### Abstract

We study the exponentially long-time behavior of solutions to linear uniformly parabolic equations that are small perturbations of transport equations with vector fields having a globally stable (attractive) equilibrium in the domain. The result is that the solutions converge to a constant, which is either the initial value at the stable point or the boundary value at the minimum of the associated quasi-potential. Problems of this type were considered by Freidlin and Wentzell and Freidlin and Koralov, using probabilistic arguments related to the theory of large deviations. Our approach, which is selfcontained, relies entirely on pde arguments, and is flexible to the extent that allows us to study a class of semilinear equations of similar structure. This note also prepares the ground for the forthcoming Part II of this work where we consider general quasilinear problems.

Original language | English |
---|---|

Pages (from-to) | 875-913 |

Number of pages | 39 |

Journal | Indiana University Mathematics Journal |

Volume | 64 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Asymptotic behavior
- Metastability
- Parabolic equation
- Stochastic perturbation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*64*(3), 875-913. https://doi.org/10.1512/iumj.2015.64.5559

**Metastability for parabolic equations with drift : Part i.** / Ishii, Hitoshi; Souganidis, Panagiotis E.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 64, no. 3, pp. 875-913. https://doi.org/10.1512/iumj.2015.64.5559

}

TY - JOUR

T1 - Metastability for parabolic equations with drift

T2 - Part i

AU - Ishii, Hitoshi

AU - Souganidis, Panagiotis E.

PY - 2015

Y1 - 2015

N2 - We study the exponentially long-time behavior of solutions to linear uniformly parabolic equations that are small perturbations of transport equations with vector fields having a globally stable (attractive) equilibrium in the domain. The result is that the solutions converge to a constant, which is either the initial value at the stable point or the boundary value at the minimum of the associated quasi-potential. Problems of this type were considered by Freidlin and Wentzell and Freidlin and Koralov, using probabilistic arguments related to the theory of large deviations. Our approach, which is selfcontained, relies entirely on pde arguments, and is flexible to the extent that allows us to study a class of semilinear equations of similar structure. This note also prepares the ground for the forthcoming Part II of this work where we consider general quasilinear problems.

AB - We study the exponentially long-time behavior of solutions to linear uniformly parabolic equations that are small perturbations of transport equations with vector fields having a globally stable (attractive) equilibrium in the domain. The result is that the solutions converge to a constant, which is either the initial value at the stable point or the boundary value at the minimum of the associated quasi-potential. Problems of this type were considered by Freidlin and Wentzell and Freidlin and Koralov, using probabilistic arguments related to the theory of large deviations. Our approach, which is selfcontained, relies entirely on pde arguments, and is flexible to the extent that allows us to study a class of semilinear equations of similar structure. This note also prepares the ground for the forthcoming Part II of this work where we consider general quasilinear problems.

KW - Asymptotic behavior

KW - Metastability

KW - Parabolic equation

KW - Stochastic perturbation

UR - http://www.scopus.com/inward/record.url?scp=84956636602&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84956636602&partnerID=8YFLogxK

U2 - 10.1512/iumj.2015.64.5559

DO - 10.1512/iumj.2015.64.5559

M3 - Article

VL - 64

SP - 875

EP - 913

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 3

ER -