Blow-up vs. global existence for quasilinear parabolic systems with a nonlinear boundary condition

Abstract.

We study the behavior of positive solutions of the system¶¶$u_t=\rm {div}(a(u)\nabla u) + f(u,v) \qquad v_t=\rm {div}(b(v) \nabla v) + g(u,v)$¶ in $\Omega$ a bounded domain with the boundary conditions ${\partial u \over \partial \eta}=r(u,v)$, ${\partial v \over \partial \eta}=s(u,v)$ on $\partial \Omega$ and the initial data $(u_0 , v_0)$. We find conditions on the functions a,b,f,g,r,s that guarantee the global existence (or finite time blow-up) of positive solutions for every $(u_0, v_0)$.