Abstract
We obtain the existence of pseudo almost automorphic solutions to the -dimensional heat equation with -pseudo almost automorphic coefficients.
Highlights
Let Ω ⊂ RN N ≥ 1 be an open bounded subset with C2 boundary ∂Ω, and let X L2 Ω be the space square integrable functions equipped with its natural · L2 Ω topology
Of concern is the study of pseudo almost automorphic solutions to the N-dimensional heat equation with divergence terms
G s − sn, u − F s, u pds −→ 0 t as n → ∞ pointwise on R for each u ∈ Y. The collection of those Sp-almost automorphic functions F : R × Y → X will be denoted by ASp R × Y
Summary
Let Ω ⊂ RN N ≥ 1 be an open bounded subset with C2 boundary ∂Ω, and let X L2 Ω be the space square integrable functions equipped with its natural · L2 Ω topology. We will make extensive use of the concept of Sp-pseudo almost automorphy combined with the techniques of hyperbolic semigroups to study the existence of pseudo almost automorphic solutions to the class of partial hyperbolic differential equations appearing in 1.3 and to the N-dimensional heat equation 1.1. In contrast with the fractional power spaces considered in some recent papers by Diagana 13 , the interpolation and Holder spaces, for instance, depend only on D A and X and can be explicitly expressed in many concrete cases Literature related to those intermediate spaces is very extensive; in particular, we refer the reader to the excellent book by Lunardi 14 , which contains a comprehensive presentation on this topic and related issues. Though to the best of our knowledge, the existence of pseudo almost automorphic solutions to the heat equation 1.1 in the case when the coefficients f, g are Sp-pseudo almost automorphic is an untreated original problem and constitutes the main motivation of the present paper
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