Electron. J. Differential Equations, Vol. 2016 (2016), No. 338, pp. 1-12.

Critical exponent for the heat equation in $\alpha$-modulation spaces

Wang Zheng, Huang Qiang, Bu Rui

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.

Submitted January 27, 2016. Published December 30, 2016.
Math Subject Classifications: 35A01, 35A02, 42B37.
Key Words: alpha-modulation space; heat equation; critical exponent.

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Wang Zheng
School of Mathematical Sciences
Zhejiang University
Hangzhou 310027, China
email: wangzheng10.17@163.com
Huang Qiang
School of Mathematical Sciences
Zhejiang University
Hangzhou 310027, China
email: huangqiang0704@163.com
Bu Rui
Department of Mathematics
Qingdao University of Science and Technology
Qingdao 266061, China
email: burui0@163.com

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