Cauchy Problems in Weighted Lebesgue Spaces

Abstract

Global solvability and asymptotics of semilinear parabolic Cauchy problems in ℝⁿ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over ℝⁿ, n ∈ ℕ. In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.