Affiliation:
1. College of Mathematics, Chengdu Normal University , Chengdu 611130 , P. R. China
Abstract
Abstract
This paper is concerned with the n-dimensional strongly coupled parabolic systems with triangular form in the cylinder
Ω
×
(
0
,
T
]
\Omega \times (0,T]
. We investigate
L
2
{L}^{2}
and Hölder regularity of the derivatives of weak solutions
(
u
1
,
u
2
)
\left({u}_{1},{u}_{2})
for the systems in the following two cases: one is that the boundedness of
u
1
{u}_{1}
and
u
2
{u}_{2}
has not been shown in existence result of solutions; the other is that the boundedness of
u
1
{u}_{1}
or
u
2
{u}_{2}
has been shown in existence result of solutions. By using difference ratios and Steklov averages methods and various estimates, we prove that if
(
u
1
,
u
2
)
\left({u}_{1},{u}_{2})
is a weak solution of the system, then for any
Ω
′
⊂
⊂
Ω
\Omega ^{\prime} \subset \hspace{-0.3em}\subset \hspace{0.33em}\Omega
and
t
′
∈
(
0
,
T
)
t^{\prime} \in \left(0,T)
,
u
1
,
u
2
{u}_{1},{u}_{2}
belong to
C
α
′
,
α
′
/
2
(
Ω
¯
′
×
[
t
′
,
T
]
)
{C}^{\alpha ^{\prime} ,\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T])
and
W
2
2
,
1
(
Ω
′
×
(
t
′
,
T
]
)
{W}_{2}^{2,1}\left(\Omega ^{\prime} \times (t^{\prime} ,T])
under certain conditions, and
u
1
,
u
2
{u}_{1},{u}_{2}
belong to
C
2
+
α
′
,
1
+
α
′
/
2
(
Ω
¯
′
×
[
t
′
,
T
]
)
{C}^{2+\alpha ^{\prime} ,1+\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T])
under stronger assumptions. Applications of these results are given to two ecological models with cross-diffusion.
Subject
General Mathematics,Statistical and Nonlinear Physics