On decay properties of solutions for degenerate Kirchhoff equations with strong damping and source terms

Abstract

We investigate the degenerate Kirchhoff equations with strong damping and source terms of the form

$$\epsilon u_{tt}-{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }\mathrm{\Delta }u-\mathrm{\Delta }{u}_t=f(u),$$

in a bounded domain. We obtain the optimal decay rate for ${\parallel \mathrm{\nabla }{u}_t\parallel }_2^2$ by deriving its decay estimate from below, provided that either ε is suitably small or the initial data satisfy the proper smallness assumption. The key ingredient in the proof is based on the work of Ono (J. Math. Anal. Appl. 381(1):229-239, 2011), with necessary modification imposed by our problem.

MSC:35L70, 35L80.

1 Introduction

In this paper, we consider the initial boundary value problem for the following degenerate Kirchhoff equations with strong damping and source terms:

(1.1)

(1.2)

(1.3)

where Ω is a bounded domain in $R^N$ ($N\ge 1$) with a smooth boundary ∂ Ω. Here, $\epsilon >0$ and $\gamma >0$ are positive constants and $f(u)$ is a nonlinear term like $|u{|}^{p-1}u$, $p>1$.

In the case $N=1$, the nonlinear vibrations of the elastic string are written in the form:

$$\rho h\frac{{\partial }^{2}u}{\partial t^2}=\{p_0+\frac{Eh}{2L}{\int }_0^L{\left(\frac{\partial u}{\partial x}\right)}^{2}dx\}\frac{{\partial }^{2}u}{\partial x^2}+f,$$

(1.4)

for $0<x<L$, $t\ge 0$; where u is the lateral deflection, x the space coordinate, t the time, E the Young modulus, ρ the mass density, h the cross section area, L the length, $p_0$ the initial axial tension and f the external force. The equation was first introduced by Kirchhoff [1] in the study of stretched strings and plates, and is called the wave equation of Kirchhoff type after his name. Moreover, it is said that (1.4) is degenerate if $p_0=0$ and nondegenerate if $p_0>0$.

A number of results on the solutions to problem (1.1)-(1.3) have been established by many authors. For example, Hosoya and Yamada [2] studied the following equation:

$$u_{tt}-M\left({\parallel \mathrm{\nabla }u\parallel }_2^2\right)\mathrm{\Delta }u+\delta u_t=0,$$

(1.5)

with $M(r)\ge M_0>0$ (the nondegenerate case). They proved the global existence of a unique solution under the small data condition in $H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega })$. Similarly, in the degenerate case ($M(r)=r^{\gamma }$), Nishihara and Yamada [3] obtained the global existence of solutions for small initial data in $H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega })$.

Concerning decay properties of solutions, Nakao [4] has derived decay estimates from above of solutions when $f(u)=0$ in (1.1). Later, Nishihara [5] established a decay estimate from below of the potential of solutions to problem (1.1) without imposing $f(u)$. Nishihara and Ono [6–8] studied detailed cases of nondegenerate type and degenerate type to problem (1.1)-(1.3). They proved the existence and uniqueness of a global solution for initial data $(u_0,u_1)\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$ and the polynomial decay of the solution. Recently, in the absence of $f(u)$ in (1.1), Ono [9] proved the optimal decay estimate for ${\parallel \mathrm{\nabla }u\parallel }_2^2$. And the decay property of the norm ${\parallel \mathrm{\Delta }u\parallel }_2^2$ for $t\ge 0$ was also given in that paper.

Motivated by these works, in this paper, we intend to give the optimal decay estimate for ${\parallel \mathrm{\nabla }{u}_t\parallel }_2$ to problem (1.1)-(1.3). In this way, we can extend the result in [8] where the author considered (1.1) with $\gamma \ge 1$; and we also improve the result of [9] in the presence of a nonlinear source term. We followed the technique skill introduced in [9] with the concept of a stable set in $H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ to derive the optimal decay rate of ${\parallel \mathrm{\nabla }{u}_t\parallel }_2$. The content of this paper is organized as follows. In Section 2, we give some lemmas which will be used later. In Section 3, we derive the global solution and its decay properties with $\gamma >0$.

2 Preliminary results

In this section, we shall give some lemmas which will be used throughout this work. We denote by ${\parallel \cdot \parallel }_p$ the $L^p$ norm over Ω.

Lemma 2.1 (The Sobolev-Poincaré inequality)

If$2\le p\le \frac{2N}{N-2}$ ($2\le p<\mathrm{\infty }$if$N=1,2$), then

$${\parallel u\parallel }_p\le B_1{\parallel \mathrm{\nabla }u\parallel }_2,\mathit{\text{for }}u\in H_0^1(\mathrm{\Omega })$$

holds with some positive constant$B_1$.

Lemma 2.2 ([4, 9])

Let $\varphi (t)$ be a nonincreasing and nonnegative function on $[0,\mathrm{\infty })$ such that

$$\varphi {(t)}^{1+r}\le {\omega }_0(\varphi (t)-\varphi (t+1))$$

with certain constants${\omega }_0\ge 0$and$r>0$. Then, the function$\varphi (t)$satisfies

$$\varphi (t)\le {(\varphi {(0)}^{-r}+{\omega }_0^{-1}r{[t-1]}^{+})}^{-\frac{1}{r}}$$

for$t\ge 0$, where we put${[a]}^{+}=max\{0,a\}$and$\frac{1}{{[a]}^{+}}=\mathrm{\infty }$if${[a]}^{+}=0$.

Now, we state the local existence for problem (1.1)-(1.3) which can be established by the arguments of [8].

Theorem 2.3 Suppose that$(u_0,u_1)\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$, then there exists a unique solution u of problem (1.1)-(1.3) satisfying

$$\begin{array}{c}u(t)\in C([0,T);H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })),\hfill \\ u_t(t)\in C([0,T);L^2(\mathrm{\Omega }))\cap L^2(0,T;H_0^1(\mathrm{\Omega })).\hfill \end{array}$$

Moreover, at least one of the following statements holds true:

• (i) $T=\mathrm{\infty }$,

• (ii) ${\parallel u_t\parallel }_2^2+{\parallel \mathrm{\Delta }u\parallel }_2^2\to \mathrm{\infty }$, as $t\to T^{-}$.

3 Decay properties

In this section, we shall show the decay properties of solutions u to problem (1.1)-(1.3). For this purpose, we define

(3.1)

(3.2)

and the energy function

$$E(t)=\frac{\epsilon }{2}{\parallel u_t\parallel }_2^2+J(t),$$

(3.3)

for $u(t)\in H_0^1(\mathrm{\Omega })$, $t\ge 0$. Then, multiplying (1.1) by $u_t$ and integrating it over Ω, we see that

$$E^{\mathrm{\prime }}(t)=-{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2.$$

(3.4)

Lemma 3.1 Let$(u_0,u_1)\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$. Suppose that p and γ satisfy

$$p>2\gamma +1\mathit{\text{and}}1<p\le \frac{N+2}{N-2}(p<\mathrm{\infty },\mathit{\text{if }}N=1,2).$$

(3.5)

If $I(0)>0$ and

$$l=B_1^{p+1}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(0))}^{\frac{p-2\gamma -1}{2(\gamma +1)}}<1,$$

(3.6)

then$I(t)>0$, for$t\in [0,T]$.

Proof Since $I(0)>0$, then there exists $t_{max}<T$ such that

$$I(t)\ge 0,t\in [0,t_{max}],$$

which gives

$$J(t)=\frac{p-2\gamma -1}{2(p+1)(\gamma +1)}{\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}+\frac{1}{p+1}I(t)\ge \frac{p-2\gamma -1}{2(p+1)(\gamma +1)}{\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}.$$

(3.7)

Thus, from (3.3) and $E(t)$ being nonincreasing by (3.4), we have that

$$\begin{array}{rcl}{\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}& \le & \frac{2(p+1)(\gamma +1)}{p-2\gamma -1}J(t)\le \frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(t)\\ \le & \frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(0)\text{on }[0,t_{max}].\end{array}$$

(3.8)

And so, exploiting Lemma 2.1, (3.8) and (3.6), we obtain

$$\begin{array}{rcl}{\parallel u\parallel }_{p+1}^{p+1}& \le & B_1^{p+1}{\parallel \mathrm{\nabla }u\parallel }_2^{p+1}\le B_1^{p+1}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(0))}^{\frac{p-2\gamma -1}{2(\gamma +1)}}{\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}\\ =& l{\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}<{\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}\text{on }[0,t_{max}].\end{array}$$

(3.9)

Therefore,

$$I(t)={\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}-{\parallel u(t)\parallel }_{p+1}^{p+1}>0\text{on }[0,t_{max}].$$

By repeating this procedure and using the fact that

$$\underset{t\to t_{max}}{lim}{B}_1^{p+1}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(t))}^{\frac{p-2\gamma -1}{2(\gamma +1)}}\le l<1,$$

we can take $t_{max}=T$. □

Lemma 3.2 Let u satisfy the assumptions of Lemma  3.1. Then there exists$0<\eta <1$such that

$${\parallel u(t)\parallel }_{p+1}^{p+1}\le (1-\eta ){\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}\mathit{\text{for }}t\in [0,T],$$

(3.10)

with$\eta =1-l$.

Proof From (3.9), we get

$${\parallel u(t)\parallel }_{p+1}^{p+1}\le l{\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)},\text{for }t\in [0,T].$$

Let $\eta =1-l$, then we have the inequality (3.10). □

Theorem 3.3 Suppose that$(u_0,u_1)\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$, $I(0)>0$, (3.5) and (3.6) hold. Then there exists a unique global solution u of problem (1.1)-(1.3) satisfying

$$\begin{array}{c}u(t)\in C([0,\mathrm{\infty });H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega }))\cap L^{\mathrm{\infty }}(0,\mathrm{\infty };H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })),\hfill \\ u_t(t)\in C([0,\mathrm{\infty });L^2(\mathrm{\Omega }))\cap L^2(0,\mathrm{\infty };H_0^1(\mathrm{\Omega }))\cap L^{\mathrm{\infty }}(0,\mathrm{\infty };L^2(\mathrm{\Omega })).\hfill \end{array}$$

Furthermore, we have the following decay estimates:

$$E(t)\le {(E{(0)}^{-\frac{\gamma }{\gamma +1}}+\frac{\gamma }{\gamma +1}{(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^{-2}{[t-1]}^{+})}^{-\frac{\gamma +1}{\gamma }}\mathit{\text{for }}t\ge 0,$$

(3.11)

where α and β are some positive constants given by (3.15).

Proof First, for $T=\mathrm{\infty }$, we can obtain the result by following the arguments as in [8], so we omit the proof. Next, we will show the decay estimate. For $t\ge 0$, integrating (3.4) over $[t,t+1]$, we have

$${\int }_t^{t+1}{\parallel \mathrm{\nabla }{u}_t(s)\parallel }_2^{2}ds=E(t)-E(t+1)\equiv D{(t)}^2.$$

(3.12)

Then there exist $t_1\in [t,t+\frac{1}{4}]$ and $t_2\in [t+\frac{3}{4},t+1]$ such that

$${\parallel \mathrm{\nabla }{u}_t(t_i)\parallel }_2^2\le 4D{(t)}^2,i=1,2.$$

(3.13)

Multiplying (1.1) by u, integrating it over $\mathrm{\Omega }\times [t_1,t_2]$, using integration by parts and Lemma 2.1, we get

$$\begin{array}{c}{\int }_{t_1}^{t_2}({\parallel \mathrm{\nabla }u(s)\parallel }_2^{2(\gamma +1)}-{\parallel u(s)\parallel }_{p+1}^{p+1})ds\hfill \\ \le \epsilon B_1^2{\int }_t^{t+1}{\parallel \mathrm{\nabla }{u}_t\parallel }_2^{2}ds+\epsilon B_1^2\sum _{i=1}^2{\parallel \mathrm{\nabla }{u}_t(t_i)\parallel }_2{\parallel \mathrm{\nabla }u(t_i)\parallel }_2\hfill \\ +{\int }_t^{t+1}{\parallel \mathrm{\nabla }{u}_t\parallel }_2{\parallel \mathrm{\nabla }u\parallel }_{2}ds.\hfill \end{array}$$

Hence, from the definition of $I(t)$ by (3.1), using (3.12), (3.13) and (3.8), we obtain

$${\int }_{t_1}^{t_2}I(s)ds\le \epsilon B_1^{2}D{(t)}^2+(4\epsilon B_1^2+1)D(t){(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(t))}^{\frac{1}{2(\gamma +1)}}.$$

(3.14)

On the other hand, integrating (3.4) over $[t,t_2]$, noting that $E(t_2)\le 2{\int }_{t_1}^{t_2}E(t)dt$ due to $t_2-t_1\ge \frac{1}{2}$, and using (3.3) and (3.7), we have that

$$\begin{array}{rcl}E(t)& =& E(t_2)+{\int }_t^{t_2}{\parallel \mathrm{\nabla }{u}_t(s)\parallel }_2^{2}ds\\ \le & 2{\int }_{t_1}^{t_2}E(s)ds+{\int }_t^{t_2}{\parallel \mathrm{\nabla }{u}_t(s)\parallel }_2^{2}ds\\ \le & (1+\epsilon B_1^2){\int }_t^{t_2}{\parallel \mathrm{\nabla }{u}_t(s)\parallel }_2^{2}ds+\frac{p-2\gamma -1}{(p+1)(\gamma +1)}{\int }_t^{t_2}{\parallel \mathrm{\nabla }u(s)\parallel }_2^{2(\gamma +1)}ds\\ +\frac{2}{(p+1)}{\int }_t^{t_2}I(s)ds.\end{array}$$

We then use the fact that ${\parallel \mathrm{\nabla }u(t)\parallel }_2^{2(\gamma +1)}\le \frac{1}{1-l}I(t)$ by (3.10), (3.12) and (3.14) to obtain

$$\begin{array}{rcl}E(t)& \le & (1+\epsilon B_1^2)D{(t)}^2+c_1{\int }_t^{t_2}I(t)dt\\ \le & \alpha D{(t)}^2+\beta D(t)E{(t)}^{\frac{1}{2(\gamma +1)}},\end{array}$$

(3.15)

where

$$\alpha =(1+\epsilon B_1^2+c_1\epsilon B_1^2)\text{and}\beta =c_1(4\epsilon B_1^2+1){\left(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}\right)}^{\frac{1}{2(\gamma +1)}}$$

(3.16)

with

$$c_1=\frac{p-2\gamma -1}{(p+1)(\gamma +1)(1-l)}+\frac{2}{(p+1)}.$$

Moreover, from (3.12) and $E(t)$ being nonincreasing by (3.4), we note that

$$D(t)\le E^{\frac{1}{2}}(t)\le E{(0)}^{\frac{\gamma }{2(\gamma +1)}}E{(t)}^{\frac{1}{2(\gamma +1)}}.$$

Thus, by Young’s inequality, (3.15) becomes

$$\begin{array}{rcl}E(t)& \le & (\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )D(t)E{(t)}^{\frac{1}{2(\gamma +1)}}\\ \le & \frac{2\gamma +1}{2(\gamma +1)}{[(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )D(t)]}^{\frac{2(\gamma +1)}{2\gamma +1}}+\frac{1}{2(\gamma +1)}E(t).\end{array}$$

This implies that

$$\begin{array}{rcl}E{(t)}^{1+\frac{\gamma }{\gamma +1}}& \le & {(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^2{D}^2(t)\\ =& {(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^2(E(t)-E(t+1)).\end{array}$$

Therefore, applying Lemma 2.2, we conclude that

$$E(t)\le {(E{(0)}^{-\frac{\gamma }{\gamma +1}}+\frac{\gamma +1}{\gamma }{(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^{-2}{[t-1]}^{+})}^{-\frac{\gamma +1}{\gamma }},\text{for }t\ge 0.$$

 □

Remark (i) Based on the estimate (3.11), we have further the following estimates:

$${\parallel \mathrm{\nabla }u\parallel }_2^2\le c_2{(1+t)}^{-\frac{1}{\gamma }},{\parallel u_t\parallel }_2^2\le c_3{(1+t)}^{-1-\frac{1}{\gamma }},\text{for }t\ge 0,$$

(3.17)

where $c_2$ and $c_3$ are some positive constants.

• (ii) If $\omega >\frac{\gamma }{\gamma +1}$, then

  $${\int }_0^{t}E{(s)}^{\omega }ds\le k_{1}E{(0)}^{\omega -\frac{\gamma }{\gamma +1}},$$

  (3.18)

with $k_1=\frac{{(\gamma +1)}^2\omega -\gamma }{((\gamma +1)\omega -\gamma )(\gamma +1)}{(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^2$. Indeed, by (3.11), we see that

$$\begin{array}{rcl}{\int }_0^{t}E{(s)}^{\omega }ds& \le & {\int }_0^t{(E{(0)}^{-\frac{\gamma }{\gamma +1}}+\frac{\gamma +1}{\gamma }{(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^{-2}{[s-1]}^{+})}^{-\frac{(\gamma +1)\omega }{\gamma }}ds\\ =& E{(0)}^{\omega }+{\int }_1^t{(E{(0)}^{-\frac{\gamma }{\gamma +1}}+\frac{\gamma +1}{\gamma }{(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^{-2}(s-1))}^{-\frac{(\gamma +1)\omega }{\gamma }}ds.\end{array}$$

A direct computation yields the result.

Next, we will improve the decay rate for ${\parallel u_t\parallel }_2^2$ given by (3.17).

Proposition 3.4 If$(u_0,u_1)\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$, then the solution u of problem (1.1)-(1.3) satisfies

$${\parallel u_t\parallel }_2^2\le c_4{(1+t)}^{-2-\frac{1}{\gamma }}\mathit{\text{and}}{\int }_0^t{\parallel \mathrm{\nabla }{u}_t\parallel }_2^{2}ds\le c_5,\mathit{\text{for }}t\ge 0,$$

(3.19)

where$c_4$and$c_5$are some positive constants.

Proof Multiplying (1.1) by $u_t$ and integrating it over Ω, we have

$$\frac{\epsilon }{2}\frac{d}{dt}{\parallel u_t\parallel }_2^2+{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2=-{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx+{\int }_{\mathrm{\Omega }}|u{|}^{p-1}u{u}_{t}dx.$$

(3.20)

We now estimate the right-hand side of (3.20). Employing Hölder’s inequality and Young’s inequality, the first term gives

$$\begin{array}{rcl}{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx& \le & {\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma +1}{\parallel \mathrm{\nabla }{u}_t\parallel }_2\\ \le & {\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}+\frac{1}{4}{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2.\end{array}$$

As for the second term, using Hölder’s inequality, Lemma 2.1, Young’s inequality and (3.8) yields

$$\begin{array}{rcl}|{\int }_{\mathrm{\Omega }}|u{|}^{p-1}u{u}_{t}dx|& \le & {\parallel u\parallel }_{p+1}^p{\parallel u_t\parallel }_{p+1}\le B_1^{p+1}{\parallel \mathrm{\nabla }u\parallel }_2^p{\parallel \mathrm{\nabla }{u}_t\parallel }_2\\ \le & B_1^{2(p+1)}{\parallel \mathrm{\nabla }u\parallel }_2^{2p}+\frac{1}{4}{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2\\ \le & B_1^{2(p+1)}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(0))}^{\frac{p-2\gamma -1}{\gamma +1}}{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}+\frac{1}{4}{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2.\end{array}$$

Combining these estimates and using (3.17), we arrive at

$$\epsilon \frac{d}{dt}{\parallel u_t\parallel }_2^2+{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2\le c_6{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}\le c_7{(1+t)}^{-2-\frac{1}{\gamma }},$$

where $c_6=2(1+B_1^{2(p+1)}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(0))}^{\frac{p-2\gamma -1}{\gamma +1}})$ and $c_7=c_6{c}_2^{2\gamma +1}$. Then, from Lemma 2.1, we have

$$\epsilon \frac{d}{dt}{\parallel u_t\parallel }_2^2+B_1^{-1}{\parallel u_t\parallel }_2^2\le c_7{(1+t)}^{-2-\frac{1}{\gamma }}.$$

Therefore, the desired estimate (3.19) is obtained. □

In order to obtain the decay estimates for problem (1.1)-(1.3), we need the function $H(t)$ and equality (3.22) as in [9]. Define

$$H(t)=\{\begin{array}{cc}\epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }}+{\parallel \mathrm{\nabla }u\parallel }_2^2,\hfill & \text{if }\gamma \ge 1;\hfill \\ \epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^2}+{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma },\hfill & \text{if }0<\gamma <1.\hfill \end{array}$$

(3.21)

Multiplying (1.1) by $\frac{2u_t}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +k)}}$ with $k\ge 0$ and integrating it over Ω, we have the following equality:

(3.22)

Proposition 3.5 Let$(u_0,u_1)\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$, $I(0)>0$, (3.5)-(3.6) hold and${\parallel \mathrm{\nabla }{u}_0\parallel }_2>0$. Suppose that${\parallel \mathrm{\nabla }u(t)\parallel }_2>0$for$0\le t<T_1$and

(3.23)

(3.24)

Then it holds that, for$0\le t<T_1$,

(3.25)

(3.26)

where constants${\alpha }_1$and${\alpha }_2$are given by (3.31) and (3.36).

Proof (i) In the case $\gamma \ge 1$, we observe from (3.21) that

$$\epsilon \frac{{\parallel u_t\parallel }_2}{{\parallel \mathrm{\nabla }u\parallel }_2}={\epsilon }^{\frac{1}{2}}{\left(\epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }}{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma -2}\right)}^{\frac{1}{2}}\le {(\epsilon H^{\gamma }(t))}^{\frac{1}{2}}.$$

(3.27)

Using (3.22) with $k=0$ yields

(3.28)

Now, we estimate the right-hand side of (3.28). So, thanks to Hölder’s inequality, Lemma 2.1, (3.8) and Young’s inequality, we see that

$$2\gamma \epsilon \frac{{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +1)}}{\parallel u_t\parallel }_2^2\le 2\gamma B_1\epsilon \frac{{\parallel u_t\parallel }_2}{{\parallel \mathrm{\nabla }u\parallel }_2}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }},$$

and

$$\begin{array}{rcl}2\frac{{\int }_{\mathrm{\Omega }}|u{|}^{p-1}u{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }}& \le & 2B_1^{p+1}{\parallel \mathrm{\nabla }u\parallel }_2^{p-\gamma }\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2}{{\parallel \mathrm{\nabla }u\parallel }_2^{\gamma }}\\ \le & 2B_1^{p+1}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(t))}^{\frac{p-\gamma }{2(\gamma +1)}}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2}{{\parallel \mathrm{\nabla }u\parallel }_2^{\gamma }}\\ \le & B_1^{2(p+1)}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(t))}^{\frac{p-\gamma }{\gamma +1}}+\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }}.\end{array}$$

Taking into account these two estimates in (3.28) and using $\epsilon \frac{{\parallel u_t\parallel }_2}{{\parallel \mathrm{\nabla }u\parallel }_2}\le {(\epsilon H^{\gamma }(t))}^{\frac{1}{2}}$ by (3.27), we obtain

$$\frac{d}{dt}H(t)+(1-2\gamma B_1{(\epsilon H^{\gamma }(t))}^{\frac{1}{2}})\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }}\le B_1^{2(p+1)}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(t))}^{\frac{p-\gamma }{\gamma +1}}.$$

If $1-2\gamma B_1{(\epsilon H^{\gamma }(0))}^{\frac{1}{2}}>0$, then there exists $T_2$ such that $0<T_2\le T_1$ and

$$1-2\gamma B_1{(\epsilon H^{\gamma }(t))}^{\frac{1}{2}}\ge 0,\text{for }0\le t\le T_2.$$

(3.29)

Hence

$$\frac{d}{dt}H(t)\le B_1^{2(p+1)}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(t))}^{\frac{p-\gamma }{\gamma +1}},$$

(3.30)

for $0\le t\le T_2$. Applying (3.18) with $\omega =\frac{p-\gamma }{\gamma +1}$, we derive that

$$H(t)\le H(0)+{\alpha }_{1}E{(0)}^{\frac{p-2\gamma }{\gamma +1}},$$

(3.31)

with

$${\alpha }_1=B_1^{2(p+1)}{\left(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}\right)}^{\frac{p-\gamma }{\gamma +1}}\frac{(\gamma +1)(p-\gamma )-\gamma }{(p-2\gamma )(\gamma +1)}{(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^2.$$

Therefore, we see that (3.29)-(3.31) hold true for $0\le t<T_1$, which gives the estimate (3.25) for $\gamma \ge 1$.

• (ii) In the case $0<\gamma <1$, we note from (3.21) that

  $$\epsilon \frac{{\parallel u_t\parallel }_2}{{\parallel \mathrm{\nabla }u\parallel }_2}\le {(\epsilon H(t))}^{\frac{1}{2}}.$$

  (3.32)

From (3.22) with $k=1-\gamma$, we have

$$\begin{array}{rcl}\frac{d}{dt}H(t)+2\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^2}& =& -2\epsilon \frac{{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^4}{\parallel u_t\parallel }_2^2-2(1-\gamma )\frac{{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(1-\gamma )}}\\ +2\frac{{\int }_{\mathrm{\Omega }}|u{|}^{p-1}u{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^2}.\end{array}$$

(3.33)

Similar to those estimates, as in the case for $\gamma \ge 1$, the right-hand side of (3.33) can be estimated as follows:

and

$$2\frac{{\int }_{\mathrm{\Omega }}|u{|}^{p-1}u{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^2}\le 2B_1^{2(p+1)}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}E(t))}^{\frac{p-1}{\gamma +1}}+\frac{1}{2}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^2}.$$

Combining these estimates, (3.33) becomes

$$\begin{array}{c}\frac{d}{dt}H(t)+(1-2B_1{(\epsilon H(t))}^{\frac{1}{2}})\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^2}\hfill \\ \le 2{(1-\gamma )}^2{\left(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}\right)}^{\frac{2\gamma }{\gamma +1}}E{(t)}^{\frac{2\gamma }{\gamma +1}}\hfill \\ +2B_1^{2(p+1)}{\left(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1}\right)}^{\frac{p-1}{\gamma +1}}E{(t)}^{\frac{p-1}{\gamma +1}}\hfill \\ =c_{8}E{(0)}^{\frac{2\gamma }{\gamma +1}}{\left(\frac{E(t)}{E(0)}\right)}^{\frac{2\gamma }{\gamma +1}}+c_{9}E{(0)}^{\frac{p-1}{\gamma +1}}{\left(\frac{E(t)}{E(0)}\right)}^{\frac{p-1}{\gamma +1}}\hfill \\ \le c_{10}{(E(t))}^{\frac{2\gamma }{\gamma +1}},\hfill \end{array}$$

where we have used the fact that $0<E(t)\le E(0)$ and $p>2\gamma +1$ on the last inequality with $c_8=2{(1-\gamma )}^2{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1})}^{\frac{2\gamma }{\gamma +1}}$, $c_9=2B_1^{2(p+1)}{(\frac{2(p+1)(\gamma +1)}{p-2\gamma -1})}^{\frac{p-1}{\gamma +1}}$ and $c_{10}=c_8+c_{9}E{(0)}^{\frac{p-2\gamma -1}{\gamma +1}}$.

If $1-2B_1{(\epsilon H(0))}^{\frac{1}{2}}>0$, then there exists $T_3$ such that $0<T_3\le T_1$ and

$$1-2B_1{(\epsilon H(t))}^{\frac{1}{2}}\ge 0,\text{for }0\le t\le T_3.$$

(3.34)

Thus

$$\frac{d}{dt}H(t)\le c_{10}{(E(t))}^{\frac{2\gamma }{\gamma +1}},$$

(3.35)

for $0\le t\le T_3$. Employing (3.18) with $\omega =\frac{2\gamma }{\gamma +1}$ gives

$$H(t)\le H(0)+{\alpha }_{2}E{(0)}^{\frac{\gamma }{\gamma +1}},\text{for }0\le t\le T_3,$$

(3.36)

with

$${\alpha }_2=\frac{(2\gamma +1)c_{10}}{\gamma +1}{(\alpha E{(0)}^{\frac{\gamma }{2(\gamma +1)}}+\beta )}^2.$$

Therefore, we see that (3.34)-(3.36) hold true for $0\le t<T_1$, which gives the estimate (3.26) for $0<\gamma <1$. □

Proposition 3.6 Under the same assumptions as in Proposition  3.5, assume further that the initial data satisfies

$$\{\begin{array}{cc}1-2(\gamma +2)B_1{(\epsilon {(H(0)+{\alpha }_{1}E{(0)}^{\frac{p-2\gamma }{\gamma +1}})}^{\gamma })}^{\frac{1}{2}}>0,\hfill & \mathit{\text{if }}\gamma \ge 1;\hfill \\ 1-2(2\gamma +1)B_1{(\epsilon (H(0)+{\alpha }_{2}E{(0)}^{\frac{\gamma }{\gamma +1}}))}^{\frac{1}{2}}>0,\hfill & \mathit{\text{if }}0<\gamma <1.\hfill \end{array}$$

(3.37)

Then it holds that

$${\parallel \mathrm{\nabla }u\parallel }_2^2\ge {\alpha }_3{(1+t)}^{-\frac{1}{\gamma }},\mathit{\text{for }}0\le t<T_1,$$

(3.38)

where${\alpha }_3$is a positive constant.

Proof (i) In the case $\gamma \ge 1$, we obtain from (3.22) with $k=2$ that

(3.39)

As in deriving the estimates for case (i) of Proposition 3.5, we get the following estimates:

$$\begin{array}{c}2(\gamma +2)\epsilon \frac{{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +3)}}{\parallel u_t\parallel }_2^2\le 2(\gamma +2)B_1\epsilon \frac{{\parallel u_t\parallel }_2}{{\parallel \mathrm{\nabla }u\parallel }_2}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}},\hfill \\ 4\frac{{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^4}\le \frac{1}{2}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}+8{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma -1)},\hfill \end{array}$$

and

$$2\frac{{\int }_{\mathrm{\Omega }}|u{|}^{p-1}u{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}\le \frac{1}{2}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}+2B_1^{2(p+1)}{\parallel \mathrm{\nabla }u\parallel }_2^{2(p-\gamma -2)}.$$

Thus, using the above estimates in (3.39), together with the fact that $\epsilon \frac{{\parallel u_t\parallel }_2}{{\parallel \mathrm{\nabla }u\parallel }_2}\le {(\epsilon H^{\gamma }(t))}^{\frac{1}{2}}$ by (3.27) and the estimate (3.25), yield

$$\begin{array}{c}\frac{d}{dt}[\epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}+\frac{1}{{\parallel \mathrm{\nabla }u\parallel }_2^2}]+\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}\hfill \\ \le 2(\gamma +2)B_1{(\epsilon H^{\gamma }(t))}^{\frac{1}{2}}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}\hfill \\ +8{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma -1)}+2B_1^{2(p+1)}{\parallel \mathrm{\nabla }u\parallel }_2^{2(p-\gamma -2)}\hfill \\ \le 2(\gamma +2)B_1{\left(\epsilon {(H(0)+{\alpha }_{1}E{(0)}^{\frac{p-2\gamma }{\gamma +1}})}^{\gamma }\right)}^{\frac{1}{2}}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}\hfill \\ +8{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma -1)}+2B_1^{2(p+1)}{\parallel \mathrm{\nabla }u\parallel }_2^{2(p-\gamma -2)},\hfill \end{array}$$

for $0\le t<T_1$.

If $1-2(\gamma +2)B_1{(\epsilon {(H(0)+{\alpha }_{1}E{(0)}^{\frac{p-2\gamma }{\gamma +1}})}^{\gamma })}^{\frac{1}{2}}>0$, then, using ${\parallel \mathrm{\nabla }u\parallel }_2^2\le c_2{(1+t)}^{-\frac{1}{\gamma }}$ by (3.17), we see that

$$\begin{array}{rcl}\frac{d}{dt}[\epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}+\frac{1}{{\parallel \mathrm{\nabla }u\parallel }_2^2}]& \le & c_{11}({(1+t)}^{-\frac{\gamma -1}{\gamma }}+{(1+t)}^{-\frac{p-\gamma -2}{\gamma }})\\ \le & 2c_{11}{(1+t)}^{-{\lambda }_1}=2c_{11}{(1+t)}^{-\frac{\gamma -1}{\gamma }},\end{array}$$

and

$$\epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +2)}}+\frac{1}{{\parallel \mathrm{\nabla }u\parallel }_2^2}\le c_{12}{(1+t)}^{\frac{1}{\gamma }},$$

which implies that

$${\parallel \mathrm{\nabla }u\parallel }_2^2\ge c_{13}{(1+t)}^{-\frac{1}{\gamma }},\text{for }0\le t<T_1,$$

with ${\lambda }_1=min\{\frac{\gamma -1}{\gamma },\frac{p-\gamma -2}{\gamma }\}=\frac{\gamma -1}{\gamma }$ and some positive constants $c_i$, $i=11\text{-}13$.

• (ii) In the case $0<\gamma <1$, it follows from (3.22) with $k=1+\gamma$ that

  

  (3.40)

Estimate the right-hand side of (3.40) as in the case (ii) of Proposition 3.5 to obtain

$$\begin{array}{c}2(2\gamma +1)\epsilon \frac{{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +2)}}{\parallel u_t\parallel }_2^2\le 2(2\gamma +1)B_1{(\epsilon H(t))}^{\frac{1}{2}}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}},\hfill \\ 2(\gamma +1)\frac{{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\mathrm{\nabla }{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(\gamma +1)}}\le 2{(\gamma +1)}^2+\frac{1}{2}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}},\hfill \end{array}$$

and

$$\frac{2{\int }_{\mathrm{\Omega }}|u{|}^{p-1}u{u}_{t}dx}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}}\le 2B_1^{2(p+1)}{\parallel \mathrm{\nabla }u\parallel }_2^{2(p-2\gamma -1)}+\frac{1}{2}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}}.$$

Back to (3.40), employing these estimates and using (3.26), we deduce that

$$\begin{array}{c}\frac{d}{dt}[\epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}}+\frac{1}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }}]+\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}}\hfill \\ \le 2(2\gamma +1)B_1{(\epsilon H(t))}^{\frac{1}{2}}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}}\hfill \\ +2{(\gamma +1)}^2+2B_1^{2(p+1)}{\left({\parallel \mathrm{\nabla }u\parallel }_2^2\right)}^{(p-2\gamma -1)}\hfill \\ \le 2(2\gamma +1)B_1{\left(\epsilon (H(0)+{\alpha }_{2}E{(0)}^{\frac{\gamma }{\gamma +1}})\right)}^{\frac{1}{2}}\frac{{\parallel \mathrm{\nabla }{u}_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}}\hfill \\ +2{(\gamma +1)}^2+2B_1^{2(p+1)}{\left({\parallel \mathrm{\nabla }u\parallel }_2^2\right)}^{(p-2\gamma -1)}\hfill \end{array}$$

for $0\le t<T_1$.

If $1-2(2\gamma +1)B_1{(\epsilon (H(0)+{\alpha }_{2}E{(0)}^{\frac{\gamma }{\gamma +1}}))}^{\frac{1}{2}}>0$, then, by ${\parallel \mathrm{\nabla }u\parallel }_2^2\le c_2{(1+t)}^{-\frac{1}{\gamma }}$, we have

$$\frac{d}{dt}[\epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}}+\frac{1}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }}]\le 2{(\gamma +1)}^2+c_{14}{(1+t)}^{-\frac{p-2\gamma -1}{\gamma }}\le c_{15},$$

and

$$\epsilon \frac{{\parallel u_t\parallel }_2^2}{{\parallel \mathrm{\nabla }u\parallel }_2^{2(2\gamma +1)}}+\frac{1}{{\parallel \mathrm{\nabla }u\parallel }_2^{2\gamma }}\le c_{16}(1+t),$$

for $0\le t<T_1$ with some positive constants $c_i$, $i=14\text{-}16$. This implies the desired estimate (3.38) for $0<\gamma <1$. □

Now, we are ready to state and prove our main result.

Theorem 3.7 Let$(u_0,u_1)\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$, $I(0)>0$and (3.5)-(3.6) hold. Then the solution u of problem (1.1)-(1.3) satisfies

$${\parallel \mathrm{\nabla }u\parallel }_2^2\le {\alpha }_4{(1+t)}^{-\frac{1}{\gamma }}\mathit{\text{and}}{\parallel u_t\parallel }_2^2\le {\alpha }_5{(1+t)}^{-2-\frac{1}{\gamma }}\mathit{\text{for }}t\ge 0.$$

(3.41)

Moreover, suppose that the initial data satisfies$u_0\ne 0$and

$$\{\begin{array}{cc}1-2(\gamma +2)B_1{(\epsilon {(H(0)+{\alpha }_{1}E{(0)}^{\frac{p-2\gamma }{\gamma +1}})}^{\gamma })}^{\frac{1}{2}}>0,\hfill & \mathit{\text{if }}\gamma \ge 1;\hfill \\ 1-6B_1{(\epsilon (H(0)+{\alpha }_{2}E{(0)}^{\frac{\gamma }{\gamma +1}}))}^{\frac{1}{2}}>0,\hfill & \mathit{\text{if }}0<\gamma <1.\hfill \end{array}$$

(3.42)

Then

$${\parallel \mathrm{\nabla }u\parallel }_2^2\ge {\alpha }_6{(1+t)}^{-\frac{1}{\gamma }},\mathit{\text{for }}t\ge 0,$$

(3.43)

with some positive constants${\alpha }_i$, $i=4,5,6$.

Proof Thanks to the decay estimates (3.11) and (3.17), we obtain (3.41). Setting

$$T_1=sup\{t\in [0,\mathrm{\infty })|{\parallel \mathrm{\nabla }u(s)\parallel }_2^2>0\text{ for }0\le s<t\},$$

then, we see that $T_1>0$ and ${\parallel \mathrm{\nabla }u(t)\parallel }_2^2>0$ for $0\le t<T_1$, because of ${\parallel \mathrm{\nabla }{u}_0\parallel }_2>0$. If $T_1<\mathrm{\infty }$, then it holds that

$${\parallel \mathrm{\nabla }u(T_1)\parallel }_2^2=0.$$

(3.44)

However, from ${\parallel \mathrm{\nabla }u\parallel }_2^2\ge {\alpha }_3{(1+t)}^{-\frac{1}{\gamma }}$ by (3.38), we observe that

$$\underset{t\to T_1}{lim}{\parallel \mathrm{\nabla }u(t)\parallel }_2^2\ge C{(1+T_1)}^{-\frac{1}{\gamma }}>0,$$

which contradicts (3.44). Hence, we have $T_1=\mathrm{\infty }$ and ${\parallel \mathrm{\nabla }u(t)\parallel }_2^2>0$ for all $t\ge 0$. Therefore, from (3.38), we have

$${\parallel \mathrm{\nabla }u(t)\parallel }_2^2\ge {\alpha }_6{(1+t)}^{-\frac{1}{\gamma }},\text{for }t\ge 0,$$

which gives the decay estimate (3.44). □