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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Matthias Birkner, José Alfredo López-Mimbela and Anton Wakolbinger
Journal: Proc. Amer. Math. Soc. 130 (2002), 2431-2442
MSC (2000): Primary 60H30, 35K57, 35B35, 60J57
Published electronically: February 4, 2002
MathSciNet review: 1897470
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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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Additional Information

Matthias Birkner
Affiliation: FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany
Email: birkner@math.uni-frankfurt.de

José Alfredo López-Mimbela
Affiliation: Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato 36000, Mexico
Email: jalfredo@cimat.mx

Anton Wakolbinger
Affiliation: FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany
Email: wakolbinger@math.uni-frankfurt.de

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06322-0
PII: S 0002-9939(02)06322-0
Keywords: Blow-up of semilinear systems, Feynman-Kac representation, symmetric stable processes
Received by editor(s): November 15, 2000
Received by editor(s) in revised form: February 28, 2001
Published electronically: February 4, 2002
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2002 American Mathematical Society