Global and non-global solutions of a fractional reaction-diffusion equation perturbed by a fractional noise
Résumé
We provide conditions implying finite-time blowup of positive weak solutions to the semilinear equation du(t, x) = [∆^α u(t, x) + Ku(t, x) + u^(1+β) (t, x)] dt + µu(t, x) dB^H_t , u(0, x) = f (x), x ∈ R^d , t ≥ 0, where α ∈ (0, 2], K ∈ R, β > 0, µ ≥ 0 and H ∈ [ 1/ 2 , 1) are constants, ∆^α is the fractional power −(−∆)^α/2 of the Laplacian, (B^H_t) is a fractional Brownian motion with Hurst parameter H, and f ≥ 0 is a bounded measurable function. To achieve this we investigate the growth of exponential functionals of the form: Integral_r_0^T exp(β(Ks+µB^H_s))/s^ (dβ/α) ds as T → ∞ with r_0 > 0. Moreover we provide sufficient conditions for the existence of a global weak solution of the above equation, as well as upper and lower bounds for the probability that the solution does not blow up in finite time.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
Loading...