Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach

摘要: We present a probabilistic approach which proves blow-up of solutions the Fujita equation ∂w/∂t = -(-Δ) α/2 w + 1+β in critical dimension d α/β. By using Feynman-Kac representation twice, we construct subsolution locally grows to infinity as t → ∞. In this way, cover results proved earlier by analytic methods. Our method also applies extend result for systems Laplacian case Escobedo and Levine (1995) α-Laplacians with possibly different parameters a.