On dual variational principles corresponding to parabolic equations

Preprint series: 06-20, Analysis

The paper is published: Jenaer Schriften zur Mathematik und Informatik, Math/Inf/13/06, Universität Jena, Germany, 2006.

MSC:
35K55 Nonlinear PDE of parabolic type
35J20 Variational methods for second-order, elliptic equations
Abstract: In this paper we consider the initial value problem for parabolic equations with nonlinearity indefinite sign $u_t = \Delta u + \lambda u + f(x)|u|^{\gamma-2}u$ in a smooth domain $\Omega\subset\rn$ with Dirichlet boundary condition. We introduce a critical value $\Lambda^\ast$ expressed in terms of a dual variational principle of a new type. We show that for any $\lambda\leq\Lambda^\ast$ there exist global positive solutions, whereas if $\lambda>\Lambda^\ast$ then any local solution blows up in finite time.