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
+(−△
)
α
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≤p≤∞
, where
α>0
,
F(u)=f(u)
or
Q(D)f(u)
with
Q(D)
being a homogeneous pseudo-differential operator of order
d∈[0,2α)
and
f(u)
is a function of
u
which behaves like
|u
|
b
u
with
b>0
.