## Stable blow-up patterns.(English)Zbl 0770.35010

Summary: For the semilinear heat equation $$u_ t=\Delta u+e^ u$$ in a convex domain $$\Omega\subset\mathbb{R}^ n$$, given any $$b\in\Omega$$ we show the existence of solutions which blow up in finite time exactly at $$b$$ and whose final profile has the form $$u(T,x)\approx-2\ln| x- b|+\ln|\ln| x-b| |+\ln 8$$, $$T$$ being the blow-up time. Using a suitable set of rescaled coordinates, this asymptotic behavior is proved to be stable with respect to small perturbations of the initial conditions.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs 35K55 Nonlinear parabolic equations 35K05 Heat equation 35B45 A priori estimates in context of PDEs
Full Text:

### References:

 [1] Bebernes, J.; Kassoy, D., A mathematical analysis for blow-up for thermal reactions, SIAM J. Appl. Math., 40, 476-484 (1981) · Zbl 0481.35048 [2] Bebernes, J.; Bressan, A.; Eberly, D., A description of blow-up for the solid fuel ignition model, Indiana Univ. Math. J., 36, 131-136 (1987) [3] Bebernes, J.; Eberly, D., Mathematical Problems from Combustion Theory (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0692.35001 [4] Bressan, A., On the asymptotic shape of blow-up, Indiana Univ. Math. J., 39, 947-960 (1990) · Zbl 0705.35014 [5] Dold, J., Analysis of the early stage of thermal runaway, Quart. J. Mech. Appl. Math., 38, 361-387 (1985) · Zbl 0569.76079 [7] Friedman, A.; McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068 [8] Galaktionov, V. A.; Posashkov, S. A., The equation $$u_t = u_{ xx } + u^{β$$ [9] Giga, Y.; Kohn, R. V., Asymptotically self-similar blowup of semilinear heat equations, Comm. Pure Appl. Math., 38, 297-319 (1985) · Zbl 0585.35051 [10] Giga, Y.; Kohn, R. V., Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36, 1-40 (1987) · Zbl 0601.35052 [11] Henry, D., Geometric aspects of semi-linear parabolic equations, (Lecture Notes in Math., Vol. 840 (1981), Springer-Verlag: Springer-Verlag New York) [12] Lacey, A., A mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math., 43, 1350-1366 (1983) · Zbl 0543.35047 [13] Liu, W., The blowup rate of solutions of semilinear heat equations, J. Differential Equations, 77, 275-308 (1989) [14] Vanderbauwhede, A., Centre manifolds, normal forms and elementary bifurcations, (Dynamics Report. Ser. Dynam. Syst. Appl., Vol. 2 (1989), Wiley: Wiley Chichester), 89-169 · Zbl 0677.58001 [15] Weissler, F. B., Single point blow-up of semilinear initial value problems, J. Differential Equations, 55, 204-224 (1984) · Zbl 0555.35061
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.