Exponential stability, change of stability and eigenvalue problems for linear time-periodic parabolic equations on ${\mathbb{R}}^N$

Abstract

The main purpose of this article is to give conditions on a nonnegative weight function $m$ which are necessary and sufficient for the zero solution of the linear time-periodic parabolic equation $${\partial }_{t}u-\mathrm{\Delta }u=-m(x,t)u\text{in }{\mathbb{R}}^N\times (0,\mathrm{\infty })$$ to be exponentially stable and to apply these results to the study of change of stability in parameter dependent time-periodic parabolic problems.