Blow-up and stability of a nonlocal diffusion-convection problem arising in Ohmic heating of foods
2002
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.
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