GLOBAL EXISTENCE OF MILD SOLUTIONS FOR THE ELASTIC SYSTEM WITH STRUCTURAL DAMPING??

Wei Shi

(Dept.of Math.,Xinjiang Agricultural University,Ulumuqi 830052,Xinjiang,PR China)

Abstract In this paper,we study the global existence of mild solutions for the semilinear initial-value problems of second order evolution equations,which can model an elastic system with structural damping.This discussion is based on the operator semigroups theory and the Leray-Schauder fixed point theorem.In addition,an example is presented to illustrate our theoretical result.

Keywords C0-semigroup;elastic systems;structural damping;mild solution;fixed point

1 Introduction

In this paper,we will investigate the following semilinear elastic systems with structural damping

The consideration of this problem seems to have been initiated by Chen and Russell[1]in 1982.They studied the following linear second order evolution equation

in a Hilbert space H with inner product(·,·),where A(the elastic operator)is a positive definite,self-adjoint operator in H,and B(the damping operator)is a positive self-adjoint operator in H.They reduced(1.2)to the first order equation in H×H,

where

Chen and Russell[1]conjectured that ABis the infinitesimal generator of an analytic semigroup on H if

and either of the following two inequalities holds for some constants β1,β2>0:

In[2,3],Huang gave the complete proofs of the two conjectures.Then,the spectral property of this systems and some fundamental results for the holomorphic property and exponential stability of the semigroup associated with these systems were discussed in[4-12].

In[13],a linear second order evolution equation in the frame of Banach spaces was explored,which can model the elastic systems with structural damping,new forms of the corresponding first order evolution equation were introduced,and sufficient conditions for analyticity and exponential stability of the associated semigroups were given.In[14],the existence results of mild solution for the elastic systems with structural damping were established by the same authors as in[13],firstly via the contraction mapping principle,secondly via the Schauder fixed theorem.In addition,by applying the method of upper and lower solutions and the operator semigroups theory,the existence results of extremalmild solutions and the asymptotic stability of solutions for the initial-value problem(1.1)were obtained,see[15,16].

In this paper,we will investigate the global existence of mild solutions for the initial-value problem(1.1)in Banach space X,via a fixed point analysis approach.By means of the Leray-Schauder fixed point theorem,we derive conditions under which a solution exists globally.The result will improve and extend some relevant results in elastic systems with structural damping.

2 Preliminaries

It is well known,when A=0,only the continuity of f is not sufficient to assure the local existence of solutions,even when X is a Hilbert space.Therefore,one has to restrict either the function f or the operator semigroup.The function f was assumed to be locally Lipschitz or monotone or completely continuous.Here we assume that T(t)(t≥0)is compact and the function f satisfies the following Caratheodory-type conditions,which do not imply that f is completely continuous:

(C1)For eath t∈[0,b],the function f:X→X is continuous,and for each x∈X,the function f:[0,b]→X is strongly measurable.

(C2)For every positive integer k there exists an hk∈L2([0,b])such that for all t∈[0,b],

Now we recall some results about the initial-value problem(1.1).

Definition 2.1[14]Let A:D(A)?X→X be a closed and linear operator,the damping coefficient beρ≥2,and f:[0,b]×X→X be a continuous function,x0∈D(A),y0∈X.A continuous solution of the integral equation

is said to be a mild solution of the initial-value problem(1.1),where

Remark 2.1In fact,by the property of C0-semigroups and(2.3),we can easily deduce that if T(t)(t≥0)is compact for t>0,then C0-semigroups S1(t)and S2(t)are also compact for t>0.

Since S1(t)(t≥ 0)and S2(t)(t≥ 0)are C0-semigroups on X,recall from[17,18]that there exist M1≥1 and M2≥1 such that

The proof of the main result is based upon an application of the following fixed point theorem.

Lemma 2.1[19,20](Leray-Schauder Fixed Point Theorem)Let S be a convex subset of a normed linear space E and assume 0∈S.Let F:S→S be a completely continuous operator and

If z(F)is bounded,then F has a fixed point.

3 Main Result

Now we present the global existence result for the initial-value problem(1.1).

Theorem 3.1Let f:[0,b]×X→ X be a function satisfying(C1)and(C2).Assume that:

(Hf)There exists a continuous function g:[0,b]→R such that

where ?:[0,∞)→(0,∞)is a continuous nondecreasing function.

(Hg)T(t)(t≥0)is compact for t>0.

Then if

where c=M2∥x0∥ +M1M2∥v0∥b,the initial-value problem(1.1)has at least one mild solution on[0,b].

ProofFirst,we must prove that the operator F:B=C([0,b],X)→B defined by

is a completely continuous operator.

Let Bk={u∈ B:∥u∥≤ k}for some k≥ 1.We first show that F maps Bkinto an equicontinuous family.Let u∈Bkand t,∈[0,b]and ε>0.Then if 0≤

The right-hand side tends to zero as?t?t→0,since the compactness of T(t)(t>0)implies the continuity in the uniform operator topology.Thus F maps Bkinto an equicontinuous family of functions.It is evident that the family Bkis uniformly bounded.

Let 0

Since S2(t)(t>0)is a compact operator semigroup,the set Uε(t)={(Fεu)(t):u ∈Bk}is precompact in X,for every 0<ε

Moreover,for every u∈Bk,we have

Therefore there are precompact sets arbitrarily close to the set{(F u)(t):u∈Bk}.Hence the set{(F u)(t):u ∈ Bk}is precompact in X.It follows from Arzela-Ascoli’s theorem that F is compact.

It remains to show that F:B→B is continuous.Letin B.Then there exists an integer q such that∥un(t)∥ ≤ q for all n and t∈ [0,b],so un∈Bqand u∈Bq.By(C1),f(t,un(t))→f(t,u(t)),for each t∈[0,b]and since∥f(t,un(t))?f(t,u(t))∥ ≤ 2hq(t)we have by dominated convergence theorem that

Thus F is continuous.This completes the proof that F is completely continuous.

Finally,we must obtain the prioribounds for the solution of the equation

From(3.1),we have

Therefore,

Denoting by w(t)the right-hand side of the above inequality we have

and

Then

This implies

This inequality implies that there is a constant K such that w(t)≤K,t∈[0,b],and hence∥u(t)∥≤ K,t∈ [0,b],where K depends on b and the functions g and ?.

Consequently,by Lemma 2.1,the operator F has a fixed point in B.This means that the initial-value problem(1.1)has a mild solution,which completes the proof.

4 Applications

In this final section,in order to demonstrate the applicability of our result,we consider the following initial-boundary value problem,which is a model for elastic system with structural damping

where b>0 is a constant,f:[0,1]×[0,b]×R→R is continuous.

Let X=Lp([0,1],R)(1

Let u(t)=u(·,t)and f(t,u(t))=f(·,t,u(·,t)),then the initial-boundary value problem(4.1)can be reformulated as the following abstract second order evolution equation initial value problem in X:

We assume that φ∈D(A)and ψ∈X,then equation(4.3)has the following decomposition form

It is well known from[15]that?A is the infinitesimal generator of a compact C0-semigroup S(t)(t≥0)on X,which satisfies

By the characterization of the infinitesimal generators of C0-semigroup,?σ1A and?σ2A generate analytic and exponentially stable semigroups S1(t)(t≥0)and S2(t)(t≥0)respectively,which satisfy

Then we obtain that

Theorem 4.1Let f:[0,b]×X→ X be a function satisfying(C1),(C2)and(Hf).Assume thatφ∈W2,p(0,1)∩(0,1),ψ∈Lp([0,1],R).Then the IBVP(4.1)has at least one mild solution on[0,b]provided

where c= ∥x0∥ + ∥v0∥b.

ProofAccording to(4.5)and(4.6),we obtain that(Hg)holds.Since f satisfies(C1),(C2),and S(t)(t≥0)is compact for t>0 on X,combining these fact with assumption(Hf),it is easy to verify that conditions in Theorem 3.1 are satisfied.Hence by Theorem 3.1,the initial-boundary value problem(4.1)has at least one mild solution on[0,b].