Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities

Abstract

In this work, we study the existence of bounded and almost automorphic solutions for evolution equations in Banach spaces. We suppose that the linear part is the infinitesimal generator of a compact C 0 -semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument. We give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on R + . We use the subvariant functional method to show that every K-minimizing mild solution is compact almost automorphic. Applications are provided for both heat and wave equations with nonlinearities in several functional spaces.

Recurrent solutions of the linearly coupled complex cubic‐quintic Ginzburg‐Landau equations

“…⋄ Semilinear abstract differential equations, 18,44,45 ⋄ Nonautonomous heat equation, 17 ⋄ Functional differential equations of neutral type, 46 ⋄ Partial functional differential equations. 47…”

“…Remark In recent years, there are some results on the almost automorphic solutions for abstract evolution equations. To this purpose, we recall the recent results: ⋄Semilinear abstract differential equations, ⋄Nonautonomous heat equation, ⋄Functional differential equations of neutral type, ⋄Partial functional differential equations …”

Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces

“…(4) is almost periodic by using the minimax principle. In [11], the authors studied the existence of compact almost automorphic solutions for some partial differential equations using a variant of the minimax principle which was introduced by Fink in [16]. We also mention the papers [14,19,23] where the authors studied almost periodic and almost automorphic solutions using spectral theory of functions.…”

Stepanov-like weighted asymptotic behavior of solutions to some stochastic differential equations in Hilbert spaces

“…Since then, this concept has undergone several interesting, natural, and powerful generalizations; for more details about this topics we refer the reader to [3][4][5][6][7][8][9][10][11][12][13][14] and references therein. Diagana [15] introduced the concept of Stepanov-like pseudo almost automorphy as a natural generalization of the pseudo almost automorphy and an implement of the Stepanov-like almost automorphy.…”