Pointwise eigenfunction estimates and mean [formula omitted]-norm blowup of a system of semilinear SPDEs with symmetric Lévy generators

Abstract

We investigate semilinear systems of the form dui(t,x)=[Aiui(t,x)+Gi(u3−i(t,x))]dt+κiui(t,x)dWt,x∈D,i=1,2,with Dirichlet boundary conditions, where D⊂Rd is a bounded smooth domain, Ai is the generator of a pure-jump symmetric Lévy process in Rd, Gi is a convex locally Lipschitz function such that Gi(z)≥z1+βi for z≥0, βi>0 and κi are constants, i=1,2, and {Wt,t≥0} is a one-dimensional standard Brownian motion. We provide conditions ensuring finite-time blowup in the mean Lp-norm sense of positive weak solutions of the system above, for any p∈[1,∞). We also obtain upper bounds for the corresponding blowup times. Our approach uses in an essential way the first eigenfunctions of the generators Ai, and some two-sided estimates for them.