Critical exponent for the heat equation in alpha-modulation spaces

Abstract

In this article, we propose a method for finding the critical exponent for heat equations in $\alpha$-modulation space $M_{p,q}^{s,\alpha}$. We define an index $\sigma (s,p,q)$, and use it to determine the critical exponent of the heat equation. Then we use this exponent to describe well and ill-posedness of the heat equation in $L^{\infty}([0,T];M_{p,q}^{s,\alpha})$. In some special case our conclusions are sharp. Furthermore, our method may be applied to other evolution equations.

Keywords

• alpha-modulation space
• heat equation
• critical exponent