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

Birkner, Matthias; López-Mimbela, José; Wakolbinger, Anton

doi:10.1090/S0002-9939-02-06322-0

Blow-up of semilinear pde’s at the critical dimension. A probabilistic approach
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by Matthias Birkner, José Alfredo López-Mimbela and Anton Wakolbinger PDF

Proc. Amer. Math. Soc. 130 (2002), 2431-2442 Request permission

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 $\alpha$.

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