Research Article | Open Access
Existence of Mild Solutions for the Elastic Systems with Structural Damping in Banach Spaces
This paper deals with the existence and uniqueness of mild solutions for a second order evolution equation initial value problem in a Banach space, which can model an elastic system with structural damping. The discussion is based on the operator semigroups theory and fixed point theorem. In addition, an example is presented to illustrate our theoretical results.
Our aim in this paper is to study the existence and uniqueness of mild solutions for the semilinear elastic system with structural damping in a Banach space , where means , is a constant; is a closed linear operator and generates a -semigroup on ; , , .
In 1982, Chen and Russell  investigated the following linear elastic system described by the second order equation 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 , with the naturally induced inner products. Then, (2) is equivalent to the first order equation in where Chen and Russell  conjectured that is the infinitesimal generator of an analytic semigroup on if and either of the following two inequalities holds for some : 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 or its closure to generate an analytic or differentiable semigroup on were discussed in [4–10], by choosing to be an operator comparable with for , based on an explicit matrix representation of the resolvent operator of or .
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 .
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.
2. Preliminaries on Linear Elastic Systems
Let be a Banach space, we consider the linear elastic system with structural damping where means , is a constant; is a closed linear operator, and generates a -semigroup on ; , , .
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 : It is clear that (16) and (17) are linear inhomogeneous initial value problems for and , respectively. Since is the infinitesimal generator of -semigroup . Furthermore, for any , (13) yield , . Thus, by operator semigroups theory , and are infinitesimal generators of -semigroups, which implies initial value problems (16) and (17) are well-posed.
Throughout this paper, we assume that and generate -semigroups and on , respectively. Note that , and is the infinitesimal generator of -semigroup . It follows that
It is well known [12, Chapter 4], when , the linear initial value problem (16) has a mild solution given by Similarly, if , 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 , the right-hand side of (21) is a continuous function on . 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 is the infinitesimal generator of -semigroup . Then a continuous solution of the integral equation is said to be a mild solution of the initial value problem (8). Where , were defined in (18) and was specified in (15).
3. Main Results
Let be the Banach space of all continuous functions with norm , . Let be the Banach space of all linear and bounded operators on . Note that and are -semigroups on . Thus, there exist and such that
In what follows, we firstly give the definition of a mild solution for the initial value problem (1) below.
Definition 3. Let is the infinitesimal generator of -semigroup . Then a continuous solution of the integral equation is said to be a mild solution of the initial value problem (1). Where , were defined in (18) and was specified in (15).
Secondly, we consider the existence and uniqueness of mild solutions for (1). To this end, we make the following assumptions: be continuous and there exists , such that be continuous and there exists a positive function such that The -semigroup is compact for .
Theorem 4. Assume that holds, is the infinitesimal generator of -semigroup . Then for every , and , the initial value problem (1) has a unique mild solution .
Proof. Define the operator by
It is obvious that the mild solution of the initial value problem (1) is equivalent to the fixed point of .
For any , (23), (27), and yield Using (27), (28), and induction on it follows easily that Hence Since Thus, for large enough and by well known extension of the contraction mapping principle, has a unique fixed point . This fixed point is the desired solution of the integral equation (24).
Theorem 5. Suppose that assumptions and hold. Then for every , and , the initial value problem (1) has at least one mild solution .
Proof. Define the operator as (27) and choose such that
Let . We proceed in two main steps.
Step 1. We show that . For that, let . Then for , we have which according to and (23) gives In view of the choice of , we obtain
Step 2. We prove that is completely continuous. Note that is a continuous mapping from to . Thus, is continuous. Next, we show that is compact. To this end, we use the Ascoli-Arzela's theorem. For that, we first prove that is relatively compact in , for . Obviously, is compact.
Let . For each and , we define the operator by Then the sets are relatively compact in since by and (18), the semigroup is compact for on . Moreover, using (23) and we have Therefore, the set is relatively compact in for all and since it is compact at we have the relatively compactness in for all .
Now, let us prove that is equicontinuous. For , we have where In fact, and tend to 0 independently of when .
Note that the function is continuous for . Thus, is uniformly continuous on and thus .
From (23) and , we have Let . By the compactness of and (18), we can easily conclude that is compact and therefore is continuous in the uniform operator topology for . Then, is also continuous in the uniform operator topology on . Thus as . Meanwhile, is bounded on . Hence, using Lebesgue dominated convergence theorem we deduce that .
Moreover, from (23) we have Hence, .
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).
4. 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 where , are all constants, is continuous.
Let , we define the linear operator in by It is well known from  that is the infinitesimal generator of a -semigroup on .
Let , , then the initial-boundary value problem (42) can be reformulated as the following abstract second order evolution equation initial value problem in :
In order to solve the initial-boundary value problem (42), we also need the following assumptions:, . The partial derivative is continuous.
Theorem 6. If the assumptions and are satisfied, then for any , the initial-boundary value problem (42) has a unique mild solution .
Proof. From the assumptions and , it is easily seen that the conditions in Theorem 4 are satisfied. Hence, by Theorem 4, for any , the problem (44) has a unique mild solution , which means is a mild solution for initial-boundary value problem (42).
The authors are grateful to the anonymous referee for his/her valuable comments and suggestions, which improve the presentation of the original paper. Research was supported by NNSFs of China (11261053, 11061031).
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