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SOME INTERESTING SPECIAL CASES OF A NON-LOCAL PROBLEM MODELLING OHMIC HEATING WITH VARIABLE THERMAL CONDUCTIVITY

Published online by Cambridge University Press:  20 January 2009

D. E. Tzanetis
Affiliation:
Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Zografou Campus, 157 80 Athens, GR ([email protected])
P. M. Vlamos
Affiliation:
Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Zografou Campus, 157 80 Athens, GR ([email protected])
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Abstract

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The non-local equation

$$ u_t=(u^3u_x)_x+\frac{\lambda f(u)}{(\int_{-1}^1f(u)\,\rd x)^{2}} $$

is considered, subject to some initial and Dirichlet boundary conditions. Here $f$ is taken to be either $\exp(-s^4)$ or $H(1-s)$ with $H$ the Heaviside function, which are both decreasing. It is found that there exists a critical value $\lambda^*=2$, so that for $\lambda>\lambda^{*}$ there is no stationary solution and $u$ ‘blows up’ (in some sense). If $0\lt\lambda\lt\lambda^{*}$, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.

AMS 2000 Mathematics subject classification: Primary 35B30; 35B35; 35B40; 35K20; 35K55; 35K99

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2001