Asymptotic behavior of a nonlocal parabolic problem in Ohmic heating process

Abstract

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 $\lambda >0,p>0$, $f$ is nonincreasing. It is found that: (a) For $0<p\le 1$, $u(x,t)$ 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 ${\lambda }^{\ast }>0$ such that for $\lambda >{\lambda }^{\ast }$, or for $0<\lambda \le {\lambda }^{\ast }$ 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 $p\ge 2$.

20pages