## 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.

(∂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 language | English |
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Number of pages | 19 |

Journal | Bernoulli Society for Mathematical Statistics and Probability |

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

## Keywords

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