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

M. Dozzi (), E. T. Kolkovska () and J. A. López-Mimbela ()
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M. Dozzi: IECN, Université de Lorraine
E. T. Kolkovska: Centro de Investigación en Matemáticas
J. A. López-Mimbela: Centro de Investigación en Matemáticas

A chapter in Modern Stochastics and Applications, 2014, pp 95-108 from Springer

Abstract: 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.

Keywords: Global Positive Solution; Blow-up Time; Random Partial Differential Equations; Fractional Brownian Motion; Exponential Functionals (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-3-319-03512-3_6

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