IUMJ

Title: Metastability for parabolic equations with drift: part 1

Authors: Hitoshi Ishii and Panagiotis E. Souganidis

Issue: Volume 64 (2015), Issue 3, 875-913

Abstract:

We study the exponentially long time behavior of solutions to linear uniformly parabolic equations which 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 self-contained, 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.