Well-posedness of the Cauchy problem for the fractional power dissipative equations

Abstract

This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation $u_t+(-\triangle)^\alpha u= F(u)$ for initial data in the Lebesgue space $L^r(\mr^n)$ with $\ds r\ge r_d\triangleq{nb}/({2\alpha-d})$ or the homogeneous Besov space $\ds\dot{B}^{-\sigma}_{p,\infty}(\mr^n)$ with $\ds\sigma=(2\alpha-d)/b-n/p$ and $1\le p\le \infty$, where $\alpha>0$, $F(u)=f(u)$ or $Q(D)f(u)$ with $Q(D)$ being a homogeneous pseudo-differential operator of order $d\in[0,2\alpha)$ and $f(u)$ is a function of $u$ which behaves like $|u|^bu$ with $b>0$.