Affiliation:
1. LMAP (UMR E2S-UPPA CNRS 5142) , Université de Pau et des Pays de l’Adour , Bat. IPRA, Avenue de l’Université F-64013 Pau , France
2. Department of Mathematics , Indian Institute of Technology Delhi , Hauz Khaz , New Delhi - 110016 , India
Abstract
Abstract
The purpose of the article is to study the existence, regularity, stabilization and blow-up results of weak solution to the following parabolic
(
p
,
q
)
{(p,q)}
-singular equation:
($\mathrm{P}_{t}$)
{
u
t
-
Δ
p
u
-
Δ
q
u
=
ϑ
u
-
δ
+
f
(
x
,
u
)
,
u
>
0
in
Ω
×
(
0
,
T
)
,
u
=
0
on
∂
Ω
×
(
0
,
T
)
,
u
(
x
,
0
)
=
u
0
(
x
)
in
Ω
,
\displaystyle{}\left\{\begin{aligned} \displaystyle{}u_{t}-\Delta_{p}u-\Delta_%
{q}u&\displaystyle=\vartheta u^{-\delta}+f(x,u),\quad u>0&&\displaystyle%
\phantom{}\text{in }\Omega\times(0,T),\\
\displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial%
\Omega\times(0,T),\\
\displaystyle u(x,0)&\displaystyle=u_{0}(x)&&\displaystyle\phantom{}\text{in }%
\Omega,\end{aligned}\right.
where Ω is a bounded domain in
ℝ
N
{\mathbb{R}^{N}}
with
C
2
{C^{2}}
boundary
∂
Ω
{\partial\Omega}
,
1
<
q
<
p
<
∞
{1<q<p<\infty}
,
0
<
δ
,
T
>
0
{0<\delta,T>0}
,
N
≥
2
{N\geq 2}
and
ϑ
>
0
{\vartheta>0}
is a parameter. Moreover, we assume that
f
:
Ω
×
[
0
,
∞
)
→
ℝ
{f:\Omega\times[0,\infty)\to\mathbb{R}}
is a bounded below Carathéodory function, locally Lipschitz with respect to the second variable uniformly in
x
∈
Ω
{x\in\Omega}
and
u
0
∈
L
∞
(
Ω
)
∩
W
0
1
,
p
(
Ω
)
{u_{0}\in L^{\infty}(\Omega)\cap W^{1,p}_{0}(\Omega)}
. We distinguish the cases as q-subhomogeneous and q-superhomogeneous depending on the growth of f (hereafter we will drop the term q). In the subhomogeneous case, we prove the existence and uniqueness of the weak solution to problem (
P
t
{\mathrm{P}_{t}}
) for
δ
<
2
+
1
p
-
1
{\delta<2+\frac{1}{p-1}}
. For this, we first study the stationary problems corresponding to (
P
t
{\mathrm{P}_{t}}
) by using the method of sub- and supersolutions and subsequently employing implicit Euler method, we obtain the existence of a solution to (
P
t
{\mathrm{P}_{t}}
). Furthermore, in this case, we prove the stabilization result, that is, the solution
u
(
t
)
{u(t)}
of (
P
t
{\mathrm{P}_{t}}
) converges to
u
∞
{u_{\infty}}
, the unique solution to the stationary problem, in
L
∞
(
Ω
)
{L^{\infty}(\Omega)}
as
t
→
∞
{t\rightarrow\infty}
. For the superhomogeneous case, we prove the local existence theorem by taking help of nonlinear semigroup theory. Subsequently, we prove finite time blow-up of solution to problem (
P
t
{\mathrm{P}_{t}}
) for small parameter
ϑ
>
0
{\vartheta>0}
in the case
δ
≤
1
{\delta\leq 1}
and for all
ϑ
>
0
{\vartheta>0}
in the case
δ
>
1
{\delta>1}
. Moreover, we prove higher Sobolev integrability of the solution to purely singular problem corresponding to the steady state of (
P
t
{\mathrm{P}_{t}}
), which is of independent interest. As a consequence of this, we improve the Sobolev regularity of solution to (
P
t
{\mathrm{P}_{t}}
) for the case
δ
<
2
+
1
p
-
1
{\delta<2+\frac{1}{p-1}}
.
Subject
General Mathematics,Statistical and Nonlinear Physics
Cited by
1 articles.
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