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

@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 -