Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations

Abstract

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.

Almost periodic solutions to some second-order nonautonomous differential equations

“…(1.1) remains an untreated question, which constitutes the main motivation of this paper. We essentially make extensive use of ideas of [4,6,7,9] and dichotomy tools, as well as the Schauder fixed point theorem to obtain our existence results.…”

Existence of weighted pseudo-almost periodic solutions to some classes of differential equations with -weighted pseudo-almost periodic coefficients

“…The setting of this Subsection follows that of Baroun et al [3] and Diagana [14]. Fix once and for all a Banach space (X, · ).…”

Pullback attractors for a class of non-autonomous thermoelastic plate systems

“…Remark 1. Following the same ideas from [5] and [19], we obtain that for all t, there exists a positive constant M (independent of t), such that…”

“…In the literature the initial boundary-value problem (1)- (2) has been extensively discussed for several authors in different contexts. For instance, Baroun et al in [5] studied the existence of almost periodic solutions for an evolution system like (1), Liu and Renardy [20] proved that the linear semigroup defined by system (1) with f ≡ 0 with clamped boundary condition for u and Dirichlet boundary condition for θ is analytic. The typical difficulties in thermoelasticity comes from the boundary condition, which make more complicated the task of getting estimates to show the exponential stability of the solutions or analyticity of the corresponding semigroup.…”