Blow-up of semilinear pde’s at the critical dimension. A probabilistic approach

Abstract

We present a probabilistic approach which proves blow-up of solutions of the Fujita equation $\partial w/\partial t = -(-\Delta )^{\alpha /2}w + w^{1+\beta }$ in the critical dimension $d=\alpha /\beta$. By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as $t\to \infty$. 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 Escobedo and Levine (1995) to the case of $\alpha$-Laplacians with possibly different parameters $\alpha$.