Asymptotic behaviour of solution for multidimensional viscoelasticity equation with nonlinear source term

Runzhang Xu; Jie Liu; Yi Niu; Shaohua Chen

doi:10.1186/1687-2770-2013-42

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Asymptotic behaviour of solution for multidimensional viscoelasticity equation with nonlinear source term

Runzhang Xu¹Email author,
Jie Liu¹,
Yi Niu² and
Shaohua Chen³

Boundary Value Problems20132013:42

https://doi.org/10.1186/1687-2770-2013-42

Received: 24 April 2012

Accepted: 11 February 2013

Published: 1 March 2013

Abstract

In this paper we study the initial-boundary value problem of the multidimensional viscoelasticity equation with nonlinear source term $u_{t t} - Δ u_{t} - \sum_{i = 1}^{N} \frac{\partial}{\partial x_{i}} σ_{i} (u_{x_{i}}) = f (u)$ . By using the potential well method, we first prove the global existence. Then we prove that when time $t \to + \infty$ , the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial data.

1 Introduction

This paper considers the initial-boundary value problem (IBVP) of the multidimensional viscoelasticity equation with nonlinear source term

(1.1)

(1.2)

(1.3)

where $u (x, t)$ is the unknown function with respect to the spacial variable $x \in Ω$ and the time variable $t, Ω \subset R^{N}$ is a bounded domain.

The viscoelasticity equation

u_{t t} - u_{x x t} = σ {(u_{x})}_{x}

(1.4)

was suggested and studied by Greenberg et al. [1, 2] from viscoelasticity mechanics in 1968. Under the condition $σ^{'} (s) > 0$ and higher smooth conditions on $σ (s)$ and the initial data, they obtained the global existence of classical solutions for the initial-boundary value problem of Eq. (1.4).

After that many authors [3–11] studied the global well-posedness of IBVP for Eq. (1.4). In [3–10] the global existence, uniqueness and stability of solution were studied thoroughly. And in [11] the blow up of solution was discussed. Furthermore, in [12–14] the global existence of solution for IBVP of some multidimensional viscoelasticity equation was considered. And in [11] the blow up of solution for IBVP of the multidimensional generalisation of Eq. (1.4) was proved. Recently, in [15] and [16], the IBVP of the multidimensional viscoelasticity equation with nonlinear damping and source terms

(1.5)

(1.6)

(1.7)

was studied, and by using the potential well method, the global existence of weak solution was proved under some assumptions on nonlinear functions $σ_{i} (s)$ , $f (s)$ , $g (s)$ and the initial data. But we do not know how the global solution behaves as the time goes to infinity, namely the asymptotic behaviour of problem (1.1)-(1.3) is still open up to now. In the present paper, we try to study this problem by the multiplier method [17–22].

The main purpose of present paper is to consider the asymptotic behaviour of solution for problem (1.1)-(1.3). Since in the proof of the asymptotic behaviour of solution the global existence theory is required, it is necessary to give the proof of global existence of solution for problem (1.1)-(1.3).

In this paper, suppose that

σ (s) = (σ_{1} (s), \dots, σ_{N} (s))

and

f (s)

satisfy the following assumptions:

where the constants in (H₁) and (H₂) are all positive and satisfy

In this paper, we first give some definitions and lemmas (Section 2). Then we prove the global existence of solution (Section 3). Finally, we prove the asymptotic behaviour of solution (Section 4).

In this paper, we denote ${∥ \cdot ∥}_{L^{p} (Ω)}$ by ${∥ \cdot ∥}_{p}$ , $∥ \cdot ∥ = {∥ \cdot ∥}_{L^{2} (Ω)}$ and $(u, v) = \int_{Ω} u v d x$ .

2 Preliminaries

In this section, we will give some definitions and prove some lemmas for problem (1.1)-(1.3).

For problem (1.1)-(1.3), we define

Remark 2.1 Note that the definitions of $J (u)$ and $I (u)$ in the present paper are different from those in [11] and [15]. The definitions given in this paper will be shown more natural and rational because they are a part of the total energy $E (t)$ .

Lemma 2.2 Let (H₁) and (H₂) hold. Set

\bar{σ_{i}} (s) = σ_{i} (s) - a s, \bar{G_{i}} (s) = \int_{0}^{s} \bar{σ_{i}} (s) d τ .

Then the following hold:

(i)

$\bar{σ_{i}} (s)$ is increasing and $s \bar{σ_{i}} (s) \geq 0$ $\forall s \in R$ ;
(ii)

$0 \leq \bar{G_{i}} (s) \leq s \bar{σ_{i}} (s)$ $\forall s \in R$ .

Proof This lemma follows from $\bar{σ_{i}} (0) = 0$ and ${\bar{σ}}_{i}^{'} (s) \geq 0$ . □

Lemma 2.3 Let (H₁) and (H₂) hold,

u \in W_{0}^{1, m + 1} (Ω)

. Then the following hold:

(i)

If $0 < {∥ \nabla u ∥}_{m + 1} < r (δ)$ , then $I_{δ} (u) > 0$ ;
(ii)

If $I_{δ} (u) < 0$ , then ${∥ \nabla u ∥}_{m + 1} > r (δ)$ ;
(iii)

If $I_{δ} (u) = 0$ , then ${∥ \nabla u ∥}_{m + 1} \geq r (δ)$ ,

where

r (δ) = {(\frac{A δ}{b C_{*}^{q + 1}})}^{\frac{1}{q - m}}, C_{*} = sup_{u \in W_{0}^{1, m + 1} (Ω) / 0} \frac{{∥ u ∥}_{q + 1}}{{∥ \nabla u ∥}_{m + 1}} .

Proof

(i)

If $0 < {∥ \nabla u ∥}_{m + 1} < r (δ)$ , then we have
$\begin{aligned} \int_{Ω} u f (u) d x & \leq \int_{Ω} | u f (u) | d x \leq b \int_{Ω} {| u |}^{q + 1} d x \\ = b {∥ u ∥}_{q + 1}^{q + 1} \leq b C_{*}^{q + 1} {∥ \nabla u ∥}_{m + 1}^{q + 1} \\ = \frac{b C_{*}^{q + 1}}{A} {∥ \nabla u ∥}_{m + 1}^{q - m} A {∥ \nabla u ∥}_{m + 1}^{m + 1} \\ < δ \sum_{i = 1}^{N} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x, \end{aligned}$

which gives

I_{δ} (u) > 0

.

(ii)

If $I_{δ} (u) < 0$ , then we have
$δ A {∥ \nabla u ∥}_{m + 1}^{m + 1} \leq δ \sum_{i = 1}^{N} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x < \int_{Ω} u f (u) d x \leq b C_{*}^{q + 1} {∥ \nabla u ∥}_{q - m}^{m + 1} {∥ \nabla u ∥}_{m + 1}^{m + 1},$

which gives

{∥ \nabla u ∥}_{m + 1} > r (δ) .

(iii)

If $I_{δ} (u) = 0$ and $u \neq 0$ , then by
$δ A {∥ \nabla u ∥}_{m + 1}^{m + 1} \leq δ \sum_{i = 1}^{N} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x < \int_{Ω} u f (u) d x \leq b C_{*}^{q + 1} {∥ \nabla u ∥}_{m + 1}^{q - m} {∥ \nabla u ∥}_{m + 1}^{m + 1},$

we get

{∥ \nabla u ∥}_{m + 1} \geq r (δ) .

□

Lemma 2.4 Let (H₁) and (H₂) hold. Then the following holds:

d \geq d_{0} = \frac{(p - l) A}{(p + 1) (l + 1)} {(\frac{A}{b C_{*}^{q + 1}})}^{\frac{m + 1}{q - m}} .

(2.1)

Proof For any

u \in N

, by Lemma 2.3, we have

{∥ \nabla u ∥}_{m + 1} \geq r (1)

and

\begin{aligned} J (u) & = \sum_{i = 1}^{N} \int_{Ω} G_{i} (u_{x_{i}}) d x - \int_{Ω} F (u) d x \\ \geq \frac{1}{l + 1} \sum_{i = 1}^{N} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x - \frac{1}{p + 1} \int_{Ω} u f (u) d x \\ = (\frac{1}{l + 1} - \frac{1}{p + 1}) \sum_{i = 1}^{N} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x + \frac{1}{p + 1} I (u) \\ = \frac{p - l}{(p + 1) (l + 1)} \sum_{i = 1}^{N} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x \\ \geq \frac{p - l}{(p + 1) (l + 1)} A {∥ \nabla u ∥}_{m + 1}^{m + 1} \\ \geq \frac{p - l}{(p + 1) (l + 1)} A r^{m + 1} (1) = \frac{(p - l) A}{(p + 1) (l + 1)} {(\frac{A}{b C_{*}^{q + 1}})}^{\frac{m + 1}{q - m}}, \end{aligned}

which gives (2.1). □

Now, for problem (1.1)-(1.3), we define

W = {u \in W_{0}^{1, m + 1} (Ω) ∣ I (u) > 0} \cup {0} .

3 Global existence of solution

In this section, we prove the global existence of weak solution for problem (1.1)-(1.3).

Definition 3.1 We call

u = u (x, t)

a weak solution of problem (1.1)-(1.3) on

Ω \times [0, T)

if

u \in L^{\infty} (0, T; W_{0}^{1, m + 1} (Ω))

,

u_{t} \in L^{\infty} (0, T; L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω))

satisfying

(i)
(ii)

$u (x, 0) = u_{0} (x) in W_{0}^{1, m + 1} (Ω); u_{t} (x, 0) = u_{1} (x) in L^{2} (Ω) .$

Theorem 3.2 Let (H₁) and (H₂) hold, $u_{0} (x) \in W_{0}^{1, m + 1} (Ω)$ , $u_{1} (x) \in L^{2} (Ω)$ . Assume that $E (0) < d$ , $u_{0} (x) \in W$ . Then problem (1.1)-(1.3) admits a global weak solution $u \in L^{\infty} (0, \infty; W_{0}^{1, m + 1} (Ω))$ and $u_{t} \in L^{\infty} (0, \infty; L^{2} (Ω)) \cap L^{2} (0, \infty; H_{0}^{1} (Ω))$ .

Proof Let

{w_{j} (x)}_{j = 1}^{\infty}

be a system of base functions in

W_{0}^{1, m + 1} (Ω)

. Construct the approximate solutions of problem (1.1)-(1.3)

u_{n} (x, t) = \sum_{j = 1}^{n} g_{j n} (t) w_{j} (x), n = 1, 2, \dots,

satisfying

(3.1)

(3.2)

(3.3)

Multiplying (3.1) by

g_{s n}^{'} (t)

and summing for s, we get

\frac{d}{d t} E_{n} (t) + {∥ \nabla u_{n t} ∥}^{2} = 0, 0 \leq t < \infty

(3.4)

and

E_{n} (t) + \int_{0}^{t} {∥ \nabla u_{n τ} ∥}^{2} d τ = E_{n} (0), 0 \leq t < \infty,

(3.5)

where

E_{n} (t) = \frac{1}{2} {∥ u_{n t} ∥}^{2} + J (u_{n}) .

From (3.2) and (3.3), we have

E_{n} (0) \to E (0)

as

n \to \infty

. Hence, for sufficiently large n, we have

E_{n} (0) < d

and

\frac{1}{2} {∥ u_{n t} ∥}^{2} + J (u_{n}) + \int_{0}^{t} {∥ \nabla u_{n τ} ∥}^{2} d τ < d, 0 \leq t < \infty .

(3.6)

On the other hand, since W is an open set in $W_{0}^{1, m + 1} (Ω)$ , Eq. (3.2) implies that for sufficiently large n, we have $u_{n} (0) \in W$ . Next, we prove that $u_{n} (t) \in W$ for $0 < t < \infty$ and sufficiently large n. If it is false, then there exists a $t_{0} > 0$ such that $u_{n} (t_{0}) \in \partial W$ , i.e. $I (u_{n} (t_{0})) = 0$ and $u_{n} (t_{0}) \neq 0$ , i.e. $u_{n} (t_{0}) \in N$ . So, by the definition of d, we get $J (u_{n} (t_{0})) \geq d$ , which contradicts (3.6).

From (3.6) we have

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \sum_{i = 1}^{N} \int_{Ω} G_{i} (u_{n x_{i}}) d x - \int_{Ω} F (u_{n}) d x + \int_{0}^{t} {∥ \nabla u_{n τ} ∥}^{2} d τ < d, 0 \leq t < \infty,

which gives

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{1}{l + 1} \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x - \frac{1}{p + 1} \int_{Ω} u_{n} f (u_{n}) d x + \int_{0}^{t} {∥ \nabla u_{n τ} ∥}^{2} d τ < d

and

which together with

u_{n} (t) \in W

gives

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{p - l}{(p + 1) (l + 1)} \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x + \int_{0}^{t} {∥ \nabla u_{n τ} ∥}^{2} d τ < d,

and

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{p - l}{(p + 1) (l + 1)} A {∥ \nabla u_{n} ∥}_{m + 1}^{m + 1} + \int_{0}^{t} {∥ \nabla u_{n τ} ∥}^{2} d τ < d, 0 \leq t < \infty .

(3.7)

From (3.7) we can get

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

Hence there exist u, $χ = (χ_{1}, χ_{2}, \dots, χ_{N})$ , η and a subsequence ${u_{ν}}$ of ${u_{n}}$ such that as $ν \to \infty$ , $u_{ν} \to u$ in $u \in L^{\infty} (0, \infty; W_{0}^{1, m + 1} (Ω))$ weak-star, and a.e. in $Q = Ω \times [0, \infty)$ , $u_{ν t} \to u_{t}$ in $L^{\infty} (0, \infty; L^{2} (Ω))$ weak-star and in $L^{2} (0, \infty; H_{0}^{1} (Ω))$ weakly, $σ_{i} (u_{ν x_{i}}) \to χ_{i} = σ_{i} (u_{x_{i}})$ in $L^{\infty} (0, \infty; L^{{(m + 1)}^{'}} (Ω))$ weak-star, ${(m + 1)}^{'} = \frac{m + 1}{m}$ , $f (u_{v}) \to η = f (u)$ in $L^{\infty} (0, \infty; L^{{(q + 1)}^{'}} (Ω))$ weak-star, ${(q + 1)}^{'} = \frac{q + 1}{q}$ .

Integrating (3.1) with respect to t, we have

(3.13)

Letting

n = ν \to \infty

in (3.13), we get

and

On the other hand, from (3.2) and (3.3), we get $u (x, 0) = u_{0} (x)$ in $W_{0}^{1, m + 1} (Ω)$ , $u_{t} (x, 0) = u_{1} (x)$ in $L^{2} (Ω)$ . Therefore u is a global weak solution of problem (1.1)-(1.3). □

4 Asymptotic behaviour of solution

In this section, we prove the main conclusion of this paper - the asymptotic behaviour of solution for problem (1.1)-(1.3).

Lemma 4.1 Let (H₁) and (H₂) hold,

u_{0} (x) \in W_{0}^{1, m + 1} (Ω)

,

u_{1} (x) \in L^{2} (Ω)

. Then, for the approximate solutions

u_{n} (x, t)

of problem (1.1)-(1.3) constructed in the proof of Theorem 3.2, the following hold:

(i)

$I (u_{n}) = {∥ u_{n t} ∥}^{2} - \frac{d}{d t} ((u_{n t}, u_{n}) + \frac{1}{2} {∥ \nabla u_{n} ∥}^{2});$

(4.1)
(ii)

Furthermore, if $E (0) < d_{0}$ and $u_{0} (x) \in W$ , then for sufficiently large n, there exists a $δ_{1} \in (0, 1)$ such that
$I (u_{n}) \geq (1 - δ_{1}) \sum_{i = 1}^{N} (σ_{i} (u_{n x_{i}}), u_{n x_{i}}) .$

(4.2)

Proof (i) Multiplying (3.1) by

g_{s n} (t)

and summing for s, we get (4.1).

(ii)

From
$E (0) < d_{0} = \frac{p - l}{(p + 1) (l + 1)} A {(\frac{A}{b C_{*}^{q + 1}})}^{\frac{m + 1}{q - m}}$

it follows that there exists a

δ_{1} \in (0, 1)

such that

E (0) < \frac{p - l}{(p + 1) (l + 1)} A {(\frac{A δ_{1}}{b C_{*}^{q + 1}})}^{\frac{m + 1}{q - m}} \equiv d (δ_{1}) .

(4.3)

From (3.2), (3.3) and (4.3), it follows that

E_{m} (0) < d (δ_{1})

for sufficiently large n. Hence from (3.5) we have

and

\sum_{i = 1}^{N} \int_{Ω} G_{i} (u_{n x_{i}}) d x - \int_{Ω} F (u_{n}) d x < d (δ_{1}),

which gives

\frac{1}{l + 1} \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x - \frac{1}{p + 1} \int_{Ω} u_{n} f (u_{n}) d x < d (δ_{1})

and

\frac{p - l}{(p + 1) (l + 1)} \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x + \frac{1}{p + 1} I (u_{n}) < d (δ_{1}),

which together with

u_{n} \in W

for sufficiently large n gives

\frac{p - l}{(p + 1) (l + 1)} A {∥ \nabla u_{n} ∥}_{m + 1}^{m + 1} < \frac{p - l}{(p + 1) (l + 1)} A {(\frac{A δ_{1}}{b C_{*}^{q + 1}})}^{\frac{m + 1}{q - m}}

and

{∥ \nabla u_{n} ∥}_{m + 1} < {(\frac{A δ_{1}}{b C_{*}^{q + 1}})}^{\frac{1}{q - m}} = r (δ_{1}) .

Hence, by Lemma 2.3, we have

I_{δ_{1}} (u_{n}) > 0

or

u_{n} = 0

. So, we have

\begin{aligned} I (u_{n}) & = \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x - \int_{Ω} u_{n} f (u_{n}) d x \\ = (1 - δ_{1}) \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x + I_{δ_{1}} (u_{n}) \geq (1 - δ_{1}) \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x . \end{aligned}

□

Theorem 4.2 Let (H₁) and (H₂) hold,

u_{0} (x) \in W_{0}^{1, m + 1} (Ω)

,

u_{1} (x) \in L^{2} (Ω)

. Assume that

E (0) < d_{0}

,

u_{0} (x) \in W

. Then, for the global weak solution u given in Theorem 3.2, there exist positive constants C and λ such that

{∥ u_{t} ∥}^{2} + {∥ \nabla u ∥}_{m + 1}^{m + 1} \leq C e^{- λ t}, 0 \leq t < \infty .

(4.4)

Proof Let

{u_{n}}

be the approximate solutions of problem (1.1)-(1.3) in the proof of Theorem 3.2, then (3.4) holds. Multiplying (3.4) by

e^{α t}

(

α > 0

), we get

\frac{d}{d t} (e^{α t} E_{n} (t)) + e^{α t} {∥ \nabla u_{n t} ∥}^{2} = α e^{α t} E_{n} (t)

and

e^{α t} E_{n} (t) + \int_{0}^{t} e^{α τ} {∥ \nabla u_{n τ} ∥}^{2} d τ = E_{n} (0) + α \int_{0}^{t} e^{α τ} E_{n} (τ) d τ, 0 \leq t < \infty .

(4.5)

From (H₂), Lemma 2.2 and Lemma 4.1, we get

where

C (r, δ_{1}) = \frac{1}{1 - δ_{1}} \frac{r}{r + 1} + \frac{1}{r + 1} .

Hence we have

(4.6)

and

(4.7)

From

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \sum_{i = 1}^{N} \int_{Ω} G_{i} (u_{n x_{i}}) d x - \int_{Ω} F (u_{n}) d x = E_{n} (t)

we get

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{1}{l + 1} \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x - \frac{1}{p + 1} \int_{Ω} u_{n} f (u_{n}) d x \leq E_{n} (t)

and

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{p - l}{(p + 1) (l + 1)} \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x + \frac{1}{p + 1} I (u_{n}) \leq E_{n} (t),

which together with

u_{n} \in W

for sufficiently large n gives

\begin{aligned} \frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{p - l}{(p + 1) (l + 1)} \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} σ_{i} (u_{n x_{i}}) d x \leq E_{n} (t), \\ \frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{p - l}{(p + 1) (l + 1)} (a {∥ \nabla u_{n} ∥}^{2} + \sum_{i = 1}^{N} \int_{Ω} u_{n x_{i}} {\bar{σ}}_{i} (u_{n x_{i}}) d x) \leq E_{n} (t), \end{aligned}

(4.8)

and

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{p - l}{(p + 1) (l + 1)} a {∥ \nabla u_{n} ∥}^{2} \leq E_{n} (t), 0 \leq t < \infty .

(4.9)

From (4.8) and the Poincaré inequality

{∥ \nabla u ∥}^{2} \geq λ_{1} {∥ u ∥}^{2}

, it follows that there exists a constant

C_{0} = C_{0} (p, l, a, λ_{1}) > 0

such that

({∥ u_{n t} ∥}^{2} + {∥ u_{n} ∥}^{2} + {∥ \nabla u_{n} ∥}^{2}) \leq C_{0} E_{n} (t), 0 \leq t < \infty .

(4.10)

From (4.5)-(4.10) it follows that there exists a

C_{0}

such that

(4.11)

Choose α such that

0 < α < min {\frac{1}{2 C_{0}}, \frac{λ_{1}}{\frac{1}{2} + C (r, δ_{1})}} .

Then from (4.11) we get

\begin{aligned} e^{α t} E_{n} (t) & \leq 2 (C_{0} α + 1) E_{n} (0) + 2 α^{2} C_{0} \int_{0}^{t} e^{α τ} E_{n} (τ) d τ \\ \leq 2 (C_{0} α + 1) d_{0} + 2 α^{2} C_{0} \int_{0}^{t} e^{α τ} E_{n} (τ) d τ \\ \leq 3 d_{0} + 2 α C_{0} \int_{0}^{t} e^{α τ} E_{n} (τ) d τ . \end{aligned}

From this and the Gronwall inequality, we get

e^{α t} E_{n} (t) \leq 3 d_{0} e^{- 2 α^{2} C_{0} t}

and

E_{n} (t) \leq 3 d_{0} e^{- λ t}, 0 \leq t < \infty, λ = α (1 - 2 C_{0} α) > 0 .

(4.12)

On the other hand, from (4.8) we get

\frac{1}{2} {∥ u_{n t} ∥}^{2} + \frac{p - l}{(p + 1) (l + 1)} A {∥ \nabla u_{n} ∥}_{m + 1}^{m + 1} \leq E_{n} (t), 0 \leq t < \infty .

Hence, there exists a

C_{1} = C_{1} (p, l, A)

such that

{∥ u_{n t} ∥}^{2} + {∥ \nabla u_{n} ∥}_{m + 1}^{m + 1} \leq C_{1} E_{n} (t), 0 \leq t < \infty .

(4.13)

Let

{u_{ν}}

be the subsequence of

{u_{n}}

in the proof of Theorem 3.2. Then from (4.13) and(4.12), we obtain

which gives (4.4), where $C = 3 d_{0} C_{1}$ , $λ = α (1 - 2 C_{0} α)$ . □

Declarations

Acknowledgements

We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11101102), Ph.D. Programs Foundation of the Ministry of Education of China (20102304120022), the Support Plan for the Young College Academic Backbone of Heilongjiang Province (1252G020), the Natural Science Foundation of Heilongjiang Province (A201014), Science and Technology Research Project of the Department of Education of Heilongjiang Province (12521401), Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities.

Authors’ Affiliations

(1)

College of Science, Harbin Engineering University

(2)

College of Automation, Harbin Engineering University

(3)

Department of Mathematics, Cape Breton University

References

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