Global existence and nonexistence for a parabolic system with nonlinear boundary conditions

Abstract

We find both necessary and sufficient conditions on the nonnegative increasing functions $f$ and $g$ so that positive solutions of $$ \begin{matrix} u_t=\Delta u+h(u,v)\qquad &v_t=\Delta v+r(u,v) \qquad &\text {in }\Omega\times (0,T)\\ \frac{\partial}{\partial\nu} u=f(v) \qquad &\frac{\partial}{\partial\nu} v=g(u)\qquad &\text {on } \partial\Omega\times (0,T) \end{matrix} $$ exist globally in time. We assume throughout that $h$ and $r$ are nonnegative, smooth and $\frac{h(u,v)}u$, $\frac{r(u,v)}v$ are globally bounded.