On some qualitative properties of parabolic problems

In this talk we investigate some special qualitative properties of parabolic problems. At the beginning of the talk we review the remarkable qualitative properties of these problems. Then we will turn to two special properties: the first property says that the number of the so-called LL-level points, or specially the number of the zeros, of the solutions must be non-increasing in time. The second property requires a similar property for the number of the local maximizers and minimizers. We show that linear equations and some special nonlinear equations fulfill the above properties in the continuous case. We use the maximum-minimum principles in the proof. Then we generate the numerical solution with the implicit Euler finite difference method and show that the obtained numerical solution satisfies the discrete versions of the above properties without any requirements on the mesh parameters. We show also some numerical test results.