Abstract
AbstractWe study the homogeneous Dirichlet problem for the equation $$\begin{aligned} u_t-{\text {div}}\left( \mathcal {F}(z,\nabla u)\nabla u\right) =f, \quad z=(x,t)\in Q_T=\Omega \times (0,T), \end{aligned}$$
u
t
-
div
F
(
z
,
∇
u
)
∇
u
=
f
,
z
=
(
x
,
t
)
∈
Q
T
=
Ω
×
(
0
,
T
)
,
where $$\Omega \subset \mathbb {R}^N$$
Ω
⊂
R
N
, is a bounded domain with $$\partial \Omega \in C^2$$
∂
Ω
∈
C
2
, and $$\mathcal {F}(z,\xi )=a(z)\vert \xi \vert ^{p(z)-2}+b(z)\vert \xi \vert ^{q(z)-2}$$
F
(
z
,
ξ
)
=
a
(
z
)
|
ξ
|
p
(
z
)
-
2
+
b
(
z
)
|
ξ
|
q
(
z
)
-
2
. The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions. It is assumed that $$\frac{2N}{N+2}<p(z),\ q(z)$$
2
N
N
+
2
<
p
(
z
)
,
q
(
z
)
, and that the modulating coefficients and growth exponents satisfy the balance conditions $$\begin{aligned} a(z)+b(z)\ge \alpha >0,\quad \vert p(z)-q(z)\vert <\frac{2}{N+2}\hbox { in }\overline{Q}_T \end{aligned}$$
a
(
z
)
+
b
(
z
)
≥
α
>
0
,
|
p
(
z
)
-
q
(
z
)
|
<
2
N
+
2
in
Q
¯
T
with $$\alpha =const$$
α
=
c
o
n
s
t
. We find conditions on the source f and the initial data $$u(\cdot ,0)$$
u
(
·
,
0
)
that guarantee the existence of a unique strong solution u with $$u_t\in L^2(Q_T)$$
u
t
∈
L
2
(
Q
T
)
and $$a\vert \nabla u\vert ^{p}+b\vert \nabla u\vert ^q\in L^\infty (0,T;L^1(\Omega ))$$
a
|
∇
u
|
p
+
b
|
∇
u
|
q
∈
L
∞
(
0
,
T
;
L
1
(
Ω
)
)
. The solution possesses the property of global higher integrability of the gradient, $$\begin{aligned} \vert \nabla u\vert ^{\min \{p(z),q(z)\}+r}\in L^1(Q_T)\quad \text {with any }r\in \left( 0,\frac{4}{N+2}\right) , \end{aligned}$$
|
∇
u
|
min
{
p
(
z
)
,
q
(
z
)
}
+
r
∈
L
1
(
Q
T
)
with any
r
∈
0
,
4
N
+
2
,
which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The global second-order differentiability of the strong solution is proven: $$\begin{aligned} D_i\left( \sqrt{\mathcal {F}(z,\nabla u)}D_j u\right) \in L^{2}(Q_T),\quad i=1,2,\ldots ,N. \end{aligned}$$
D
i
F
(
z
,
∇
u
)
D
j
u
∈
L
2
(
Q
T
)
,
i
=
1
,
2
,
…
,
N
.
The same results are obtained for the equation with the regularized flux $$\mathcal {F}(z,\sqrt{\epsilon ^2+(\xi ,\xi )})\xi $$
F
(
z
,
ϵ
2
+
(
ξ
,
ξ
)
)
ξ
.
Funder
Ministerio de Ciencia, Innovación y Universidades
Czech Science Foundation
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis
Cited by
1 articles.
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