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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
  • J. A. López-Mimbela
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 90)

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.

Notes

Acknowledgements

This research was partially supported by the CNRS-CONACyT research grant “Blow-up of parabolic stochastic partial differential equations”. M. Dozzi acknowledges the European Commission for partial support by the grant PIRSES 230804 “Multifractionality”; he also acknowledges the hospitality of CIMAT, Guanajuato, where part of this work was done. E.T. Kolkovska and J.A. López-Mimbela acknowledge the hospitality of Institut Elie Cartan, Université de Lorraine.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • M. Dozzi
    • 1
  • E. T. Kolkovska
    • 2
  • J. A. López-Mimbela
    • 2
  1. 1.IECN, Université de LorraineVandoeuvre-lès-NancyFrance
  2. 2.Centro de Investigación en MatemáticasGuanajuatoMexico

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