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Pointwise eigenfunction estimates and mean Lp-norm blowup of a system of semilinear SPDEs with symmetric Lévy generators

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  • Guerrero, Eugenio
  • López-Mimbela, José Alfredo

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.

Suggested Citation

  • Guerrero, Eugenio & López-Mimbela, José Alfredo, 2019. "Pointwise eigenfunction estimates and mean Lp-norm blowup of a system of semilinear SPDEs with symmetric Lévy generators," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 47-54.
    • Handle: RePEc:eee:stapro:v:149:y:2019:i:c:p:47-54
    DOI: 10.1016/j.spl.2019.01.015
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