Affiliation:
1. Department of Mathematical Sciences, Indian Institute of Technology (IIT-BHU) , Varanasi , 221005, Uttar Pradesh , India
2. Department of Mathematics, University of Oviedo, c/Federico García Lorca 18 , Oviedo , 33007, Asturias , Spain
Abstract
Abstract
We consider the homogeneous Dirichlet problem for the parabolic equation
u
t
−
div
(
∣
∇
u
∣
p
(
x
,
t
)
−
2
∇
u
)
=
f
(
x
,
t
)
+
F
(
x
,
t
,
u
,
∇
u
)
{u}_{t}-{\rm{div}}({| \nabla u| }^{p\left(x,t)-2}\nabla u)=f\left(x,t)+F\left(x,t,u,\nabla u)
in the cylinder
Q
T
≔
Ω
×
(
0
,
T
)
{Q}_{T}:= \Omega \times \left(0,T)
, where
Ω
⊂
R
N
\Omega \subset {{\mathbb{R}}}^{N}
,
N
≥
2
N\ge 2
, is a
C
2
{C}^{2}
-smooth or convex bounded domain. It is assumed that
p
∈
C
0
,
1
(
Q
¯
T
)
p\in {C}^{0,1}\left({\overline{Q}}_{T})
is a given function and that the nonlinear source
F
(
x
,
t
,
s
,
ξ
)
F\left(x,t,s,\xi )
has a proper power growth with respect to
s
s
and
ξ
\xi
. It is shown that if
p
(
x
,
t
)
>
2
(
N
+
1
)
N
+
2
p\left(x,t)\gt \frac{2\left(N+1)}{N+2}
,
f
∈
L
2
(
Q
T
)
f\in {L}^{2}\left({Q}_{T})
,
∣
∇
u
0
∣
p
(
x
,
0
)
∈
L
1
(
Ω
)
{| \nabla {u}_{0}| }^{p\left(x,0)}\in {L}^{1}\left(\Omega )
, then the problem has a solution
u
∈
C
0
(
[
0
,
T
]
;
L
2
(
Ω
)
)
u\in {C}^{0}\left(\left[0,T];\hspace{0.33em}{L}^{2}\left(\Omega ))
with
∣
∇
u
∣
p
(
x
,
t
)
∈
L
∞
(
0
,
T
;
L
1
(
Ω
)
)
{| \nabla u| }^{p\left(x,t)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega ))
,
u
t
∈
L
2
(
Q
T
)
{u}_{t}\in {L}^{2}\left({Q}_{T})
, obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties:
∣
∇
u
∣
2
(
p
(
x
,
t
)
−
1
)
+
r
∈
L
1
(
Q
T
)
,
for any
0
<
r
<
4
N
+
2
,
∣
∇
u
∣
p
(
x
,
t
)
−
2
∇
u
∈
L
2
(
0
,
T
;
W
1
,
2
(
Ω
)
)
N
.
{| \nabla u| }^{2\left(p\left(x,t)-1)+r}\in {L}^{1}\left({Q}_{T}),\hspace{1.0em}\hspace{0.1em}\text{for any\hspace{0.5em}}0\lt r\lt \frac{4}{N+2}\text{}\hspace{0.1em},\hspace{1.0em}{| \nabla u| }^{p\left(x,t)-2}\nabla u\in {L}^{2}{\left(0,T;{W}^{1,2}\left(\Omega ))}^{N}.
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献