Affiliation:
1. Institute of Applied System Analysis, Jiangsu University , Zhenjiang , Jiangsu, 212013 , P. R. China
Abstract
Abstract
In the present article, we are concerned with the following problem:
v
t
=
Δ
v
+
∣
x
∣
β
e
v
,
x
∈
R
N
,
t
>
0
,
v
(
x
,
0
)
=
v
0
(
x
)
,
x
∈
R
N
,
\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}{v}_{t}=\Delta v+| x{| }^{\beta }{e}^{v},\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\hspace{0.33em}t\gt 0,\\ v\left(x,0)={v}_{0}\left(x),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right.
where
N
≥
3
N\ge 3
,
0
<
β
<
2
0\lt \beta \lt 2
, and
v
0
{v}_{0}
is a continuous function in
R
N
{{\mathbb{R}}}^{N}
. We prove the existence and asymptotic behavior of forward self-similar solutions in the case where
v
0
{v}_{0}
decays at the rate
−
(
2
+
β
)
log
∣
x
∣
-\left(2+\beta )\log | x|
as
∣
x
∣
→
∞
| x| \to \infty
. Particularly, we obtain the optimal decay bound for initial value
v
0
{v}_{0}
.
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