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THE QUENCHING OF SOLUTIONS OF A REACTION–DIFFUSION EQUATION WITH FREE BOUNDARIES

Part of: Parabolic equations and systems Partial differential equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  22 January 2016

NINGKUI SUN
NINGKUI SUN*
Affiliation:
Department of Mathematics, Tongji University, Shanghai 200092, China email [email protected]
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Abstract

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This paper concerns the quenching phenomena of a reaction–diffusion equation $u_{t}=u_{xx}+1/(1-u)$ in a one dimensional varying domain $[g(t),h(t)]$, where $g(t)$ and $h(t)$ are two free boundaries evolving by a Stefan condition. We prove that all solutions will quench regardless of the choice of initial data, and we also show that the quenching set is a compact subset of the initial occupying domain and that the two free boundaries remain bounded.

Keywords

reaction–diffusion equationquenchingfree-boundary problem

MSC classification

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35R35: Free boundary problems 35B99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 110 - 120
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Acker, A. and Walter, W., ‘On the global existence of solutions of parabolic differential equations with a singular nonlinear term’, Nonlinear Anal. 2 (1978), 499–505.Google Scholar
Chan, C. Y. and Kaper, Hans G., ‘Quenching for semilinear singular parabolic problems’, SIAM J. Math. Anal. 20 (1989), 558–566.Google Scholar
Du, Y. and Guo, Z. M., ‘Spreading-vanishing dichotomy in a diffusion logistic model with a free boundary, II’, J. Differential Equations 250 (2011), 4336–4366.Google Scholar
Du, Y. and Lin, Z. G., ‘Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary’, SIAM J. Math. Anal. 42 (2010), 377–405.Google Scholar
Du, Y. and Lou, B. D., ‘Spreading and vanishing in nonlinear diffusion problems with free boundaries’, J. Eur. Math. Soc. 17 (2015), 2673–2724.Google Scholar
Friedman, A. and Mcleod, B., ‘Blow-up of positive solutions of semilinear heat equations’, Indiana Univ. Math. J. 34 (1985), 425–447.Google Scholar
Guo, J.-S., ‘On the quenching behavior of the solution of a semilinear parabolic equation’, J. Math. Anal. Appl. 151 (1990), 58–79.Google Scholar
Guo, J.-S. and Hu, B., ‘Quenching problem for a singular semilinear heat equation’, Nonlinear Anal. 30 (1997), 905–910.Google Scholar
Kawarada, H., ‘On solutions of initial-boundary problem for u t = u xx + 1/(1 - u)’, Publ. RIMS, Kyoto Univ. 10 (1975), 729–736.Google Scholar
Ke, L. and Ning, S., ‘Quenching for degenerate parabolic equations’, Nonlinear Anal. 34 (1998), 1123–1135.Google Scholar
Levine, H. A. and Montgomery, J. T., ‘The quenching of solutions of some nonlinear parabolic equations’, SIAM J. Math. Anal. 11 (1980), 842–847.Google Scholar
Montenegro, M., ‘Complete quenching for singular parabolic equations’, J. Math. Anal. Appl. 384 (2011), 591–596.Google Scholar
Rubinstein, L. I., The Stefan Problem (American Mathematical Society, Providence, RI, 1971).Google Scholar
Sun, N. K., ‘Blow-up and asymptotic behavior of solutions for reaction-diffusion equations with free boundaries’, J. Math. Anal. Appl. 428 (2015), 838–854.Google Scholar