Abstract
In this paper we study the initialboundary value problem of the multidimensional viscoelasticity equation with nonlinear source term . By using the potential well method, we first prove the global existence. Then we prove that when time , the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial data.
1 Introduction
This paper considers the initialboundary value problem (IBVP) of the multidimensional viscoelasticity equation with nonlinear source term
where is the unknown function with respect to the spacial variable and the time variable is a bounded domain.
The viscoelasticity equation
was suggested and studied by Greenberg et al.[1,2] from viscoelasticity mechanics in 1968. Under the condition and higher smooth conditions on and the initial data, they obtained the global existence of classical solutions for the initialboundary value problem of Eq. (1.4).
After that many authors [311] studied the global wellposedness of IBVP for Eq. (1.4). In [310] the global existence, uniqueness and stability of solution were studied thoroughly. And in [11] the blow up of solution was discussed. Furthermore, in [1214] 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
was studied, and by using the potential well method, the global existence of weak solution was proved under some assumptions on nonlinear functions , , 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 [1722].
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 and satisfy the following assumptions:
where the constants in (H_{1}) and (H_{2}) 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).
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 and 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 .
Lemma 2.2Let (H_{1}) and (H_{2}) hold. Set
Then the following hold:
Proof This lemma follows from and . □
Lemma 2.3Let (H_{1}) and (H_{2}) hold, . Then the following hold:
where
Proof
which gives
we get
□
Lemma 2.4Let (H_{1}) and (H_{2}) hold. Then the following holds:
Proof For any , by Lemma 2.3, we have and
which gives (2.1). □
Now, for problem (1.1)(1.3), we define
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 a weak solution of problem (1.1)(1.3) on if , satisfying
(i)
(ii)
Theorem 3.2Let (H_{1}) and (H_{2}) hold, , . Assume that, . Then problem (1.1)(1.3) admits a global weak solutionand.
Proof Let be a system of base functions in . Construct the approximate solutions of problem (1.1)(1.3)
satisfying
Multiplying (3.1) by and summing for s, we get
and
where
From (3.2) and (3.3), we have as . Hence, for sufficiently large n, we have and
On the other hand, since W is an open set in , Eq. (3.2) implies that for sufficiently large n, we have . Next, we prove that for and sufficiently large n. If it is false, then there exists a such that , i.e. and , i.e.. So, by the definition of d, we get , which contradicts (3.6).
From (3.6) we have
which gives
and
and
From (3.7) we can get
Hence there exist u, , η and a subsequence of such that as , in weakstar, and a.e. in , in weakstar and in weakly, in weakstar, , in weakstar, .
Integrating (3.1) with respect to t, we have
and
On the other hand, from (3.2) and (3.3), we get in , in . 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 4.1Let (H_{1}) and (H_{2}) hold, , . Then, for the approximate solutionsof problem (1.1)(1.3) constructed in the proof of Theorem 3.2, the following hold:
(i)
(ii) Furthermore, ifand, then for sufficiently largen, there exists asuch that
Proof (i) Multiplying (3.1) by and summing for s, we get (4.1).
(ii) From
it follows that there exists a such that
From (3.2), (3.3) and (4.3), it follows that for sufficiently large n. Hence from (3.5) we have
and
which gives
and
which together with for sufficiently large n gives
and
Hence, by Lemma 2.3, we have or . So, we have
□
Theorem 4.2Let (H_{1}) and (H_{2}) hold, , . Assume that, . Then, for the global weak solutionugiven in Theorem 3.2, there exist positive constantsCandλsuch that
Proof Let 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 (), we get
and
From (H_{2}), Lemma 2.2 and Lemma 4.1, we get
where
Hence we have
and
From
we get
and
which together with for sufficiently large n gives
and
From (4.8) and the Poincaré inequality , it follows that there exists a constant such that
From (4.5)(4.10) it follows that there exists a such that
Choose α such that
Then from (4.11) we get
From this and the Gronwall inequality, we get
and
On the other hand, from (4.8) we get
Hence, there exists a such that
Let be the subsequence of in the proof of Theorem 3.2. Then from (4.13) and(4.12), we obtain
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The work presented here was carried out in collaboration between all authors. RZ organised this paper and found the topic of this paper. He introduced this problem and suggested the methods and the outline of the proofs. JL finished the proof of the global existence and YN finished the long time behaviour part. SC discussed all the problems arising in the research and provided many good ideas for proving the problems. All authors have contributed to, seen and approved the manuscript.
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.
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