### Abstract

Consider the following equation ∂tut(x) = 1 2 ∂xxut(x) + λσ(ut(x))W˙ (t, x) on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if λ is large enough. But if λ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what λ is. We also provide various extensions.

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

Pages | 551-571 |

Number of pages | 21 |

Journal | Alea |

Volume | 12 |

Issue number | 2 |

Publication status | Published - 2015 |

Externally published | Yes |

### Fingerprint

### Keywords

- stochastic partial differential equations
- Dirichlet boundary condition

### Cite this

*Alea*,

*12*(2), 551-571.

}

*Alea*, vol. 12, no. 2, pp. 551-571.

**On the behaviour of stochastic heat equations on bounded domains.** / Foondun, Mohammud; Nualart, Eulalia.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the behaviour of stochastic heat equations on bounded domains

AU - Foondun, Mohammud

AU - Nualart, Eulalia

PY - 2015

Y1 - 2015

N2 - Consider the following equation ∂tut(x) = 1 2 ∂xxut(x) + λσ(ut(x))W˙ (t, x) on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if λ is large enough. But if λ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what λ is. We also provide various extensions.

AB - Consider the following equation ∂tut(x) = 1 2 ∂xxut(x) + λσ(ut(x))W˙ (t, x) on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if λ is large enough. But if λ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what λ is. We also provide various extensions.

KW - stochastic partial differential equations

KW - Dirichlet boundary condition

UR - http://www.scopus.com/inward/record.url?scp=84953730452&partnerID=8YFLogxK

UR - http://alea.impa.br/english/index_v13.htm

M3 - Article

VL - 12

SP - 551

EP - 571

JO - Alea

T2 - Alea

JF - Alea

SN - 1980-0436

IS - 2

ER -