\prime }{|}^2\mathsf{\text{d}}A$. Using (2.11) and the Hölder's inequality to estimate the boundary integral $\left|{\int }_{{\Gamma }_{\tau }}{\nabla }^{\prime }u\mathsf{\text{d}}A\right|$ in (2.10), we obtain

$$\begin{array}{c}\underset{0}{\overset{T}{\int }}{\left(u_{x_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t\\ \le \underset{0}{\overset{T}{\int }}\left\{M_1\left|\right|{\Delta }^{\prime }u|{|}_{{\Gamma }_{\tau }}+{\gamma }^{\frac{1}{2p}}{M}_2{\left(\underset{\partial {\Gamma }_{\tau }}{\int }|{\nabla }^{\prime }u{|}^{2p}\mathsf{\text{d}}A\right)}^{\frac{1}{2p}}\right\}\left|\right|u_{x_n{x}_n}|{|}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t,\end{array}$$

(2.12)

where $M_1={\lambda }^{-\frac{1}{2}}+\frac{I^{1∕2}}{2m^{1∕2}}$, $M_2=\frac{1}{2}r{L}^{\left(2p-1\right)∕2p}{m}^{-1∕2}{\gamma }^{-1∕2p}$, such that r = sup _τ r _τ , λ = inf _τ λ_τ, I = sup _τ I _τ , L = sup _τ L _τ and m = inf _τ |Γ _τ | in which L _τ is the area of ∂Γ _τ . From (1.6) the inequality (2.12) yields

$$\begin{array}{c}\underset{0}{\overset{T}{\int }}{\left(u_{x_n},u_{x_n{x}_n}\right)}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t\\ \le M_1\underset{0}{\overset{T}{\int }}\left|\right|\Delta u|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t+M_2\underset{0}{\overset{T}{\int }}{\left(\underset{\partial {\Gamma }_{\tau }}{\int }{\nabla }^{\prime }uf\left({\nabla }^{\prime }u\right)\mathsf{\text{d}}A\right)}^{\frac{1}{2p}}|\left|u_{x_n{x}_n}\right|{|}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t.\end{array}$$

(2.13)

Consequently

$$\begin{array}{c}{\left(\underset{\partial {\Gamma }_{\tau }}{\int }{\nabla }^{\prime }uf\left({\nabla }^{\prime }u\right)\mathsf{\text{d}}s\right)}^{\frac{1}{2p}}{\left(\underset{{\Gamma }_{\tau }}{\int }{u}_{x_n{x}_n}^2\mathsf{\text{d}}A\right)}^{\frac{1}{2}}\\ ={\left[{\left(\underset{\partial {\Gamma }_{\tau }}{\int }{\nabla }^{\prime }uf\left({\nabla }^{\prime }u\right)\mathsf{\text{d}}s\right)}^{\frac{1}{p+1}}{\left(\underset{{\Gamma }_{\tau }}{\int }{u}_{x_n{x}_n}^2\mathsf{\text{d}}A\right)}^{\frac{p}{p+1}}\right]}^{\frac{p+1}{2p}}\\ \le {\left[\frac{{\mu }^p}{1+p}\underset{\partial {\Gamma }_{\tau }}{\int }{\nabla }^{\prime }uf\left({\nabla }^{\prime }u\right)\mathsf{\text{d}}s+\frac{p}{\mu \left(1+p\right)}\underset{{\Gamma }_{\tau }}{\int }{u}_{x_n{x}_n}^2\mathsf{\text{d}}A\right]}^{\frac{p+1}{2p}},\end{array}$$

where the Young's inequality

$${\alpha }^{\epsilon }{\beta }^{1-\epsilon }={\left(\alpha \gamma \right)}^{\epsilon }{\left[\beta {\gamma }^{\frac{-\epsilon }{1-\epsilon }}\right]}^{\left(1-\epsilon \right)}\le \epsilon \alpha \gamma +\left(1-\epsilon \right)\beta {\gamma }^{\frac{-\epsilon }{1-\epsilon }},$$

for 0 < ε < 1, $\mu =p^{\frac{1}{p+1}}$ and γ = μ^p have been used. Therefore,

$$\begin{array}{c}{\left(\underset{\partial {\Gamma }_{\tau }}{\int }{\nabla }^{\prime }uf\left({\nabla }^{\prime }u\right)\mathsf{\text{d}}s\right)}^{\frac{1}{2p}}{\left(\underset{{\Gamma }_{\tau }}{\int }{u}_{x_n{x}_n}^2\mathsf{\text{d}}A\right)}^{\frac{1}{2}}\\ \le {\left[N\left(p\right)\left(\underset{\partial {\Gamma }_{\tau }}{\int }{\nabla }^{\prime }uf\left({\nabla }^{\prime }u\right)\mathsf{\text{d}}s+\underset{{\Gamma }_{\tau }}{\int }{u}_{x_n{x}_n}^2\mathsf{\text{d}}A\right)\right]}^{\frac{p+1}{2p}},\end{array}$$

(2.14)

where

$$N\left(p\right)=\frac{p^{\frac{p}{p+1}}}{\left(1+p\right)}.$$

By using (2.13) and (2.14), we get

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{T}{\int }}{(u_{x_n},u_{x_n{x}_n})}_{{\Gamma }_{\tau }}}\text{d}t\le M_1{\displaystyle \underset{0}{\overset{T}{\int }}|}\left|\Delta u\right|{|}_{{\Gamma }_{\tau }}^2\text{d}t\\ +M_2\tilde{N}(p){\displaystyle \underset{0}{\overset{T}{\int }}{\left({\displaystyle \underset{\partial {\Gamma }_{\tau }}{\int }{\nabla }^{\prime }}uf({\nabla }^{\prime }u)\text{d}s+{\displaystyle \underset{{\Gamma }_{\tau }}{\int }{u}_{x_n{x}_n}^2}\text{d}A\right)}^{\frac{p+1}{2p}}}\text{d}t,\end{array}$$

(2.15)

where $\tilde{N}(p)={\left[N(p)\right]}^{\frac{p+1}{2p}}$. From (2.15), it is easy to see

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{T}{\int }}{(u_{x_n},u_{x_n{x}_n})}_{{\Gamma }_{\tau }}}\text{d}t\le M_1{\displaystyle \underset{0}{\overset{T}{\int }}|}\left|\Delta u\right|{|}_{{\Gamma }_{\tau }}^2\text{d}t\\ +M_{2}C\tilde{N}(p){\left({\displaystyle \underset{0}{\overset{T}{\int }}{\displaystyle \underset{\partial {\Gamma }_{\tau }}{\int }\nabla }}uf(\nabla u)\text{d}s\text{d}t+{\displaystyle \underset{0}{\overset{T}{\int }}|}\left|\Delta u\right|{|}_{{\Gamma }_{\tau }}^2\text{d}t\right)}^{\frac{p+1}{2p}},\end{array}$$

where C is a positive constant.

Next, we exploit Poincaré inequality to estimate

$$\underset{0}{\overset{T}{\int }}{\left(u,u_{t{x}_n}\right)}_{{\Gamma }_{\tau }}\mathsf{\text{d}}t\le \frac{{\rho }^{-1}}{2}\underset{0}{\overset{T}{\int }}\left|\right|\Delta u|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t+\frac{1}{2}\underset{0}{\overset{T}{\int }}||\nabla u_t|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t,$$

(2.17)

where ρ is the Poincaré constant.

Now, from the inequalities (2.5-2.7), (2.16), and (2.17), one can find

$$\begin{array}{l}E(\tau )\le {\displaystyle \underset{0}{\overset{T}{\int }}\left[\frac{\sigma }{2}\left|\right|u_t|{|}_{{\Gamma }_{\tau }}^2+\frac{3}{2}\sigma ||\nabla u_t|{|}_{{\Gamma }_{\tau }}^2+\sigma \left(\frac{1}{2}+M_1+\frac{{\rho }^{-1}}{2}\right)\left|\right|\Delta u|{|}_{{\Gamma }_{\tau }}^2\right]}\text{d}t\\ +{\displaystyle \underset{0}{\overset{T}{\int }}{\displaystyle \underset{\partial {\Gamma }_{\tau }}{\int }\nabla }}uf(\nabla u)\text{d}s\text{d}t+\sigma M_{2}C\tilde{N}(p)\{{\displaystyle \underset{0}{\overset{T}{\int }}{\displaystyle \underset{\partial {\Gamma }_{\tau }}{\int }\nabla }}uf(\nabla u)\text{d}s\text{d}t\\ {+{\displaystyle \underset{0}{\overset{T}{\int }}\left[\left|\right|\nabla u_t|{|}_{{\Gamma }_{\tau }}^2\text{d}t+|\left|\Delta u\right|{|}_{{\Gamma }_{\tau }}^2+\left|\right|u_t|{|}_{{\Gamma }_{\tau }}^2\right]}\text{d}t\}}^{\frac{p+1}{2p}}.\end{array}$$

Upon inserting (2.1) into the right hand side of (2.18), we may write an inequality in the form

$$E(\tau )\le \sigma \left(\frac{5}{2}+M_1+\frac{{\rho }^{-1}}{2}\right)E^{\prime }(\tau )+\sigma M_{2}C\tilde{N}(p){[E^{\prime }(\tau )]}^{\frac{p+1}{2p}}.$$

(2.19)

At this point, by the inequality (2.19), the function $\psi \left(z\right)={\alpha }_{1}z+{\alpha }_2{z}^{\frac{p+1}{2p}}$ satisfies in the hypothesis of the Lemma. Therefore, we have proved the following theorem.

Theorem 1 Let u(x, t) be a nontrivial solution of (1.1) - (1.4) with h _i (x', t) = 0, i = 1, 2 under the conditions (1.5) and (1.6). Then

$$\underset{\tau \to +\infty }{\lim \inf }E(\tau ){\tau }^{-\frac{p+1}{1-p}}>0,p\in (\frac{1}{2},1),$$

and

$$\underset{\tau \to +\infty }{\lim \inf }E(\tau )\exp (-\frac{\tau }{c})>0,p\in [1,+\infty ),$$

where

$$c=\max \left\{\sigma \left(\frac{5}{2}+M_1+\frac{{\rho }^{-1}}{2}\right),\sigma M_{2}C\tilde{N}(p)\right\}.$$

Theorem 2 Consider the equation (1.1) subject to the conditions u(x', 0, t) = h₁(x', t) and $\frac{\partial u}{\partial v}\left(x^{\prime },0,t\right)=h_2\left(x^{\prime },t\right)$ for x' ∈ Γ₀. If E(+∞) is finite, then

$$\underset{\tau \to +\infty }{lim}\left(\underset{0}{\overset{T}{\int }}\left|\right|u_t|{|}_{R_{\tau }}^2\mathsf{\text{d}}t+\underset{0}{\overset{T}{\int }}||\nabla u_t|{|}_{R_{\tau }}^2\mathsf{\text{d}}t+\underset{0}{\overset{T}{\int }}\left|\right|\Delta u|{|}_{R_{\tau }}^2\mathsf{\text{d}}t\right)=0.$$

(2.20)

proof By the same manner followed in theorem 1, it is easy to find the inequality

$$\begin{array}{c}\left(a-\delta \right)\underset{0}{\overset{T}{\int }}\left|\right|u_t|{|}_{R_{\tau }}^2\mathsf{\text{d}}t+\underset{0}{\overset{T}{\int }}||\nabla u_t|{|}_{R_{\tau }}^2\mathsf{\text{d}}t+\delta \underset{0}{\overset{T}{\int }}\left|\right|\Delta u|{|}_{R_{\tau }}^2\mathsf{\text{d}}t\le \frac{1}{2}\underset{0}{\overset{T}{\int }}|\left|u_t\right|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t\\ +\left(\frac{3}{2}+\frac{\delta }{2}\right)\underset{0}{\overset{T}{\int }}\left|\right|\nabla u_t|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t+\left[1+\delta \left(1+{\lambda }_{\tau }^{-1}+\frac{1}{2}{\lambda }_{\tau }^{-2}\right)\right]\underset{0}{\overset{T}{\int }}|\left|\Delta u\right|{|}_{{\Gamma }_{\tau }}^2\mathsf{\text{d}}t,\end{array}$$

where λ _τ is the Poincaré constant. Choosing δ ∈ (0, a), η = min{a -δ, δ, 1} and

$$\tilde{\gamma }={\eta }^{-1}max\left\{\frac{3}{2}+\frac{\delta }{2},1+\delta \left(1+\underset{\tau }{\overset{-1}{\lambda }}+\frac{1}{2}{\lambda }_{\tau }^{{}^{-2}}\right)\right\},$$

we obtain

$$Ẽ\left(\tau \right)\le -\tilde{\gamma }{Ẽ}^{\prime }\left(\tau \right),$$

(2.21)

where

$$Ẽ\left(\tau \right)=\underset{0}{\overset{T}{\int }}\left|\right|u_t|{|}_{R_{\tau }}^2\mathsf{\text{d}}t+\underset{0}{\overset{T}{\int }}||\nabla u_t|{|}_{R_{\tau }}^2\mathsf{\text{d}}t+\underset{0}{\overset{T}{\int }}\left|\right|\Delta u|{|}_{R_{\tau }}^2\mathsf{\text{d}}t.$$

Thus, (2.20) follows from (2.21).   ■