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

We present a probabilistic approach which proves blow-up of solutions of the Fujita equation ∂ w/∂ t = -(-Δ )α /2w + w1+β in the critical dimension d=α /β. By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as t→ ∞. 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 α-Laplacians with possibly different parameters α.