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ASYMPTOTIC BEHAVIOUR AND BLOW-UP FOR A NONLINEAR DIFFUSION PROBLEM WITH A NON-LOCAL SOURCE TERM

Published online by Cambridge University Press:  01 July 2004

N. I. Kavallaris
N. I. Kavallaris
Affiliation:
Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Zografu Campus, 15780 Athens, Greece ([email protected])
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Abstract

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In this work, the behaviour of solutions for the Dirichlet problem of the non-local equation

$$ u_t=\varDelta(\kappa(u))+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^p},\quad \varOmega\subset\mathbb{R}^N,\quad N=1,2, $$

is studied, mainly for the case where $f(s)=\mathrm{e}^{\kappa(s)}$. More precisely, the interplay of exponent $p$ of the non-local term and spatial dimension $N$ is investigated with regard to the existence and non-existence of solutions of the associated steady-state problem as well as the global existence and finite-time blow-up of the time-dependent solutions $u(x,t)$. The asymptotic stability of the steady-state solutions is also studied.

AMS 2000 Mathematics subject classification: Primary 35K60. Secondary 35B40

Keywords

non-local parabolic problemsasymptotic behaviourblow-up
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 2 , June 2004 , pp. 375 - 395
Copyright
Copyright © Edinburgh Mathematical Society 2004