Singular behavior in nonlinear parabolic equations

Abstract

In this paper, we study the well-posedness of the initial-boundary value problems of some quasilinear parabolic equations, namely, nonlinear heat equations and the porous medium equation in the fast-diffusion case. We establish nonunique- ness (local in time) and/or nonregularizing effect of these equations in some critical cases. The key which leads to the resolution of these problems is to study some singular solutions of the elliptic counterparts of these parabolic problems (the so-called M-solutions of the Lane-Emden equations in astrophysics). Introduction. In this article we are concerned first of all with the existence or nonexistence of singular solutions of certain semilinear elliptic boundary value problems, and secondly with the application of these results to the study of some related parabolic problems. To begin with, consider the problem