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Blow-up with logarithmic nonlinearities

Ferreira de Pablo, Raúl and Pablo, Arturo de and Rossi, Julio D. (2007) Blow-up with logarithmic nonlinearities. Journal of Differential Equations, 240 (1). pp. 196-215. ISSN 0022-0396

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Abstract

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition, ut = uxx − _(u + 1) logp(u + 1) (x, t) € R+ × (0, T),−ux(0, t) = (u + 1) logq(u + 1)(0, t) t € (0, T),u(x, 0) = u0(x) x € R+, with p, q, _ > 0. We describe in terms of p, q and when the solution is global in time and when it blows up in finite time. For blow-up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behaviour close to the blow-up time, showing that a phenomenon of asymptotic simplification takes place. We finally study the appearance of extinction in finite time.

Item Type:Article
Uncontrolled Keywords:Blow-up; Asymptotic behaviour; Nonlinear boundary conditions
Subjects:Sciences > Mathematics > Differential equations
ID Code:12515
Deposited On:05 Apr 2011 08:51
Last Modified:06 Feb 2014 09:26

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