Existence of Mild Solutions for the Elastic Systems with Structural Damping in Banach Spaces

and Applied Analysis 3 For every h ∈ L1([0, a],X), the right-hand side of (21) is a continuous function on [0, a]. It is natural to consider it as a generalized solution of (8) even it is not differentiable and dose not strictly satisfy the equation. We therefore define the following. Definition 2. Let −A is the infinitesimal generator of C 0 semigroup T(t) (t ≥ 0). Then a continuous solution u(t) of the integral equation u (t) = S 2 (t) x 0 + ∫ t 0 S 2 (t − s) S 1 (s) v 0 ds


Introduction
Our aim in this paper is to study the existence and uniqueness of mild solutions for the semilinear elastic system with structural damping ..
In 1982, Chen and Russell [1] investigated the following linear elastic system described by the second order equation ..

𝑢 (𝑡) + 𝐴𝑢 (𝑡) = 0
(2) in a Hilbert space  with inner (⋅, ⋅), where  (the elastic operator) and  (the damping operator) are positive definite selfadjoint operators in .They reduced (2) to the first order equation in  × Let  = D( 1/2 ), H =  ×  with the naturally induced inner products.Then, (2) is equivalent to the first order equation in where Chen and Russell [1] conjectured that A  is the infinitesimal generator of an analytic semigroup on H if  ( 1/2 ) ⊂  () (6) and either of the following two inequalities holds for some  1 ,  2 > 0: 1 ( 1/2 , ) ≤ (, ) ≤  2 ( 1/2 , ) ,  ∈  ( In the same paper they obtained some results in this direction. The complete proofs of the two conjectures were given by Huang [2,3].Then, other sufficient conditions for A  or its closure A  to generate an analytic or differentiable semigroup on H were discussed in [4][5][6][7][8][9][10], by choosing  to be an operator comparable with   for 0 <  ≤ 1, based on an explicit matrix representation of the resolvent operator of A  or A  .However, so far as we know, among the previous works, little is concerned with an elastic system with structural damping in a Banach space.Motivated by previous works, in this paper, we investigate the existence and uniqueness of mild solutions for the elastic system (1) in a frame of Banach spaces.To this end, we firstly introduce the concept of mild solutions for system (1), which is based on the discussion about associated linear system.Secondly, we prove the existence and uniqueness of mild solutions for the semilinear elastic system (1) in a Banach space X.
The paper is organized as follows.In Section 2, we discuss the associated linear elastic system and give its definition of mild solutions.In Section 3, we study the existence and uniqueness of mild solutions for the semilinear elastic system (1).An example to illustrate our theoretical results is given in Section 4.

Preliminaries on Linear Elastic Systems
Let X be a Banach space, we consider the linear elastic system with structural damping ..
For the second order evolution equation ..
it has the following decomposition That is, It follows from ( 9) and (11) that By (12), we have Let which means So we reduce the linear elastic system (8) to the following two abstract Cauchy problems in Banach space X: It is clear that ( 16) and ( 17) are linear inhomogeneous initial value problems for − 1 A and − 2 A, respectively.Since −A is the infinitesimal generator of  0 -semigroup () ( ≥ 0).Furthermore, for any  ≥ 2, (13) yield  1 > 0,  2 > 0. Thus, by operator semigroups theory [11], − 1 A and − 2 A are infinitesimal generators of  0 -semigroups, which implies initial value problems ( 16) and ( 17) are well-posed.Throughout this paper, we assume that − 1 A and − 2 A generate  0 -semigroups  1 () ( ≥ 0) and  2 () ( ≥ 0) on X, respectively.Note that  1 > 0,  2 > 0 and −A is the infinitesimal generator of  0 -semigroup () ( ≥ 0).It follows that It is well known [12,Chapter 4], when ℎ ∈  1 ([0, ], X), the linear initial value problem (16) has a mild solution  given by Similarly, if  ∈ ([0, ], X), then the mild solution of the linear initial value problem (17) expressed by Substituting ( 19) into (20), we get From the argument above, we obtain the following corollary.For every ℎ ∈  1 ([0, ], X), the right-hand side of ( 21) is a continuous function on [0, ].It is natural to consider it as a generalized solution of (8) even it is not differentiable and dose not strictly satisfy the equation.We therefore define the following.
Definition 2. Let −A is the infinitesimal generator of  0semigroup () ( ≥ 0).Then a continuous solution () of the integral equation is said to be a mild solution of the initial value problem (8).
Proof.Define the operator  : (, X) → (, X) by It is obvious that the mild solution of the initial value problem ( 1) is equivalent to the fixed point of .
Step 1.We show that (  ) ⊂   .For that, let  ∈   .Then for  ∈ , we have In view of the choice of , we obtain Step 2. We prove that  is completely continuous.Note that  :  → (⋅, (⋅)) is a continuous mapping from   to (, X).Thus,  :   →   is continuous.Next, we show that  is compact.To this end, we use the Ascoli-Arzela's theorem.
Let  ∈ (0, ].For each  ∈ (0, ) and  ∈   , we define the operator   by Then the sets {(  )() :  ∈   } are relatively compact in X since by (3) and (18), the semigroup  2 () ( ≥ 0) is compact for  > 0 on X.Moreover, using ( 23) and (2) we have Therefore, the set {()() :  ∈   } is relatively compact in X for all  ∈ (0, ] and since it is compact at  = 0 we have the relatively compactness in X for all  ∈ .Now, let us prove that (  ) is equicontinuous.For 0 ≤  1 <  2 ≤ , we have where Hence, lim In short, we have show, that (  ) is relatively compact for  ∈ , { :  ∈   } is a family of equicontinuous functions.It follows from Ascoli-Arzela's theorem that  is compact.By Schauder fixed point theorem  has a fixed point  ∈   , which obviously is a mild solution to (1).

An Example
In order to illustrate our main results, we consider the following initial-boundary value problem, which is a model for elastic system with structural damping  In order to solve the initial-boundary value problem (42), we also need the following assumptions: (1)  ∈  2, (0, 1) ∩  Proof.From the assumptions (1) and (2), it is easily seen that the conditions in Theorem 4 are satisfied.Hence, by Theorem 4, for any  ≥ 2, the problem (44) has a unique mild solution  ∈ ([0, ], X), which means  is a mild solution for initial-boundary value problem (42).

Corollary 1 .
If ℎ ∈  1 ([0, ], X), then the initial value problem (8) has at most one solution.If it has a solution, this solution is given by (21).