The Osgood condition for stochastic partial differential equations

Mohammud Foondun, Eulalia Nualart

Research output: Contribution to journalArticle

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 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 Rd . 
LanguageEnglish
Number of pages19
JournalAnnals of Probability
Publication statusAccepted/In press - 28 Jul 2019

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Stochastic Partial Differential Equations
Space-time White Noise
Colored Noise
Blow-up Solution
Dirichlet Boundary Conditions
Nonexistence
Continuous Function
Initial conditions
Non-negative
Sufficient
If and only if
Line
Theorem
Partial differential equations
Dirichlet
Blow-up
Boundary conditions

Keywords

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

Cite this

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title = "The Osgood condition for stochastic partial differential equations",
abstract = "We study the following equation (∂u(t,x))/∂t  = ∆u(t,x) + b(u(t,x)) + σW (t,x),t > 0where 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 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 Rd . ",
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The Osgood condition for stochastic partial differential equations. / Foondun, Mohammud; Nualart, Eulalia.

In: Annals of Probability, 28.07.2019.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The Osgood condition for stochastic partial differential equations

AU - Foondun, Mohammud

AU - Nualart, Eulalia

PY - 2019/7/28

Y1 - 2019/7/28

N2 - We study the following equation (∂u(t,x))/∂t  = ∆u(t,x) + b(u(t,x)) + σW (t,x),t > 0where 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 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 Rd . 

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

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KW - space-time white noise

KW - spatial colored noise

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JO - Annals of Probability

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