Local solvability of a fully nonlinear parabolic equation

Abstract

This paper is concerned with the existence of local (in time) positive solutions to the Cauchy-Neumann problem in a smooth bounded domain of R^N for some fully nonlinear parabolic equation involving the positive part function r $\in$ R $\mapsto$ (r)₊: = r ∨ 0. To show the local solvability, the equation is reformulated as a mixed form of two different sorts of doubly nonlinear evolution equations in order to apply an energy method. Some approximated problems are also introduced and the global (in time) solvability is proved for them with an aid of convex analysis, an energy method and some properties peculiar to the nonlinearity of the equation. Moreover, two types of comparison principles are also established, and based on these, the local existence and the finite time blow-up of positive solutions to the original equation are concluded as the main results of this paper.