Halbgruppen und semilineare Anfangs-Randwertprobleme

Abstract

A semilinear parabolic initial-boundary-value problem of order 2m in a possibly unbounded domain Ωx(O,T), Ω⊂Rⁿ, is considered within the framework of the L_p-and C^α-theory. In the first case a proof is given of the existence of a “strict” solution of the corresponding evolution equation. In the second case one can guarantee a classical solution, provided the homogeneous linear parabolic equation has a unique classical solution. Only local solvability is considered. The nonlinearity is a Hölder-continuous function of the derivatives up to the order 2m-1 of the unknown solution. The principal tool is the semigroup-theory in L_p(Ω) as well as in C^α($\bar \Omega$). In the latter case the semigroup is not strongly continuous, but it has sufficiently good properties to use it for existence proofs of classical solutions.