Tohoku Mathematical Journal

Nonconstant selfsimilar blow-up profile for the exponential reaction-diffusion equation

Abstract

We study the blow-up profile of radial solutions of a semilinear heat equation with an exponential source term. Our main aim is to show that solutions which can be continued beyond blow-up possess a nonconstant selfsimilar blow-up profile. For some particular solutions we determine this profile precisely.

Article information

Source
Tohoku Math. J. (2), Volume 60, Number 3 (2008), 303-328.

Dates
First available in Project Euclid: 3 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1223057730

Digital Object Identifier
doi:10.2748/tmj/1223057730

Mathematical Reviews number (MathSciNet)
MR2453725

Zentralblatt MATH identifier
1158.35056

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35B40: Asymptotic behavior of solutions

Citation

Fila, Marek; Pulkkinen, Aappo. Nonconstant selfsimilar blow-up profile for the exponential reaction-diffusion equation. Tohoku Math. J. (2) 60 (2008), no. 3, 303--328. doi:10.2748/tmj/1223057730. https://projecteuclid.org/euclid.tmj/1223057730

References

• P. Baras and L. Cohen, Complete blow-up after $T_\max$ for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987), 142--174.
• J. Bebernes and S. Bricher, Final time blow-up profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal. 23 (1992), 852--869.
• J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989.
• A. Bressan, Stable blow-up patterns, J. Differential Equations 98 (1992), 57--75.
• J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity 7 (1994), 539--575.
• P. Brunovský and B. Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Anal. 10 (1986), 179--193.
• X. Chen, M. Fila and J.-S. Guo, Boundedness of global solutions of a supercritical parabolic equation, Nonlinear Anal. 68 (2008), 621--628.
• X.-Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of reaction-diffusion equations on a ball, J. Reine Angew. Math. 472 (1996), 17--51.
• D. Eberly, On the nonexistence of solutions to the Kassoy problem in dimensions 1 and 2, J. Math. Anal. Appl. 129 (1988), 401--408.
• D. Eberly and W. Troy, On the existence of logarithmic-type solutions to the Kassoy-Kapila problem in dimensions $3\leq N\leq 9$, J. Differential Equations 70 (1987), 309--324.
• M. Fila and H. Matano, Connecting equilibria by blow-up solutions, Discrete Contin. Dynam. Systems 6 (2000), 155--164.
• M. Fila, H. Matano and P. Poláčik, Existence of $L^1$-connections between equilibria of a semilinear parabolic equation, J. Dynam. Differential Equations 14 (2002), 463--491.
• M. Fila, H. Matano and P. Poláčik, Immediate regularization after blow-up, SIAM J. Math. Anal. 37 (2005), 752--776.
• M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations 4 (1999), 163--196.
• S. Filippas and R. Kohn, Refined asymptotics for the blow-up of $u_t - \Delta u = u^p$, Comm. Pure Appl. Math. 45 (1992), 821--869.
• A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425--447.
• I. M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. 29 (1963), 295--381.
• Y. Giga and R. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297--319.
• M. A. Herrero and J. J. L. Velázquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré, Anal. Non Linéaire 10 (1993), 131--189.
• M. A. Herrero and J. J. L. Velázquez, Plane structures in thermal runaway, Israel J. Math. 81 (1993), 321--341.
• D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal. 49 (1972), 241--269.
• A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43 (1983), 1350--1366.
• A. A. Lacey and D. E. Tzanetis, Global, unbounded solutions to a parabolic equation, J. Differential Equations 101 (1993), 80--102.
• H. Matano and F. Merle, On non-existence of type II blow-up for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), 1494--1541.
• J. Matos, Convergence of blow-up solutions of nonlinear heat equations in the supercritical case, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 1197--1227.
• J. Matos, Self-similar blow up patterns in supercritical heat equations, Commun. Appl. Anal. 5 (2001), 455--483.
• P. Poláčik, personal communication.
• J. L. Vázquez, Domain of existence and blowup for the exponential reaction-diffusion equation, Indiana Univ. Math. J. 48 (1999), 677--709.
• J. J. L. Velázquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), 441--464.
• J. J. L. Velázquez, Higher-dimensional blow up for semilinear parabolic equations, Comm. Partial Differential Equations 17 (1992), 1567--1696.