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A semilinear heat equation with singular initial data

Published online by Cambridge University Press:  14 November 2011

Minkyu Kwak
Minkyu Kwak
Affiliation:
Department of Mathematics, Chonnam National University, Kwangju, 500-757, Korea
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Abstract

We first prove existence and uniqueness of non-negative solutions of the equation

in in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the form

where g = ga satisfies

After uniqueness is proved, the asymptotic behaviour of solutions of

is studied. In particular, we show that

The case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 745 - 758
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Baek, J. S. and Kwak, M.. Singular solutions of semilinear parabolic equations in several space dimensions. J. Korean Math. Soc. 34 (1997), 1049–64.Google Scholar
2Ban, H. J. and Kwak, M.. Asymptotic behavior of singular solutions of semilinear parabolic equations. Honam Math. J. 17 (1995), 107–18.Google Scholar
3Brezis, H. and Friedman, A.. Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. 62 (1983), 73–97.Google Scholar
4Brezis, H., Peletier, L. A. and Terman, D.. A very singular solution of the hard equation with absorption. Arch. Rational Mech. Anal. 95 (1986), 185–209.CrossRefGoogle Scholar
5Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (Malabar, FL: R. E. Krieger, 1987).Google Scholar
6Escobedo, M. and Kavian, O.. Variational problems related to self-similar solutions of the heat equation. Nonlinear Anal. 11 (1987), 1103–33.CrossRefGoogle Scholar
7Escobedo, M., Kavian, O. and Matano, H.. Large time behavior of solutions of a dissipative semilinear heat equation. Comm. Partial Differential Equations 20 (1995), 1427–52.CrossRefGoogle Scholar
8Friedman, A.. Partial differential equations of parabolic type (Englewood Cliffs, NJ: Prentice-Hall, 1994).Google Scholar
9Hartman, P.. Ordinary differential equations (New York: John Wiley, 1964).Google Scholar
10Kamin, S., Peletier, L. A. and Vazquez, J. L.. Classification of singular solutions of nonlinear heat equation. Duke Math. J. 58 (1989), 601–15.CrossRefGoogle Scholar
11Kamin, S. and Vazquez, J. L.. Singular solutions of some nonlinear parabolic equations (IMA Preprint Series 834, 1991).Google Scholar
12Kwak, M. and Yu, K.. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete Contin. Dynam. Systems 2 (1996), 483–96.CrossRefGoogle Scholar
13Oswald, L.. Isolated positive singularities for a nonlinear heat equation. Houston J. Math. 14 (1988), 543–72.Google Scholar
14Porter, M. H. and Weinberger, H. F.. Maximum principles in differential equations (New York: Springer, 1984).CrossRefGoogle Scholar