Almost automorphic solutions for nonautonomous parabolic evolution equations

Abstract

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.

$({\mathrm R},{\mathcal B})$-Multidimensional-almost automorphic functions with applications to integral and partial differential equations

“…The concept of bi-almost automorphy was introduced in literature in [11,Definition 2.7] to study the asymptotic behavior of nonautonomous evolution equations in Banach spaces, see also [5,9] and references therein. The research study [9] by A. Chávez, S. Castillo and M. Pinto where the authors have used the notion of bi-almost automorphy of the Green functions in their investigation of almost automorphic solutions of abstract differential equations with piecewise constant arguments.…”

“…Here, the pivot space is denoted in general by X which equals to the pivot product space in [9]. Furthermore, in [5], the authors proved the existence and uniqueness of µpseudo almost automorphic solutions to a class of nonautonomous evolution equations with inhomogeneous boundary conditions, using the notion of bialmost automorphic Green functions. In addition, the authors established sufficient weak conditions on the initial data of the equation insuring the bi-almost automorphy of the associated Green function, see [5].…”

Stepanov pseudo‐almost periodic functions and applications

“…In the literature, we found several works devoted to the existence and uniqueness of 𝜇-pseudo-almost periodic solutions to semilinear evolution equations and inclusions; we quote previous studies. [4][5][6][7][8][11][12][13][14][15][16][17][18][19][20][21][22][23] In that case, a solution (at least in a mild sense) is usually represented by an integral operator (this holds under a certain decay of the associated linear system). More specifically, for given an integral operator solution, namely, (u)(t) = ∫ R (t, s)𝑓 (s, u(s))ds, t ∈ R, (1.1) where the input parameters, ((t, s)) t≥s which represents the Green function (resp.…”

Compact almost automorphic solutions for semilinear parabolic evolution equations

“…The concept of almost automorphy introduced by Bochner [7] is not restricted just to continuous functions. One can generalize that notion to measurable functions with some suitable conditions of integrability, namely, Stepanov almost automorphic functions, see [5,7,13]. That is a Stepanov almost automorphic function is neither continuous nor bounded necessarily.…”

“…The study of existence of almost periodic and almost automorphic solutions to equation (1.1) in infinite dimensional Banach spaces was deeply investigated in the last decades, we refer to [4,5,9,10,24,25,26]. Recently, in [4], authors studied equation (1.1), in the parabolic context, that is when (A, D(A)) generates an analytic semigroup (T (t)) t≥0 on a Banach space X which has an exponential dichotomy on R and f is Stepanov almost periodic of order 1 ≤ p < ∞ and Lipschitzian with respect to x.…”