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International Journal of Partial Differential Equations
Volume 2013 (2013), Article ID 207581, 5 pages
Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel
Received 18 February 2013; Revised 5 April 2013; Accepted 16 April 2013
Academic Editor: Pavel A. Krutitskii
Copyright © 2013 Michael Gil'. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a generator of an exponentially stable operator semigroup in a Banach space, and let be a linear bounded variable operator. Assuming that is sufficiently small in a certain sense for the equation , we derive exponential stability conditions. Besides, we do not require that for each , the “frozen” autonomous equation is stable. In particular, we consider evolution equations with periodic operator coefficients. These results are applied to partial differential equations.
1. Introduction and Statement of the Main Result
In this paper, we investigate stability of linear nonautonomous equations in a Banach space, which can be considered as integrally small perturbations of autonomous equations. The stability theory of evolution equations in a Banach space is well developed, compare and confare with  and references therein, but the problem of stability analysis of evolution equations continues to attract the attention of many specialists despite its long history. It is still one of the most burning problems, because of the absence of its complete solution. One of the basic methods for the stability analysis is the direct Lyapunov method. By that method, many strong results were established, compare and confare with [2, 3]. But finding the Lyapunov functionals is usually a difficult mathematical problem. A fundamental approach to the stability of diffusion parabolic equations is the method of upper and lower solutions. A systematical treatment of that approach is given in . In , stability conditions are established by a normalizing mapping. Note that a normalizing mapping enables us to use more complete information about the equation than a usual (number) norm. In , the “freezing” method for ordinary differential equations is extended to equations in a Banach space. About the recent results, see the interesting papers [7–11]. In particular, in  the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces is investigated. In the paper , a Rolewicz’s type theorem of in-solid function spaces is proved. Dragan and Morozan  established criteria for exponential stability of linear differential equations on ordered Banach spaces. Paper  deals with the stability and controllability of hyperbolic type abstract evolution equations. Pucci and Serrin  investigated the asymptotic stability for nonautonomous wave equations.
Certainly, we could not survey the whole subject here and refer the reader to the previously listed publications and references given therein.
Let be a complex Banach space with a norm and the unit operator . For a bounded operator , is the operator norm.
Everywhere below is a linear operator in with a domain , generating a strongly continuous semigroup ; that is, in the strong topology, and is a linear bounded variable operator mapping into itself. Put . In the present paper, we establish stability conditions for the equation as follows: It should be noted that in the previously pointed papers it is assumed that for each , the “frozen” autonomous equation is stable. We do not require that condition. The aim of this paper is to generalize the main result from the paper , which deals with finite dimensional equations.
A solution of (1), for a given is a function having a strong derivative, satisfying (1) and . We will investigate (1) as a perturbation of the following equation: Put We say that (1) is exponentially stable if there is an , such that for any solution with . Now we are in a position to formulate the main result of the paper.
Theorem 1. Let Then, (1) is exponentially stable.
The proof of this theorem is divided into a series of lemmas which are presented in the next section. To the best of our knowledge, Theorem 1 is new even in the case of bounded operators. In Section 3 we consider particular cases of Theorem 1. In Section 4, the previously pointed results are applied to a partial differential equation. For the brevity, we restrict ourselves by a scalar equation with the periodic boundary condition, but our results enable us to consider coupled systems of equations and other boundary conditions, for example, the Dirichlet condition.
We need the following simple result.
Lemma 2. Let , and be functions whose values are bounded linear operators. Assume that is integrable and and have integrable derivatives on . Then, with the notation , one has
Proof. Clearly, Integrating this equality and taking into account that , we arrive at the required result.
Lemma 3. One has
Lemma 4. Let condition hold. Then, , , where is a solution of the following equation:
Proof. Thanks to the previous lemma, Hence Then by the well-known (comparison) Lemma from  we have the required result.
Let Then (14) implies Due to the previous lemma we get the following.
Proof of Theorem 1. Assume that then . If then and thanks to the previous lemma, (1) is stable. But condition (5) implies that or Thus (5) implies the inequality , and therefore, from (5), condition (21) follows. This proves the stability. To prove the exponential stability we use the well-known Theorem 4.1 [13, p. 116] (see also Theorem 2.44 [1, p. 49]). It asserts that the finiteness of the -norm of implies the inequality where , . So for , the semigroup generated by satisfies the inequality , , and therefore it also has a finite -norm. Substitute the equality into (1). Then we obtain the equation Denote the Cauchy operator of (26) by . Repeating our above arguments with instead of and the equation instead of (2), due to Lemma 5 we can assert that is bounded. Now (25) implies This proves the theorem.
3. A particular Case of Theorem 1
To illustrate Theorem 1, consider the following equation: where is a constant operator and is a scalar real piece-wise continuous function bounded on . So, . Without any loss of generality, assume that and with the notation we obtain Due to (24), Thus, denoting due to Theorem 1, we arrive at the following result.
Corollary 6. If the inequality holds, then (28) is exponentially stable.
For example, let . Then, and Thus, (34) takes the following form:
4. Equations with Periodic Boundary Conditions
Consider the problem with a positive constant and a real differentiable function ; is the same as in the previous section.
Take , where is the Hilbert space of real functions defined on with the scalar product and the norm . Set then we have Let be a solution of (2) with defined by (41). Then, we obtain Hence, . Thus, . In addition, , Due to condition (34), we obtain the following.
Corollary 7. If the inequality holds, then (37) is exponentially stable.
For example, let . Then and (45) takes the form
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