Blow-up and global solutions to L^p norm preserving non-local flows

TL;DR

This paper investigates the behavior of specific non-local heat flows that preserve the L^p norm, demonstrating conditions for global existence and providing examples of blow-up in the supremum norm despite norm preservation.

Contribution

It introduces and analyzes two types of L^p norm preserving non-local heat flows, establishing their global solutions and illustrating potential blow-up scenarios.

Findings

Both flows have global solutions under studied conditions.

One flow can blow up in L^{ } norm despite preserving L^p norm.

The paper provides explicit examples of blow-up behavior.

Abstract

In this paper, we study global existence and blow up properties to $L^p$ norm preserving non-local heat flows. We first study two kinds of $L^p$ norm preserving non-local flows and prove that these flows have the global solutions. Finally, we give a example to show that one kind of this heat flow may blow up in $L^{\infty}$ norm though its $L^p$ norm is preserved.