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Approaching an extinction point in one-dimensional semilinear heat-equations with strong absorption

Herrero, Miguel A. and Velázquez, J.J. L. (1992) Approaching an extinction point in one-dimensional semilinear heat-equations with strong absorption. Journal of Mathematical Analysis and Applications, 170 (2). pp. 353-381. ISSN 0022-247X

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Abstract

This paper deals with the Cauchy problem
u(t)-u(xx)+u(p)=0; -infinity<x<+infinity, t>o,
u(x, 0)=u(0)(x); -infinity<x<+infinity,
where 0<p<1 and u(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.

Item Type:Article
Uncontrolled Keywords:One-dimensional semilinear heat equations; interfaces; extinction point
Subjects:Sciences > Mathematics > Differential equations
ID Code:17346
References:

S. B. ANGENENT, The zero set of a solution of a parabolic equation, J. R&e Angew. Math. 390 (1988), 79-96.

S. B. ANGENENT AND B. FIEDLER, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc. 307 (1988), 545-568.

H. BREZIS AND A. FRIEDMAN, Estimates on the support of solutions of parabolic variational inequalities, Illinois J. Math. 20 (1976), 82-98.

C. BANDLE AND I. STAKGOLD, The formation of the dead core in a parabolic reaction-diffusion equation, Trans. Amer. Math. Soc. 286 (1984), 275-293.

X. CHEN, H. MATANO, AND M. MIMURA, Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption, to appear.

L. C. EVANS AND B. F. KNERR, Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois J. Math. 23 (1979) 153-166.

A. FRIEDMAN AND M. A. HERRERO, Extinction properties of semilinear heat equations with strong absorption, J. Math. Anal. Appl. 124 (1987), 530-546.

V. A. GALAKTIONOV AND S. A. POSASHKOV, Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations, Differentsial’nye Uravneniya 22, No. 7 (1986), 1165-1173.

V. A. GALAKTIONOV, M. A. HERRERO AND J. J. L. VELAZQUEZ, The structure of solutions near an extinction point in a semilinear heat equation with strong absorption: A formal approach, in “Nonlinear Diffusion Equations and Their Equilibrium States” (N. G. Lloyd et al., Eds.), pp. 215-236, Birkhäuser-Verlag, Basel, 1992.

Y. GICA AND R. V. KOHN, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985) 297-319.

R. E. GRUNDY AND L. A. PELETIER, Short time behaviour of a singular solution to the heat equation with absorption, Proc. Roy. Sot. Edinburgh Sect. A 107 (1987), 271-288.

M. A. HERRERO AND J. J. L. VELÁZQUEZ, On the dynamics of a semilinear heat equation with strong absorption, Comm. Partial Differential Equations 14, No. 12 (1989), 1653-1715.

M. A. HERRERO, AND J. J. L. VELÁZQUEZ, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincaré, to appear.

M. A. HERRERO AND J. J. L. VELÁZQUEZ, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations 5, No. 5 (1992), 973-997.

A. S. KALASHNIKOV, The propagation of disturbances in problems of nonlinear heat conduction with strong absorption, U.S.S.R. Comput. Math. and Math. Phys. 14 (1974), 70-85.

H. G. KAPER AND M. K. KWONG, Comparison theorems and their use in the analysis of some nonlinear diffusion problems, in “Transport Theory, Invariant Embedding and Integral Equations,” pp. 165-178, Dekker, New York, 1989.

H. G. KAPER AND M. K. KWONG, Concavity of solutions of certain Emden-Fowler equations, Differential Integral Equations 1 (1988), 327-340.

PH. ROSENAU AND S. KAMIN, Thermal waves in an absorbing and convecting medium, Phys. D 8 (1983), 273-283.

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