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February 2021 The Osgood condition for stochastic partial differential equations
Mohammud Foondun, Eulalia Nualart
Bernoulli 27(1): 295-311 (February 2021). DOI: 10.3150/20-BEJ1240


We study the following equation \begin{equation*}\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x)+b\bigl(u(t,x)\bigr)+\sigma \dot{W}(t,x),\quad t>0,\end{equation*} where $\sigma $ is a positive constant and $\dot{W}$ is a space–time white noise. The initial condition $u(0,x)=u_{0}(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman in (Phys. D 238 (2009) 209–215), our first result shows that the solution blows up in finite time if and only if for some $a>0$, \begin{equation*}\int _{a}^{\infty }\frac{1}{b(s)}\,\mathrm{d}s<\infty,\end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $\mathbf{R}^{d}$.


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Mohammud Foondun. Eulalia Nualart. "The Osgood condition for stochastic partial differential equations." Bernoulli 27 (1) 295 - 311, February 2021.


Received: 1 August 2019; Revised: 1 April 2020; Published: February 2021
First available in Project Euclid: 20 November 2020

zbMATH: 07282852
MathSciNet: MR4177371
Digital Object Identifier: 10.3150/20-BEJ1240

Keywords: fractional stochastic heat equation , space–time white noise , spatial colored noise

Rights: Copyright © 2021 Bernoulli Society for Mathematical Statistics and Probability


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Vol.27 • No. 1 • February 2021
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