Global and nonglobal solutions of a system of nonautonomous semilinear equations with ultracontractive Lévy generators

We consider a semilinear system of the form ∂ui(t,x)/∂t=k(t)Aui(t,x)+ui′βi(t,x), with Dirichlet boundary conditions on a bounded open set D⊂RdD⊂Rd, where k:[0,∞)→[0,∞)k:[0,∞)→[0,∞) is continuous, AA is the infinitesimal generator of a symmetric Lévy process Z≡{Z(t)}t≥0Z≡{Z(t)}t≥0, βi>1βi>1, i∈{1,2}i∈{1,2} and i′=3−ii′=3−i. We give conditions on D and on the Lévy measure of Z under which our system possesses global positive solutions or exhibits blow-up in finite time. Our approach is based on the intrinsic ultracontractivity property of the semigroup generated by the process Z killed on leaving D.