The Osgood condition for stochastic partial differential equations

Mohammud Foondun, Eulalia Nualart

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We study the following equation 
(∂u(t,x))/∂t  = ∆u(t,x) + b(u(t,x)) + σW (t,x),t > 0

where σ is a positive constant and W is a space-time white noise. The initial condition u(0, x) = u0(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, our first result shows that the solution blows up in finite time if and only if 
which is the well-known Osgood condition. We also consider the same equation on thewhole line and show that the above condition is sufficient for the nonexistence of globalsolutions. Various other extensions are provided; we look at equations with fractionalLaplacian and spatial colored noise in Rd. 
Original languageEnglish
Pages (from-to)295-311
Number of pages19
JournalBernoulli - Journal of the Bernoulli Society
Publication statusPublished - 1 Feb 2021


  • fractional stochastic heat equation
  • space-time white noise
  • spatial colored noise

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