by **
Y. Il'yasov**

**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
- 35B32 Bifurcation [See also 37Gxx, 37K50]
- 35B38 Critical points

**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.

**Keywords:** *parabolic equations, variational principle*

**Upload:** 2006-11-16

**Update:** 2006
-11
-16

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