Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems

Benhamoud, Tayeb, Zaouche, Elmehdi, and Bousselsal, Mahmoud. "Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems." Mathematica Bohemica 149.4 (2024): 533-548. <http://eudml.org/doc/299643>.

@article{Benhamoud2024,
abstract = {This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_t-M(\int _\{\Omega \}\phi u \{\rm d\}x)\{\rm div\} (A(x,t,u)\nabla u)=g(x,t,u)$ in $\Omega \times (0,T)$, where $\Omega $ is a bounded domain of $\mathbb \{R\}^\{n\}$$(n\ge 1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n\times n$ matrix of variable coefficients depending on $u$ and $M\colon \mathbb \{R\}\rightarrow \mathbb \{R\}$, $\phi \colon \Omega \rightarrow \mathbb \{R\}$, $g\colon \Omega \times (0,T)\times \mathbb \{R\}\rightarrow \mathbb \{R\}$ are given functions. We consider two different assumptions on $g$. The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if $A(x,t,u)=a(x,t)$ depends only on the variable $(x,t)$, we investigate two uniqueness theorems and give a continuity result depending on the initial data.},
author = {Benhamoud, Tayeb, Zaouche, Elmehdi, Bousselsal, Mahmoud},
journal = {Mathematica Bohemica},
keywords = {nonlocal nonlinear parabolic problem; Schauder fixed point theorem; weak solution; existence; uniqueness},
language = {eng},
number = {4},
pages = {533-548},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems},
url = {http://eudml.org/doc/299643},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Benhamoud, Tayeb
AU - Zaouche, Elmehdi
AU - Bousselsal, Mahmoud
TI - Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 4
SP - 533
EP - 548
AB - This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_t-M(\int _{\Omega }\phi u {\rm d}x){\rm div} (A(x,t,u)\nabla u)=g(x,t,u)$ in $\Omega \times (0,T)$, where $\Omega $ is a bounded domain of $\mathbb {R}^{n}$$(n\ge 1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n\times n$ matrix of variable coefficients depending on $u$ and $M\colon \mathbb {R}\rightarrow \mathbb {R}$, $\phi \colon \Omega \rightarrow \mathbb {R}$, $g\colon \Omega \times (0,T)\times \mathbb {R}\rightarrow \mathbb {R}$ are given functions. We consider two different assumptions on $g$. The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if $A(x,t,u)=a(x,t)$ depends only on the variable $(x,t)$, we investigate two uniqueness theorems and give a continuity result depending on the initial data.
LA - eng
KW - nonlocal nonlinear parabolic problem; Schauder fixed point theorem; weak solution; existence; uniqueness
UR - http://eudml.org/doc/299643
ER -