Unbounded Perturbation of the Exponential Dichotomy for Evolution Equations

Abstract

In this paper we prove that the exponential dichotomy for evolution equations in Banach spaces is not destroyed, if we perturb the equation by``small'' unbounded linear operator. This is done by employing a skew-product semiflow technique and a perturbation principle from linear operator theory. Finally, we apply these results to partial parabolic equations and functional differential equations. AcademicPress, Inc.

Notes on spectrum and exponential decay in nonautonomous evolutionary equations

“…For instance, delay differential equations were considered in [12]. Building on previous results from [4,9,10], in our Section 3 we determine the dichotomy spectrum for linear evolutionary equations whose infinitesimal generator is sectorial with compact resolvent. Canonical examples include uniformly elliptic differential operators or the poly-Laplacian under the standard boundary conditions.…”

“…Nonetheless, we feel the present examples and results are both handy and of independent interest when dealing with nonautonomous parabolic PDEs, their geometric theory and beyond. Our approach to nonautonomous dynamics is via evolution families and 2-parameter semigroups, rather than skew-product semiflows as used in [4,9,10]. We feel this is more appropriate in the present situation since one can omit e.g.…”

“…uniform continuity properties of the coefficient functions (in order to guarantee a compact base space). Finally, compared to [4,9,10] more general time-dependencies and a wider flexibility on the differential operator is allowed.…”

On the uniform exponential stability of linear skew‐product semiflows

“…Also, it is interesting to see that almost all interesting infinite dimensional situations, as for instance flows originating from partial differential equations and functional differential equations, only yield strongly continuous cocycles. In this context, there has been studied the dichotomy of linear skew-product semiflows defined on compact spaces (see [2,3,4,5]), and on a locally compact spaces, respectively (see [15]). The idea of associating an evolution semigroup in the expanded case of exponential stability or dichotomy of linear skew-product flow on locally compact metric space Θ , has its origins in the works of Latushkin and Stepin [13], respectively Latushkin, Montgomery-Smith and Schnaubelt [14].…”

Stabilizability and controllability of systems associated to linear skew-product semiflows

“…This approach led to the generalization of some classical theorems of dichotomy and stability (see [2]- [6], [10], [11], [13], [14]). …”