Critical exponent for evolution equation in Modulation space

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.