Finite-time blowup and existence of global positive solutions of a semi-linear SPDE with fractional noise

M Dozzi (1) , E. T. Kolkovska (2) , J. A. Lopez (2)

1 IECL - Institut Élie Cartan de Lorraine
2 CIMAT - Centro de Investigación en Matemáticas

We consider stochastic equations of the prototype du(t, x) = delta u(t, x) + c*u(t, x) + u(t, x)^(1+ beta)) dt + k*u(t, x) dB^(H)_t on a smooth domain D in IR^d , with Dirichlet boundary condition, where beta > 0, c and k are constants and (B^(H)_ t , t in IR+) is a real-valued fractional Brownian motion with Hurst index H > 1/2. By means of an associated random partial differential equation lower and upper bounds for the blowup time are given. Sufficient conditions for blowup in finite time and for the existence of a global solution are deduced in terms of the parameters of the equation. For the case H = 1/2 (i.e. for Brownian motion) estimates for the probability of blowup in finite time are given in terms of the laws of exponential functionals of Brownian motion.

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

M Dozzi, E. T. Kolkovska, J. A. Lopez. Finite-time blowup and existence of global positive solutions of a semi-linear SPDE with fractional noise. Modern Stochastics and Applications, pp.95-108, 2014, 978-3-319-03511-6. ⟨hal-01095624⟩