GLOBAL EXISTENCE AND BLOW-UP FOR NONAUTONOMOUS SYSTEMS WITH NON-LOCAL SYMMETRIC GENERATORS AND DIRICHLET CONDITIONS

We study a semilinear system of the form partial derivative ui(t, x)/partial derivative t = k(i)(t)A(i)u(i)(t,x) + u(i) (beta i)(t, x), t > 0, x is an element of D, u(i)(0, x) = f(i)(x), x is an element of D, u(i)|D-c 0, where D subset of R-d is a bounded open domain, ki : [0,infinity) -> [0,infinity) is continuous, A(i) is the infinitesimal generator of a symmetric jump-type process Z(i) {Z(i) (t)}(t >= 0), beta(i) > 1, i is an element of{1,2} and i' = 3-i. Under some assumptions on the infinitesimal generator A(i)(D) of the subprocess Zi killed upon leaving D, i = 1,2, we give sufficient conditions for global existence or finite-time blow-up of the positive mild solutions of our system. This paper can be considered as a continuation of the article [16].