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Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption. (English) Zbl 0799.35111
Summary: This paper deals with the Cauchy problem $$u\sb t- u\sb{xx}+ u\sp p=0; \quad -\infty<x<+ \infty, \quad t>0, \qquad u(x,0)= u\sb 0(x),$$ where $0<p<1$ and $u\sb 0(x)$ is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time $T$, and interfaces separating the regions where $u(x,t)>0$ and $u(x,t)=0$ appear when $t$ is close to $T$. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur.

MSC:
35K55Nonlinear parabolic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
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References:
[1] Angenent, S. B.: The zero set of a solution of a parabolic equation. J. reine angew. Math. 390, 79-96 (1988) · Zbl 0644.35050
[2] Angenent, S. B.; Fiedler, B.: The dynamics of rotating waves in scalar reaction diffusion equations. Trans. amer. Math. soc. 307, 545-568 (1988) · Zbl 0696.35086
[3] Brezis, H.; Friedman, A.: Estimates on the support of solutions of parabolic variational inequalities. Illinois J. Math. 20, 82-98 (1976)
[4] Bandle, C.; Stakgold, I.: The formation of the dead core in a parabolic reaction-diffusion equation. Trans. amer. Math. soc. 286, 275-293 (1984) · Zbl 0519.35042
[5] X. Chen, H. Matano, and M. Mimura, Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption, to appear. · Zbl 0814.35045
[6] Evans, L. C.; Knerr, B. F.: Instantaneous shrinking of the support of non-negative solutions to certain nonlinear parabolic equations and variational inequalities. Illinois J. Math. 23, 153-166 (1979) · Zbl 0403.35052
[7] Friedman, A.; Herrero, M. A.: Extinction properties of semilinear heat equations with strong absorption. J. math. Anal. appl. 124, 530-546 (1987) · Zbl 0655.35050
[8] Galaktionov, V. A.; Posashkov, S. A.: Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations. Differentsial’nye uravneniya 22, No. No. 7, 1165-1173 (1986) · Zbl 0632.35028
[9] Galaktionov, V. A.; Herrero, M. A.; Velázquez, J. J. L: The structure of solutions near an extinction point in a semilinear heat equation with strong absorption: A formal approach. Nonlinear diffusion equations and their equilibrium states, 215-236 (1992) · Zbl 0792.35079
[10] Giga, Y.; Kohn, R. V.: Asymptotically self-similar blow-up of semilinear heat equations. Comm. pure appl. Math. 38, 297-319 (1985) · Zbl 0585.35051
[11] Grundy, R. E.; Peletier, L. A.: Short time behaviour of a singular solution to the heat equation with absorption. Proc. roy. Soc. Edinburgh sect. A 107, 271-288 (1987) · Zbl 0642.35045
[12] Herrero, M. A.; Velázquez, J. J. L: On the dynamics of a semilinear heat equation with strong absorption. Comm. partial differential equations 14, No. No. 12, 1653-1715 (1989) · Zbl 0697.35019
[13] M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincaré, to appear.
[14] Herrero, M. A.; Velázquez, J. J. L: Flat blow-up in one-dimensional semilinear heat equations. Differential integral equations 5, No. No. 5, 973-997 (1992) · Zbl 0767.35036
[15] Kalashnikov, A. S.: The propagation of disturbances in problems of nonlinear heat conduction with strong absorption. U.S.S.R. comput. Math. and math. Phys. 14, 70-85 (1974)
[16] Kaper, H. G.; Kwong, M. K.: Comparison theorems and their use in the analysis of some nonlinear diffusion problems. Transport theory, invariant embedding and integral equations, 165-178 (1989)
[17] Kaper, H. G.; Kwong, M. K.: Concavity of solutions of certain Emden-Fowler equations. Differential integral equations 1, 327-340 (1988) · Zbl 0723.34026
[18] Rosenau, Ph; Kamin, S.: Thermal waves in an absorbing and convecting medium. Phys. D 8, 273-283 (1983) · Zbl 0542.35043