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2002 Blow-up and stability of a nonlocal diffusion-convection problem arising in Ohmic heating of foods
N. I. Kavallaris, D. E. Tzanetis
Differential Integral Equations 15(3): 271-288 (2002).

Abstract

We study the blow-up and stability of solutions of the equation $u_t+u_x=u_{xx}+{\lambda} f(u)/ (\int^1_0 f(u)\,dx )^2$ with certain initial and boundary conditions. When $f$ is a decreasing function, we show that if $\int^{\infty}_0f(s)\,ds <\infty$, then there exists a ${\lambda}^{*}>0$ such that for ${\lambda}>{\lambda}^{*}$, or for any $0 <{\lambda}\leq {\lambda}^{*}$ but with initial data sufficiently large, the solutions blow up in finite time. If $\int^{\infty}_0f(s)\,ds=\infty$, then the solutions are global in time. The stability of solutions in both cases is discussed. We also study the case of $f$ being increasing.

Citation

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N. I. Kavallaris. D. E. Tzanetis. "Blow-up and stability of a nonlocal diffusion-convection problem arising in Ohmic heating of foods." Differential Integral Equations 15 (3) 271 - 288, 2002.

Information

Published: 2002
First available in Project Euclid: 21 December 2012

zbMATH: 1011.35022
MathSciNet: MR1871290

Subjects:
Primary: 35K57
Secondary: 35B35 , 35B40 , 35K20 , 35K55

Rights: Copyright © 2002 Khayyam Publishing, Inc.

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Vol.15 • No. 3 • 2002
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