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

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.