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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
Abstract
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
Keywords
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- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 44 , Issue 3 , October 2001 , pp. 585 - 595
- Copyright
- Copyright © Edinburgh Mathematical Society 2001
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