Critical exponent for evolution equation in Modulation space

TL;DR

This paper introduces a method to determine the critical exponent for evolution equations in modulation spaces, exemplified on the fractional heat equation, establishing conditions for well-posedness and ill-posedness.

Contribution

It presents a novel approach to identify critical exponents in modulation spaces, applicable to various evolution equations beyond the fractional heat equation.

Findings

Well-posedness when $\sigma(s,q)$ exceeds the critical exponent.

Ill-posedness when $\sigma(s,q)$ is below the critical exponent.

Method applicable to other evolution equations.

Abstract

In this paper, we propose a method to find the critical exponent for certain evolution equations in modulation spaces. We define an index $\sigma (s,q)$, and use it to determine the critical exponent of the fractional heat equation as an example. We prove that when $\sigma (s,q)$ is greater than the critical exponent, this equation is locally well posed in the space $C(0,T;M_{p,q}^{s})$; and when $\sigma (s,q)$ is less than the critical exponent, this equation is ill-posed in the space $C(0,T;M_{2,q}^{s})$. Our method may further be applied to some other evolution equations.