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

In this paper we study the initial-boundary value problem of the multidimensional viscoelasticity equation with nonlinear source term u_{tt}-\mathrm{\Delta }{u}_t-{\sum }_{i=1}^N\frac{\partial }{\partial x_i}{\sigma }_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 +\mathrm{\infty }, the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial data.

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 \mathrm{\Omega } and the time variable t,\mathrm{\Omega }\subset {\mathbb{R}}^N 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 {\sigma }^{\prime }(s)>0 and higher smooth conditions on \sigma (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 {\sigma }_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 \sigma (s)=({\sigma }_1(s),\dots ,{\sigma }_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 {\parallel \cdot \parallel }_{L^p(\mathrm{\Omega })} by {\parallel \cdot \parallel }_p, \parallel \cdot \parallel ={\parallel \cdot \parallel }_{L^2(\mathrm{\Omega })} and (u,v)={\int }_{\mathrm{\Omega }}uv\mathrm{d}x.

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

Then the following hold:

1. (i)

   \overline{{\sigma }_i}(s) is increasing and s\overline{{\sigma }_i}(s)\ge 0 \mathrm{\forall }s\in \mathbb{R};

2. (ii)

   0\le \overline{G_i}(s)\le s\overline{{\sigma }_i}(s) \mathrm{\forall }s\in \mathbb{R}.

Proof This lemma follows from \overline{{\sigma }_i}(0)=0 and {\overline{\sigma }}_i^{\prime }(s)\ge 0. □

Lemma 2.3 Let (H₁) and (H₂) hold, u\in W_0^{1,m+1}(\mathrm{\Omega }). Then the following hold:

1. (i)

   If , then I_{\delta }(u)>0;

2. (ii)

   If , then {\parallel \mathrm{\nabla }u\parallel }_{m+1}>r(\delta );

3. (iii)

   If I_{\delta }(u)=0, then {\parallel \mathrm{\nabla }u\parallel }_{m+1}\ge r(\delta ),

where

Proof

1. (i)

   If , then we have

which gives I_{\delta }(u)>0.

1. (ii)

   If , then we have

which gives

1. (iii)

   If I_{\delta }(u)=0 and u\ne 0, then by

we get

 □

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

Proof For any u\in \mathcal{N}, by Lemma 2.3, we have {\parallel \mathrm{\nabla }u\parallel }_{m+1}\ge r(1) and

which gives (2.1). □

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

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 \mathrm{\Omega }\times [0,T) if u\in L^{\mathrm{\infty }}(0,T;W_0^{1,m+1}(\mathrm{\Omega })), u_t\in L^{\mathrm{\infty }}(0,T;L^2(\mathrm{\Omega }))\cap L^2(0,T;H_0^1(\mathrm{\Omega })) satisfying

1. (i)
   
2. (ii)
   u(x,0)=u_0(x)\text{in }{W}_0^{1,m+1}(\mathrm{\Omega });u_t(x,0)=u_1(x)\text{in }{L}^2(\mathrm{\Omega }).

Theorem 3.2 Let (H₁) and (H₂) hold, u_0(x)\in W_0^{1,m+1}(\mathrm{\Omega }), u_1(x)\in L^2(\mathrm{\Omega }). Assume that , u_0(x)\in W. Then problem (1.1)-(1.3) admits a global weak solution u\in L^{\mathrm{\infty }}(0,\mathrm{\infty };W_0^{1,m+1}(\mathrm{\Omega })) and u_t\in L^{\mathrm{\infty }}(0,\mathrm{\infty };L^2(\mathrm{\Omega }))\cap L^2(0,\mathrm{\infty };H_0^1(\mathrm{\Omega })).

Proof Let {\{w_j(x)\}}_{j=1}^{\mathrm{\infty }} be a system of base functions in W_0^{1,m+1}(\mathrm{\Omega }). Construct the approximate solutions of problem (1.1)-(1.3)

satisfying

(3.1)

(3.2)

(3.3)

Multiplying (3.1) by g_{sn}^{\prime }(t) and summing for s, we get

and

where

From (3.2) and (3.3), we have E_n(0)\to E(0) as n\to \mathrm{\infty }. Hence, for sufficiently large n, we have and

On the other hand, since W is an open set in W_0^{1,m+1}(\mathrm{\Omega }), 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 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)\ne 0, i.e. u_n(t_0)\in \mathcal{N}. So, by the definition of d, we get J(u_n(t_0))\ge d, which contradicts (3.6).

From (3.6) we have

which gives

and

which together with u_n(t)\in W gives

and

From (3.7) we can get

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

Hence there exist u, \chi =({\chi }_1,{\chi }_2,\dots ,{\chi }_N), η and a subsequence \{u_{\nu }\} of \{u_n\} such that as \nu \to \mathrm{\infty }, u_{\nu }\to u in u\in L^{\mathrm{\infty }}(0,\mathrm{\infty };W_0^{1,m+1}(\mathrm{\Omega })) weak-star, and a.e. in Q=\mathrm{\Omega }\times [0,\mathrm{\infty }), u_{\nu t}\to u_t in L^{\mathrm{\infty }}(0,\mathrm{\infty };L^2(\mathrm{\Omega })) weak-star and in L^2(0,\mathrm{\infty };H_0^1(\mathrm{\Omega })) weakly, {\sigma }_i(u_{\nu x_i})\to {\chi }_i={\sigma }_i(u_{x_i}) in L^{\mathrm{\infty }}(0,\mathrm{\infty };L^{{(m+1)}^{\prime }}(\mathrm{\Omega })) weak-star, {(m+1)}^{\prime }=\frac{m+1}{m}, f(u_v)\to \eta =f(u) in L^{\mathrm{\infty }}(0,\mathrm{\infty };L^{{(q+1)}^{\prime }}(\mathrm{\Omega })) weak-star, {(q+1)}^{\prime }=\frac{q+1}{q}.

Integrating (3.1) with respect to t, we have

(3.13)

Letting n=\nu \to \mathrm{\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}(\mathrm{\Omega }), u_t(x,0)=u_1(x) in L^2(\mathrm{\Omega }). Therefore u is a global weak solution of problem (1.1)-(1.3). □

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}(\mathrm{\Omega }), u_1(x)\in L^2(\mathrm{\Omega }). 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:

1. (i)
   I(u_n)={\parallel u_{nt}\parallel }^2-\frac{\mathrm{d}}{\mathrm{d}t}((u_{nt},u_n)+\frac{1}{2}{\parallel \mathrm{\nabla }{u}_n\parallel }^2);

   (4.1)
2. (ii)

   Furthermore, if and u_0(x)\in W, then for sufficiently large n, there exists a {\delta }_1\in (0,1) such that

   I(u_n)\ge (1-{\delta }_1)\sum _{i=1}^N({\sigma }_i(u_{n{x}_i}),u_{n{x}_i}).

   (4.2)

Proof (i) Multiplying (3.1) by g_{sn}(t) and summing for s, we get (4.1).

1. (ii)

   From

it follows that there exists a {\delta }_1\in (0,1) 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 u_n\in W for sufficiently large n gives

and

Hence, by Lemma 2.3, we have I_{{\delta }_1}(u_n)>0 or u_n=0. So, we have

 □

Theorem 4.2 Let (H₁) and (H₂) hold, u_0(x)\in W_0^{1,m+1}(\mathrm{\Omega }), u_1(x)\in L^2(\mathrm{\Omega }). Assume that , 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

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^{\alpha t} (\alpha >0), we get

and

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

where

Hence we have

(4.6)

and

(4.7)

From

we get

and

which together with u_n\in W for sufficiently large n gives

and

From (4.8) and the Poincaré inequality {\parallel \mathrm{\nabla }u\parallel }^2\ge {\lambda }_1{\parallel u\parallel }^2, it follows that there exists a constant C_0=C_0(p,l,a,{\lambda }_1)>0 such that

From (4.5)-(4.10) it follows that there exists a C_0 such that

(4.11)

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 C_1=C_1(p,l,A) such that

Let \{u_{\nu }\} 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=3d_0{C}_1, \lambda =\alpha (1-2C_0\alpha ). □

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.

Affiliations

1. College of Science, Harbin Engineering University, Harbin, 150001, People’s Republic of China

   • Runzhang Xu
   •  & Jie Liu

2. College of Automation, Harbin Engineering University, Harbin, 150001, People’s Republic of China

   • Yi Niu

3. Department of Mathematics, Cape Breton University, Sydney, B1P 6L2, Canada

   • Shaohua Chen

Corresponding author

Correspondence to Runzhang Xu.

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.

Reprints and Permissions