The global feature of unbounded solutions to a nonlinear parabolic equation

Abstract

where CI >O. The above equation consists of several interesting physical models, for instance, the case 0 < CI < 1 corresponds to a special case of the well-known porous media equation (see [9]); the case 0: = 2 arises from plasma physics, as demonstrated in [2] and [S]. It has been known (see [6, 3, 21) for some time that the Dirichlet initialboundary value problem defined in a finite domain or the Cauchy initial value problem defined in R1 of the above equation may have solutions that blow-up in finite time. But the problem of finding the blow-up set and asymptotic behaviour when its solutions evolve to their blow-up time does not have a complete answer yet. In fact, only the case 0 <M < 1 has been studied in [4] but with a different formula from ours. In addition, the interesting problem concerning the similarity solutions (scaling invariant solutions) of the above equation is still open for 012 1. Therefore the purpose of this paper is to study the blow-up set and asymptotic behaviour of the solution to (1) and the existence of related similarity solutions.