Blow-up of semilinear pde's at the critical dimension. A probabilistic approach期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Matthias Birkner José Alfredo Ló pez-Mimbela Anton Wakolbinger

Institution:	FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany ; Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato 36000, Mexico ; FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany

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 $\alpha$ .

Keywords:	Blow-up of semilinear systems Feynman-Kac representation symmetric stable processes

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