Existence of Infinitely Many Solutions for a Nonlinear Parabolic Equation

摘要

The purpose of this paper is to study the well-posedness of the model initial boundary value problem for the simplest case of a nonmonotone, piecewise linear, coercive phi which is decreasing on a single finite interval (a,b). Our result, as stated in the abstract, is that the problem has infinitely many solutions, whenever the initial function has f' > a, and therefore, the problem is apparently not well-posed in general. However, numerical computations suggest that there should be a natural way to single out a unique solution and it is hoped that imposing additional physical motivated assumptions will lead to a well-posed problem and further insight into the general situation of nonmonotone constitutive functions phi.