Semilinear parabolic problems define semiflows on $C^{k}$ spaces

Abstract:

Linear parabolic problems of a general class are proved to determine analytic semigroups on certain closed subspaces of ${C^k}(\overline \Omega )$ ($k$ integer); ${C^k}(\overline \Omega )$ denotes the space of functions whose derivatives or order $\leqslant k$ are bounded and uniformly continuous, with the usual supremum norm; the closed subspaces where the semigroups are obtained, denoted by ${\hat C^k}(\overline \Omega )$, are determined by the boundary conditions and a possible condition at infinity. One also obtains certain embedding relations concerning the fractional power spaces associated to these semigroups. Usually, results of this type are based upon the theory of solution of elliptic problems, while this work uses the corresponding theory for parabolic problems. The preceding results are applied to show that certain semilinear parabolic problems define semiflows on spaces of the type ${\hat C^k}(\overline \Omega )$.