Cauchy problems for parabolic equations in Sobolev-Slobodeckii and Holder spaces on uniformly regular Riemannian manifolds

摘要

In this paper, we establish optimal solvability results-maximal regularity theorems-for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds (M, g) with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in Amann (Math Nachr 186: 5-56, 1997). This allows for a unified treatment of Sobolev-Slobodeckii and little Hlder spaces. In the flat case (M, g) = (R-m, |dx|(2)), we recover classical results for Petrowskii-parabolic Cauchy problems.