Blow-up of Semilinear PDE’s at the Critical Dimension. A Probabilistic Approach

Abstract. We present a probabilistic approach which proves blow-up of so-lutions of the Fujita equation @w=@t = ()=2w + w1+ in the critical dimension d = =. By using the Feynman-Kac representation twice, we con-struct a subsolution which locally grows to innity as t! 1. In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Es-cobedo and Levine (1995) to the case of -Laplacians with possibly dierent parameters