### Abstract

(∂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 .

Language | English |
---|---|

Number of pages | 19 |

Journal | Annals of Probability |

Publication status | Accepted/In press - 28 Jul 2019 |

### Fingerprint

### Keywords

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

### Cite this

*Annals of Probability*.

}

**The Osgood condition for stochastic partial differential equations.** / Foondun, Mohammud; Nualart, Eulalia.

Research output: Contribution to journal › Article

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 .

KW - fractional stochastic heat equation

KW - space-time white noise

KW - spatial colored noise

UR - https://www.imstat.org/journals-and-publications/annals-of-probability/

M3 - Article

JO - Annals of Probability

T2 - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

ER -