The Osgood condition for stochastic partial differential equations
Foondun, Mohammud and Nualart, Eulalia (2021) The Osgood condition for stochastic partial differential equations. Bernoulli - Journal of the Bernoulli Society. pp. 295-311. ISSN 1350-7265 (https://doi.org/10.3150/20-BEJ1240)
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Abstract
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.
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Item type: Article ID code: 69333 Dates: DateEvent1 February 2021Published28 July 2019Accepted29 July 2018SubmittedSubjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 13 Aug 2019 10:51 Last modified: 16 Sep 2024 00:48 URI: https://strathprints.strath.ac.uk/id/eprint/69333