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

Miao, Changxing; Yuan, Baoquan; Zhang, Bo

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

Authors

Miao, Changxing

Yuan, Baoquan

Zhang, Bo

Description

This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation $u_t+(-\triangle)^αu= F(u)$ for initial data in the Lebesgue space $L^r(\mr^n)$ with $\ds r\ge r_d\triangleq{nb}/({2α-d})$ or the homogeneous Besov space $\ds\dot{B}^{-σ}_{p,\infty}(\mr^n)$ with $\dsσ=(2α-d)/b-n/p$ and $1\le p\le \infty$, where $α>0$, $F(u)=f(u)$ or $Q(D)f(u)$ with $Q(D)$ being a homogeneous pseudo-differential operator of order $d\in[0,2α)$ and $f(u)$ is a function of $u$ which behaves like $|u|^bu$ with $b>0$.
30pages