Asymptotic properties of PDEs in compact spaces

TL;DR

This paper investigates the asymptotic behavior of PDE solutions in compact spaces by developing new function spaces, proving key theorems, and demonstrating the theory's applicability through an example.

Contribution

It introduces novel function spaces and analytical tools for studying PDEs in compactified domains, advancing the understanding of their asymptotic properties.

Findings

Established a version of the Ascoli-Arzelà Theorem for new function spaces

Proved fixed point index results for existence and multiplicity of solutions

Demonstrated the applicability with a concrete example

Abstract

In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of Ascoli-Arzel\`a Theorem, develop the fixed point index results necessary to prove existence and multiplicity of solutions in these spaces and also illustrate the applicability of the theory with an example.