The paper is concerned with the existence of almost periodic solutions to the so-called semilinear thermoelastic plate systems. For that, the strategy consists of seeing these systems as a particular case of the semilinear parabolic evolution equationswhere A(t) for t ∈ R is a family of sectorial linear operators on a Banach space X satisfying the so-called Acquistapace-Terreni conditions, and f is a function defined on a real interpolation space X α for α ∈ (0, 1). Under some reasonable assumptions it will be shown that ( * ) has a unique almost periodic solution. We then make use of the previous result to obtain the existence and uniqueness of an almost periodic solution to the thermoelastic plate systems.
In this work, we study the existence and uniqueness of an almost automorphic solution to semilinear nonautonomous parabolic evolution equations with inhomogeneous boundary conditions using the exponential dichotomy. We assume that the homogeneous problem satisfies the "Acquistapace-Terreni" conditions and that the forcing terms are Stepanov-like almost automorphic. An example is given for illustration.
We show the existence and uniqueness of the (asymptotically) almost periodic solution to parabolic evolution equations with inhomogeneous boundary values on R and R±, if the data are (asymptotically) almost periodic. We assume that the underlying homogeneous problem satisfies the 'Acquistapace-Terreni' conditions and has an exponential dichotomy. If there is an exponential dichotomy only on half intervals (−∞, −T ] and [T, ∞), then we obtain a Fredholm alternative of the equation on R in the space of functions being asymptotically almost periodic on R+ and R−.
Mathematics Subject Classification (2000). Primary 47D06.
This paper deals with semilinear evolution equations with unbounded observation operators. Sufficient conditions are given guaranteeing that the output function of a semilinear system is in L 2 loc ([0, ∞); Y ). We prove that the Lebesgue extension of the observation operators are invariant under nonlinear globally Lipschitz continuous perturbations. Further, relations between the corresponding Λ-extensions are studied. We show that exact observability of linear autonomous system is conserved under small Lipschitz perturbations. The obtained results are illustrated by several examples. (2000). Primary 47H20; Secondary 93C73, 93B07, 93C25.
Mathematics Subject Classification
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