Refined asymptotics for the blow-up of \(u_ t-{\Delta}u=u^ p\). (English) Zbl 0784.35010

Summary: This work is concerned with positive, blowing-up solutions of the semilinear heat equation \(u_ t-\Delta u=u^ p\) in \(\mathbb{R}^ n\). Our main contribution is a sort of center manifold analysis for the equation in similarity variables, leading to refined asymptotics for \(u\) in a backward space-time parabola near any blow-up point. We also explore a connection between the asymptotics of \(u\) and the local geometry of the blow-up set.


35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI


[1] , A description of self similar blow up for dimensions n{\(\deg\)} 3, Ann. Inst. H. Poincaré Analyse Nonlinéaire 5, 1988, pp. 1–22.
[2] Berger, Comm. Pure Appl. Math. 41 pp 841– (1988)
[3] Bressan, Indiana Univ. Math. J. 39 pp 947– (1990)
[4] Bressan, J. Diff. Eqns.
[5] Chen, J. Diff. Eqns. 78 pp 160– (1989)
[6] Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981. · doi:10.1007/978-1-4612-5929-9
[7] and , On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincaré Analyse Nonlinéaire, in press.
[8] Blow up of solutions of nonlinear parabolic equations, pp. 301–318 in: Nonlinear Diffusion Equations and their Equilibrium States, Vol. 1, et al. eds., Springer-Verlag, New York, 1988. · doi:10.1007/978-1-4613-9605-5_19
[9] Friedman, Indiana Univ. Math. J. 34 pp 425– (1985)
[10] Galaktionov, USSR Comp. Math. and Math. Physics 31 pp 399– (1991)
[11] Giga, Comm. Pure Appl. Math. 38 pp 297– (1985)
[12] Giga, Indiana Univ. Math. J. 36 pp 1– (1987)
[13] Giga, Comm. Pure Appl. Math. 42 pp 297– (1989)
[14] and , Elliptic Partial Differential Equations of the Second Order, Second Edition, Springer-Verlag, Berlin, 1983. · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0
[15] Geometric theory of semilinear parabolic equations, Lecture Notes in Math. No. 840, Springer-Verlag, Berlin, 1981. · doi:10.1007/BFb0089647
[16] Herrero, Ann. Inst. H. Poincaré, Analyse Nonlinéaire
[17] and , Flat blow-up in one-dimensional semilinear parabolic equations, Integral and Differential Eqns., in press.
[18] Herrero, Comm. P.D.E.
[19] and , in preparation.
[20] and , editors, Dynamics Reported, Vol. 2, Wiley, 1989. · Zbl 0659.00009 · doi:10.1007/978-3-322-96657-5
[21] Blowup behavior for semilinear heat equations: multidimensional case, IMA preprint 711, Nov. 1990.
[22] Orthogonal Polynomials, A.M.S. Colloquium Publications, Vol. XXIII, Providence, 1967.
[23] Local behavior near blowup points for semilinear parabolic equations, J. Diff. Eqns., in press.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.