BLOW-UP VS. GLOBAL EXISTENCE FOR QUASI-LINEAR PARABOLIC-SYSTEMS WITH A NONLINEAR BOUNDARY-CONDITION

We study the behavior of positive solutions of the system u(t) = div(a (u)del u) + f(u, v) v(t) = div(b(v)del v) + g(u, v) in Omega a bounded domain with the boundary conditions partial derivative u/partial deri vative eta = r(u, v), partial derivative v/partial derivative eta = s( u, v) on partial derivative Omega and the initial data (u(o), v(o)). W e 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 e very (u(o), v(o)).