Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. DYNKIN, E. B. (1965). Markov processes, vol. 1. Springer Verlag, Berlin.MR 33:1887
  • 2. ESCOBEDO, M. AND LEVINE, H. (1995). Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch. Rational Mech. Anal. 129, 47-100.MR 96d:35063
  • 3. FREIDLIN, M. (1985). Functional integration and partial differential equations. Princeton University Press.MR 87g:60066
  • 4. FUJITA, H. (1966). On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u + u^{1+\alpha}$. J. Fac. Sci. Univ. Tokyo Sect. I 13, 109-124.MR 35:5761
  • 5. GUEDDA, M., AND KIRANE, M. (1999). A note on nonexistence of global solutions to a nonlinear integral equation. Bull. Belg. Math. Soc. Simon Stevin 6, 491-497.MR 2000k:35127
  • 6. KOBAYASHI, K., SIRAO, T. AND TANAKA, H. (1977). On the growing up problem for semilinear heat equations. J. Math. Soc. Japan 29, 407-424.MR 56:9076
  • 7.OPEZ-MIMBELA, J.A. AND WAKOLBINGER, A. (1998). Length of Galton-Watson trees and blow-up of semilinear systems. J. Appl. Prob. 35, 802-811.MR 2000k:60171
  • 8.OPEZ-MIMBELA, J.A. AND WAKOLBINGER, A. (2000). A probabilistic proof of non-explosion of a non-linear PDE system. J. Appl. Prob. 37, 635-641. MR 2001m:60197
  • 9. PORTNOY, S. (1975). Transience and solvability of a non-linear diffusion equation. Ann. Probab. 3, 465-477.MR 52:15688
  • 10. PORTNOY, S. (1976). On solutions to $u_t=\Delta u + u^{2}$ in two dimensions. J. Math. Anal. Appl. 55, 291-294.MR 53:13838
  • 11. STROOCK, D. (1993). Probability theory, an analytic view. Cambridge University Press.MR 95f:60003
  • 12. SUGITANI, S. (1975). On nonexistence of global solutions for some nonlinear integral equations. Osaka J. Math. 12, 45-51.MR 57:10247
  • 13. WANG, L. (2001). The blow-up for weakly coupled reaction-diffusion systems. Proc. Amer. Math. Soc. 129, 89-95 (electronic) MR 2001j:35167

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60H30, 35K57, 35B35, 60J57

Retrieve articles in all journals with MSC (2000): 60H30, 35K57, 35B35, 60J57

Additional Information

Matthias Birkner
Affiliation: FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany

José Alfredo López-Mimbela
Affiliation: Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato 36000, Mexico

Anton Wakolbinger
Affiliation: FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany

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