Global existence and finite-time blow-up for a class of nonlocal parabolic problems

Abstract

An analysis of the nonlocal parabolic equation $$ u_t-\Delta u=\frac{\delta f(u)}{(\int_\Omega f(u))^p},\quad x\in\Omega, t>0, \tag"(P)" $$ and its associated steady state equation $$ -\Delta u=\frac{\delta f(u)}{(\int_\Omega f(u))^p}, \quad x\in \Omega, \tag"(S)" $$ with Dirichlet boundary conditions is given, assuming $\Omega\subset\Bbb R^n$ is a smooth bounded domain, $\delta>0$, $p\ge0$, and $f$ is positive and locally Lipschitz continuous. Existence-nonexistence results are proven for (S) when $f(u)=e^u$ or $e^{-u}$ and $\Omega$ is a ball or star-shaped domain. For $f(u)\ge c>0$, $n=1$, and $p\ge1$, we prove that (P) has a global bounded solution for any nonnegative initial data $u_0(x)$ and any $\delta>0$. For $f(u)=e^u$, $n=2$, $\Omega=B_1(0)$, $u_0(x)$ radially symmetric, nonnegative, if $p>1$, (P) has a unique, globally bounded solution for any $\delta>0$. If $p=1$, $0<\delta<8\pi$, (P) again has a bounded global solution. For $f(u)=e^u$, $n=1$ or 2, if $p<1$ and $\delta>\delta^*$ where $\delta^*$ is critical value for existence-nonexistence for (S), then the solution $u$ of (P) blows up in finite time.