Asymptotic behavior of a nonlocal parabolic problem in Ohmic heating process
In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem \[ u_{t}=Δu+\displaystyle\frac{λf(u)}{\big(\int_Ωf(u)dx\big)^{p}}, x\in Ω, t>0, \] with homogeneous Dirichlet boundary condition, where $λ>0, p>0$, $f$ is nonincreasing. It is found that: (a) For $0 0$; (b) For $1 0$; (c) For $p=2$, if $0<λ<2|\partialΩ|^2$, then $u(x,t)$ is globally bounded, if $λ=2|\partialΩ|^2$, 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 $x\inΩ$, if $λ>2|\partialΩ|^2$, there is no stationary solution and $u(x,t)$ blows up in finite time for all $x\inΩ$; (d) For $p>2$, there exists a $λ^*>0$ such that for $λ>λ^*$, or for $0<λ\leqλ^*$ 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 $p\geq2$.