Analyticity of the maximal solution of an abstract nonlinear parabolic equation

Abstract

where u(t, x): [0, 7’j X R” + R satisfies other suitable conditions. Problem (P) is studied under the assumption that af/au generates an analytic semigroup in E and has domain F, which corresponds to a parabolicity condition on equation (P’). Problems of this kind have been studied by monotonicity methods (see Brezis 111, Crandall & Ligget [3] and Crandall & Pazy [4]) to obtain global but weak solutions of (P). Here we find a maximally defined strict solution by means of a linearization method and by virtue of maximal regularity results for the linear case. This method has been used by Da Prato & Grisvard in [5] to obtain strict solutions of(P); with arguments which seem to be more simple and suitable to our purpose we get also the analyticity of u with respect to (1, W) when f is analytic. In section 1 some notation and assumptions are given. Problem (P) is studied in section 2 where we prove existence and uniqueness of the strict solution of (P), which is defined on a maximal time interval contained in [0, T] and it is continuous with respect to (t, ~0). In section 3 we give conditions to get global existence of the solution. In section 4 we study analyticity with respect to (t, ug) of the maximally defined solution of (P) and in section 5 a simple example is given. In a subsequent paper we will show how this theory applies to the study of the analyttctty of the solutions of nonlinear parabol.ic partial differential equations in spaces of Holder continuous functions.