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Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

Published online by Cambridge University Press:  16 July 2002

N. I. KAVALLARIS ,
C. V. NIKOLOPOULOS  and
D. E. TZANETIS
N. I. KAVALLARIS
Affiliation:
Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
C. V. NIKOLOPOULOS
Affiliation:
Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
D. E. TZANETIS
Affiliation:
Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
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Abstract

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 13 , Issue 3 , June 2002 , pp. 337 - 351
Copyright
2002 Cambridge University Press

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