Finite-Time Blowup and Existence of Global Positive Solutions of a Semi-linear Stochastic Partial Differential Equation with Fractional Noise

Abstract

We consider stochastic equations of the prototype$$\displaystyle{\mathrm{d}u(t,x) = \left (\Delta u(t,x) +\gamma u(t,x) + u{(t,x)}^{1+\beta }\right )\mathrm{d}t +\kappa u(t,x)\,\mathrm{d}B_{ t}^{H}}$$ on a smooth domain \(D \subset {\mathbb{R}}^{d}\), with Dirichlet boundary condition, where β > 0, γ and κ are constants and \(\{B_{t}^{H}\), t ≥ 0} is a real-valued fractional Brownian motion with Hurst index H > 1∕2. By means of the associated random partial differential equation, obtained by the transformation \(v(t,x) = u(t,x)\exp \{\kappa B_{t}^{H}\}\), lower and upper bounds for the blowup time of u 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.