|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'500'096
Articles rated: 2609
18 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
A qualitative study of (p,q) Singular parabolic equations: local existence, Sobolev regularity and asymptotic behaviour | | Jacques Giacomoni
; Deepak Kumar
; K. Sreenadh
; | | Date: |
26 Aug 2020 | | Abstract: | The purpose of the article is to study the existence, regularity,
stabilization and blow up results of weak solution to the following parabolic
$(p,q)$-singular equation:
egin{equation*}
(P_t); left{egin{array}{rllll}
u_t-Delta_{p}u -Delta_{q}u & = vth ; u^{-de}+ f(x,u), ; u>0 ext{ in }
Om imes (0,T), \ u&=0 quad ext{ on } paOm imes (0,T),
u(x,0)&= u_0(x) ; ext{ in }Om,
end{array}
ight.
end{equation*}
where $Om$ is a bounded domain in $mathbb{R}^N$ with $C^2$ boundary
$paOm$, $1<q<p< infty$, $0<de, T>0$, $Nge 2$ and $vth>0$ is a parameter.
Moreover, we assume that $f:Om imes [0,infty) o mb R$ is a bounded below
Carathéodory function, locally Lipschitz with respect to the second variable
uniformly in $xinOm$ and $u_0in L^infty(Om)cap W^{1,p}_0(Om)$. 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 $(P_t)$ for $de<2+1/(p-1)$. For this, we first study the stationary
problems corresponding to $(P_t)$ by using the method of sub and super
solutions and subsequently employing implicit Euler method, we obtain the
existence of a solution to $(P_t)$. Furthermore, in this case, we prove the
stabilization result, that is, the solution $u(t)$ of $(P_t)$ converges to
$u_infty$, the unique solution to the stationary problem, in $L^infty(Om)$
as $t
ainfty$. 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 $(P_t)$ for small parameter
$vartheta>0$ in the case $deleq 1$ and for all $vth>0$ in the case $de>1$. | | Source: | arXiv, 2008.11627 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER