Abstract
This paper deals with the abstract evolution equations in
L
s
{L}^{s}
-spaces with critical temporal weights. First, embedding and interpolation properties of the critical
L
s
{L}^{s}
-spaces with different exponents
s
s
are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term
f
f
and its average
Φ
f
\Phi f
both lie in an
L
1
/
s
s
{L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s}
-space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator
F
(
t
,
u
)
F\left(t,u)
has a growth number
ρ
≥
s
+
1
\rho \ge s+1
, and its asymptotic behavior acts like
α
(
t
)
/
t
\alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t
as
t
→
0
t\to 0
for some bounded function
α
(
t
)
\alpha \left(t)
like
(
−
log
t
)
−
p
{\left(-\log t)}^{-p}
with
2
≤
p
<
∞
2\le p\lt \infty
.
Long-time asymptotics for an integrable evolution equation with a 3×3 Lax pair
