Metastability for parabolic equations with drift: Part i

Hitoshi Ishii, Panagiotis E. Souganidis

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    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 languageEnglish
    Pages (from-to)875-913
    Number of pages39
    JournalIndiana University Mathematics Journal
    Volume64
    Issue number3
    DOIs
    Publication statusPublished - 2015

    Keywords

    • Asymptotic behavior
    • Metastability
    • Parabolic equation
    • Stochastic perturbation

    ASJC Scopus subject areas

    • Mathematics(all)

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