### Abstract

We consider the stochastic heat equation of the following form: where L is the generator of a L ́evy process and ̇F is a spatially-colored, temporally white, Gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE. For the most part, we work under the assumptions that the initial data u0 is a bounded and measurable function and σ is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also weakly intermittent. In addition, we study the same equation in the case where Lu is replaced by its massive/dispersive analogue Lu - λu, where λ ∈ R. We also accurately describe the effect of the parameter λ on the intermittence of the solution in the case where σ(u) is proportional to u [the 'parabolic Anderson model']. We also look at the linearized version of our stochastic PDE, that is, the case where σ is identically equal to one [any other constant also works]. In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.

Original language | English |
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Pages (from-to) | 409-458 |

Number of pages | 50 |

Journal | Transactions of the American Mathematical Society |

Volume | 365 |

Issue number | 1 |

DOIs | |

Publication status | Published - 8 Aug 2012 |

Externally published | Yes |

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### Keywords

- stochastic heat equation
- weakly intermittent

### Cite this

*Transactions of the American Mathematical Society*,

*365*(1), 409-458. https://doi.org/10.1090/S0002-9947-2012-05616-9