On certain classes of mild solutions of the scalar Li\'enard equation revisited

TL;DR

This paper investigates the existence, uniqueness, and exponential decay of almost periodic, asymptotically almost periodic, and pseudo almost periodic mild solutions to the scalar Lie9nard equation using Green functions and a Massera-type principle.

Contribution

It introduces a novel approach employing Green functions and an abstract framework to analyze the asymptotic behavior of solutions, extending previous results on the Lie9nard equation.

Findings

Established existence and uniqueness of various almost periodic solutions.

Proved exponential decay of solutions using Gronwall's inequality.

Extended the theory to a broader class of parabolic evolution equations.

Abstract

In this work we revisit the existence, uniqueness and exponential decay of some classes of mild solutions which are almost periodic (AP-), asymptotically almost periodic (AAP-) and pseudo almost periodic (PAP-) of the scalar Lin\'eard equation by employing the notion of Green function and Massera-type principle. First, by changing variable we convert this equation to a system of first order differential equations. Then, we transform the problem into a framework of an abstract parabolic evolution equation which associates with an evolution family equipped an exponential dichtonomy and the corresponding Green function is exponentially almost periodic. After that, we prove a Massera-type principle that the corresponding linear equation has a uniqueness AP-, AAP- and PAP- mild solution if the right hand side and the coefficient functions are AP-, AAP- and PAP- functions, respectively. The well-posedness of semilinear equation is proved by using fixed point arguments and the exponential decay of mild solutions are obtained by using Gronwall's inequality. Although our work revisits some previous works on well-posedness of the Lin\'eard equation on these type solutions but provide a difference view by using Green function and go farther on the aspects of asymptotic behaviour of solutions and the construction of abstract theory. Our abstract results can be also applied to other parabolic evolution equations.