Abstract
This paper considers the Cauchy problem for fast diffusion equation with nonlocal source $u_{t}=\Delta u^{m}+ (\int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx )^{\frac{p-1}{q}}u^{r+1}$
u
t
=
Δ
u
m
+
(
∫
R
n
u
q
(
x
,
t
)
d
x
)
p
−
1
q
u
r
+
1
, which was raised in [Galaktionov et al. in Nonlinear Anal. 34:1005–1027, 1998]. We give the critical Fujita exponent $p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}$
p
c
=
m
+
2
q
−
n
(
1
−
m
)
−
n
q
r
n
(
q
−
1
)
, namely, any solution of the problem blows up in finite time whenever $1< p\le p_{c}$
1
<
p
≤
p
c
, and there are both global and non-global solutions if $p>p_{c}$
p
>
p
c
.
We consider the dead-core problem for the fast diffusion equation with spatially dependent coefficient and obtain precise estimates on the single-point final dead-core profile. The proofs rely on maximum principle and require much delicate computation.
Extinction properties of solutions for a fast diffusion equation with nonlocal source
