Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations

Michael Gil'

doi:10.1155/2013/207581

Abstract

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.

Highlights

Introduction and Statement of the MainResultIn this paper, we investigate stability of linear nonautonomous equations in a Banach space, which can be considered as integrally small perturbations of autonomous equations
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 [1] and references therein, but the problem of stability analysis of evolution equations continues to attract the attention of many specialists despite its long history

Summary

Introduction and Statement of the Main Result

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 [1] 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. Dragan and Morozan [9] established criteria for exponential stability of linear differential equations on ordered Banach spaces. 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

Proofs

A particular Case of Theorem 1

Equations with Periodic Boundary Conditions