An Interpolation Approach to Pseudo Almost Periodic Solutions for Parabolic Evolution Equations

Pham Truong Xuan, Le The Sac, Vu Thi Thuy Ha

Published 2021-09-11Version 1

In this work we study the existence, uniqueness and polynomial stability of the pseudo almost periodic mild solutions of a class of linear and semi-linear parabolic evolution equations on the whole line $\mathbb{R}$ on interpolation spaces. We consider the cases where we have polynomial stability of the semigroups of the corresponding linear equations. This allows us to prove the boundedness of the solution operator for the linear equations in appropriate interpolation spaces and then we show that this operator preserves the pseudo almost periodic property of functions. We will use the fixed point argument to obtain the existence and stability of the pseudo almost periodic mild solutions for the semi-linear equations. The abstract results will be applied to the semi-linear diffusion equations with rough coefficients.