Global existence for the p-Sobolev flow

TL;DR

This paper proves the global existence and analyzes qualitative properties of the p-Sobolev flow, a nonlinear parabolic equation related to Sobolev inequalities, generalizing the classical Yamabe flow.

Contribution

It establishes the global existence of solutions for the p-Sobolev flow and explores its qualitative behavior, extending known results to a broader class of nonlinear flows.

Findings

Proved global existence of p-Sobolev flow solutions.

Analyzed qualitative properties of the flow.

Connected the flow to classical Yamabe flow when p=2.

Abstract

In this paper, we study a doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow. In the special case p=2 our theory includes the classical Yamabe flow on a bounded domain in Euclidean space. Our main aim is to prove the global existence of the p-Sobolev flow together with its qualitative properties.