Pseudo almost periodic weak solutions of a semilinear elliptic equation

Desheng Ji,Chuanyi Zhang

doi:10.1186/1687-1847-2014-46

Abstract

In this paper, pseudo almost periodic functions on R N , with N an integer larger than 1, are introduced and some basic properties of them are studied. As an application, we investigate the pseudo almost periodicity of a weak solution of the semilinear elliptic equation −Δu+ ∑ j = 1 N c j ∂ j u+f(x,u)=h(x). In addition, a pseudo almost periodic forced pendulum equation is considered as an example.MSC:35B15, 35J61.

Highlights

1 Introduction The notion of pseudo almost periodic functions on R was introduced by Zhang [ – ] in the early s and it is a natural generalization of Bohr almost periodicity
To the best of our knowledge, there is no literature systematically treating this yet. We will introduce these functions and study some of their basic properties. To illustrate their potential use, we study the pseudo almost periodicity of a weak solution of the following semilinear elliptic equation: N
We firstly show the almost periodicity of a bounded weak solution υ of ( . )

Summary

Introduction

The notion of pseudo almost periodic functions on R was introduced by Zhang [ – ] in the early s and it is a natural generalization of Bohr almost periodicity. The notion of Bohr almost periodicity is suitable for functions on RN , with N >. It is a natural idea to introduce pseudo almost periodic functions on RN and these functions may have potential use in differential equations with more than one spatial variables. We will introduce these functions and study some of their basic properties. To illustrate their potential use, we study the pseudo almost periodicity of a weak solution of the following semilinear elliptic equation: N. where cj’s are real constants, f and h are pseudo almost periodic functions. Section is devoted to the pseudo almost periodic functions on RN. We investigate pseudo almost periodicity of a weak solution of We investigate pseudo almost periodicity of a weak solution of ( . ) and consider a pseudo almost periodic forced pendulum equation as an example

Bohr almost periodic functions

Almost periodic and pseudo almost periodic weak solutions

Pseudo almost periodic forced pendulum equation Consider the equation