Critical exponent for evolution equations in modulation spaces

In this paper, we propose a method to find the critical exponent for certain evolution equations in modulation spaces. We define an index σ(s,q)σ(s,q), and use it to determine the critical exponent of the fractional heat equation as an example. We prove that when σ(s,q)σ(s,q) is greater than the critical exponent, this equation is locally well posed in the space C(0,T;Mp,qs); and when σ(s,q)σ(s,q) is less than the critical exponent, this equation is ill-posed in the space C(0,T;M2,qs). Our method may further be applied to some other evolution equations.