On the Exponential Decay of Solutions for Some Kirchhoff-Type Modelling Equations with Strong Dissipation

This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equations with strong dissipative and source terms 2 2 2 ( ) | | , , 0 tt t u u u a u b u u x t            in a bounded domain, where , 0 a b  and 2   are constants. We obtain the global existence of solutions by constructing a stable set in 1 0 ( ) H  and show the energy exponential decay estimate by applying a lemma of V. Komornik.

  are constants.We obtain the global existence of solutions by constructing a stable set in 1 0 ( ) H  and show the energy exponential decay estimate by applying a lemma of V. Komornik.

Introduction
Let  be a bounded domain in n R with smooth boundary  .In this paper, we investigate the existence and the energy exponential decay estimate of global solutions for the initial boundary value problem of the following Kirchhoff-type equation with strong dissipative and source terms in a bounded domain where , 0 a b  and 2 with 0 1 m  constant.When 1 n  , the equation (1.1) describes a small am- plitude vibration of an elastic string ( [1]).The original equation is where 0 x L   and 0, t  ( , ) u x t is the lateral displacement at the space coordinate x and the time t,  is the mass density, h is the cross-section area, L is the length, 0 P is the initial axial tension,  is the resistance modulus, E is the Young modulus and f is the external force.
Many authors have studied the existence and uniqueness of solutions of (1.1)-(1.3)by using various methods.When , 0 a b  , and ( ) , 1, r s s r    K. Nishihara and Y.
Yamada [2] have proved the existence and the polynomial decay of global solution under the assumptions that the initial data 0 u and 1 u are sufficiently small and 0 0 u  .However, the method in [2] can not be applied directly to the case that the equations have the blow-up term . M. Aassila and A. Benaissa [3] extend the global existence part of [2] to the case where ( ) . K. Ono and K. Nishihara [4] have proved the global existence and decay structure of solutions of (1.1)-(1.3)without small condition of data using Galerkin method.K. Ono [5] and the asymptotic behavior as t   , where either 0 The case ( ) 0 s r    has been considered by M.
They proved that, if the initial data are small enough, the problem (1.1)-( 1.3) has a global solution which decays exponentially as t   .
In this paper, we prove the global existence for the problem (1.1)-(1.3)by applying the potential well theory introduced by D. H. Sattinger [9] and L. Payne and D. H. Sattinger [10].Meanwhile, we obtain the exponential decay estimate of global solutions by using the different method from paper [8].
We adopt the usual notation and convention.Let For simplicity of notations, hereafter we denote by Moreover, M denotes various positive constants depending on the known constants and it may be different at each appearance.

Preliminary
In order to state and prove our main results, we first define the following functionals Then we define the stable set S by We denote the total energy functional associated with is the total energy of the initial data.Lemma 2.1 Let q be a number with 2 , q    Then there exists a constant C depending on  and q such that 1 0 1 0 ( ) , ( ) Lemma 2.2 [11] Let   : Hence, we have from Lemma 2.1 that we get from the definition of d that 0. d  In order to prove the existence of global solutions for the problem ( From the definition of S and the continuity of It follows from (1.4) and (2.1) that which contradicts the definition of d .Therefore, the case ( ( )) 0 K u t  is impossible as well.Thus, we conclude that ( ) u t S  on [0, ).T

Main Results and Proof
Therefore, we have from (3.1) that Hence, we get The above inequality and the continuation principle lead to the existence of global solution, that is, T   .
Therefore, the solution   where 0 M  is a constant.Proof Multiplying by u on both sides of the Equa- tion (1.1) and integrating over By exploiting Lemma 2.1 and (3.6), we easily arrive at We obtain from (3.6) and (3.7) that It follows from (3.5), (3.8) and (3.9) that We have from Lemma 2.1 and (3.2) that Substituting the estimate (3.11) into (3.10),we conclude that The proof of Theorem 3.2 is finished.

Acknowledgments
This Research was supported by Natural Science Foun- space with the norm

u t is a global 3 . 2
solution of the problem (1.1)-(1.3).The following Theorem shows the exponential decay estimate of global solutions for problem (1.1)-(1.3).Theorem If the hypotheses in Theorem 3.1 are valid, then the global solutions of problem (1.1)-(1.3)has the following exponential decay property 1.1)-(1.3),we need the following Lemma.
3.12)We get from Lemma 2.1 and Lemma 2.3 that S E t dt ME S