Well-posedness and stability for a class of solutions of semi-linear diffusion equations with rough coefficients

TL;DR

This paper establishes the existence, uniqueness, and polynomial stability of pseudo almost periodic solutions for semi-linear diffusion equations with rough coefficients, using semigroup theory and fixed point methods.

Contribution

It introduces a novel approach to analyze semi-linear diffusion equations with rough coefficients, proving stability and existence of pseudo almost periodic solutions in interpolation spaces.

Findings

Proved boundedness of solution operators in interpolation spaces.

Showed preservation of pseudo almost periodicity by the solution operator.

Established existence and stability of solutions using fixed point arguments.

Abstract

In this work we study the existence, uniqueness and polynomial stability of the pseudo almost periodic mild solutions of semi-linear diffusion equations with rough coefficients in certain interpolation spaces. First, we rewirte the equations in abstract parabolic equation. Then, we use the polynomial stability of the semigroups of the corresponding linear equations to prove the boundedness of the solution operator for the linear equations in appropriate interpolation spaces. 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 to obtain our desired results.