Affiliation:
1. Department of Mathematics and Statistics, Masaryk University , Building 08, Kotlářská 2, Brno 611 37 , Czech Republic
2. Mathematics Department, University of Oviedo , c/Federico García Lorca 18, 33007 , Oviedo Spain
Abstract
Abstract
We study the homogeneous Dirichlet problem for the parabolic equations
u
t
−
div
(
A
(
z
,
∣
∇
u
∣
)
∇
u
)
=
F
(
z
,
u
,
∇
u
)
,
z
=
(
x
,
t
)
∈
Ω
×
(
0
,
T
)
,
{u}_{t}-{\rm{div}}\left({\mathcal{A}}\left(z,| \nabla u| )\nabla u)=F\left(z,u,\nabla u),\hspace{1.0em}z=\left(x,t)\in \Omega \times \left(0,T),
with the double phase flux
A
(
z
,
∣
∇
u
∣
)
∇
u
=
(
∣
∇
u
∣
p
(
z
)
−
2
+
a
(
z
)
∣
∇
u
∣
q
(
z
)
−
2
)
∇
u
{\mathcal{A}}\left(z,| \nabla u| )\nabla u=(| \nabla u{| }^{p\left(z)-2}+a\left(z)| \nabla u{| }^{q\left(z)-2})\nabla u
and the nonlinear source
F
F
. The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions
a
a
,
p
p
, and
q
q
are Lipschitz-continuous,
a
(
z
)
a\left(z)
is nonnegative, and may vanish on a set of nonzero measure. The exponents
p
p
, and
q
q
satisfy the balance conditions
2
N
N
+
2
<
p
−
≤
p
(
z
)
≤
q
(
z
)
<
p
(
z
)
+
r
∗
2
\frac{2N}{N+2}\lt {p}^{-}\le p\left(z)\le q\left(z)\lt p\left(z)+\frac{{r}^{\ast }}{2}
with
r
∗
=
r
∗
(
p
−
,
N
)
{r}^{\ast }={r}^{\ast }\left({p}^{-},N)
,
p
−
=
min
Q
¯
T
p
(
z
)
{p}^{-}={\min }_{{\overline{Q}}_{T}}\hspace{0.33em}p\left(z)
. It is shown that under suitable conditions on the growth of
F
(
z
,
u
,
∇
u
)
F\left(z,u,\nabla u)
with respect to the second and third arguments, the problem has a solution
u
u
with the following properties:
u
t
∈
L
2
(
Q
T
)
,
∣
∇
u
∣
p
(
z
)
+
δ
∈
L
1
(
Q
T
)
for every
0
≤
δ
<
r
∗
,
∣
∇
u
∣
s
(
z
)
,
a
(
z
)
∣
∇
u
∣
q
(
z
)
∈
L
∞
(
0
,
T
;
L
1
(
Ω
)
)
with
s
(
z
)
=
max
{
2
,
p
(
z
)
}
.
\begin{array}{l}{u}_{t}\in {L}^{2}\left({Q}_{T}),\hspace{1.0em}| \nabla u{| }^{p\left(z)+\delta }\in {L}^{1}\left({Q}_{T})\hspace{1.0em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}0\le \delta \lt {r}^{\ast },\\ | \nabla u{| }^{s\left(z)},\hspace{0.33em}a\left(z)| \nabla u{| }^{q\left(z)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega ))\hspace{1em}{\rm{with}}\hspace{0.33em}s\left(z)=\max \left\{2,p\left(z)\right\}.\end{array}
Uniqueness is proven under stronger assumptions on the source
F
F
. The same results are established for the equations with the regularized flux
A
(
z
,
(
ε
2
+
∣
∇
u
∣
2
)
1
/
2
)
∇
u
{\mathcal{A}}(z,{({\varepsilon }^{2}+| \nabla u{| }^{2})}^{1\text{/}2})\nabla u
,
ε
>
0
\varepsilon \gt 0
.