Differential and Integral Equations

Blow-up and stability of a nonlocal diffusion-convection problem arising in Ohmic heating of foods

N. I. Kavallaris and D. E. Tzanetis

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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.

Article information

Source
Differential Integral Equations, Volume 15, Number 3 (2002), 271-288.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060861

Mathematical Reviews number (MathSciNet)
MR1871290

Zentralblatt MATH identifier
1011.35022

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35K20: Initial-boundary value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations

Citation

Kavallaris, N. I.; Tzanetis, D. E. Blow-up and stability of a nonlocal diffusion-convection problem arising in Ohmic heating of foods. Differential Integral Equations 15 (2002), no. 3, 271--288. https://projecteuclid.org/euclid.die/1356060861


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