A VIABILITY RESULT FOR SEMILINEAR REACTION-DIFFUSION SYSTEMS

We consider a reaction-diffusion system of the form {u'(t) = Au(t) + F(u(t), v(t)), t >= 0 u'(t) = Bv(t) + G(u(t), v(t)), t >= 0 u(0) = xi, v(0) = eta, where X and Y are real Banach spaces, K is a nonempty and locally closed subset in X x Y, A : D(A) subset of X -> X, B : D(B) subset of Y -> Y are the generators of two C-0-semigroups, {S-A(t) : X -> X; t >= 0} and {S-B(t) : Y -> Y; t >= 0} respectively, F : K -> X, G : K -> Y are continuous such that A + F and/or B + G are locally of beta-compact type. We prove some necessary and sufficient conditions in order that for each (xi, eta) is an element of K, problem above has at least one mild solution (u, v) : [0, T] -> K.