Title: Complete quenching for a degenerate parabolic problem with a localised nonlinear source

Authors: C.Y. Chan; X.O. Jiang

Addresses: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, USA. ' Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, USA

Abstract: This paper studies a degenerate semilinear parabolic first initial boundary value problem with a nonlinear reaction f(u(b, t)) taking place only at the single site b with lim_u→c – f(u) = ∞ for some positive constant c. It is shown that there exists some t_q ≤ ∞ such that for 0 ≤ t < t_q, the problem has a unique nonnegative solution u before u(b, t) reaches c⁻, u is a strictly increasing function of t, and if t_q is finite, then u(b, t) reaches c⁻ at t_q. The problem is shown to have a unique a^* such that a unique global solution u exists for a ≤ a^* while for a > a^*, u(b, t) reaches c⁻ at a finite t_q. A formula relating a^*, b and f is given, and no quenching in infinite time is deduced. It is also shown that when u(b, t) reaches c at a finite t_q, u_q blows up everywhere. A computational method is devised to compute the finite t_q. For illustration, an example is given.

Keywords: localised nonlinear sources; critical length; quenching set; computational methods; boundary value problems.

DOI: 10.1504/IJDSDE.2012.045994

International Journal of Dynamical Systems and Differential Equations, 2012 Vol.4 No.1/2, pp.56 - 69

Published online: 10 Dec 2014 *

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