Critical exponents for a system of heat equations coupled in a non-linear boundary condition

In this paper we consider a system of heat equations u(t) = Delta u, upsilon(t) = Delta upsilon in an unbounded domain Omega subset of R(N) coupled through the Neumann boundary conditions u(c) = upsilon(p), upsilon(upsilon) = u(q), where p > 0, q > 0, pq > 1 and nu is the exterior unit normal on partial derivative Omega. It is shown that for several types of domain there exists a critical exponent such that all of positive solutions blow up in a finite time in subcritical case (including the critical case) while there exist positive global solutions in the supercritical case if initial data are small.