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 $\alpha$.References
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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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2431-2442
- MSC (2000): Primary 60H30, 35K57, 35B35, 60J57
- DOI: https://doi.org/10.1090/S0002-9939-02-06322-0
- MathSciNet review: 1897470