Asymptotic behavior of a nonlocal parabolic problem in Ohmic heating process

In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem $$u_{t}=\Delta u+\displaystyle\frac{\lambda f(u)}{\big(\int_{\Omega}f(u)dx\big)^{p}}, x\in \Omega, t>0,$$ with homogeneous Dirichlet boundary condition, where $\lambda>0, p>0$, $f$ is nonincreasing. It is found that: (a) For $0<p\leq1$, $u(x,t)$ is globally bounded and the unique stationary solution is globally asymptotically stable for any $\lambda>0$; (b) For $1<p<2$, $u(x,t)$ is globally bounded for any $\lambda>0$; (c) For $p=2$, if $0<\lambda<2|\partial\Omega|^2$, then $u(x,t)$ is globally bounded, if $\lambda=2|\partial\Omega|^2$, there is no stationary solution and $u(x,t)$ is a global solution and $u(x,t)\to\infty$ as $t\to\infty$ for all $x\in\Omega$, if $\lambda>2|\partial\Omega|^2$, there is no stationary solution and $u(x,t)$ blows up in finite time for all $x\in\Omega$; (d) For $p>2$, there exists a $\lambda^*>0$ such that for $\lambda>\lambda^*$, or for $0<\lambda\leq\lambda^*$ and $u_0(x)$ sufficiently large, $u(x,t)$ blows up in finite time. Moreover, some formal asymptotic estimates for the behavior of $u(x,t)$ as it blows up are obtained for $p\geq2$.