Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems - Tayeb Benhamoud; Elmehdi Zaouche; Mahmoud Bousselsal (Mathematica Bohemica)
This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_t-M(\int_{Ω}φ u {\rm d}x){\rm div} (A(x,t,u)\nabla u)=g(x,t,u)$ in $Ω×(0,T)$, where $Ω$ is a bounded domain of $\mathbb{R}^n$ $(n≥1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n× n$ matrix of variable coefficients depending on $u$ and $M \mathbb{R}→\mathbb{R}$, $φ Ω→\mathbb{R}$, $g Ω×(0,T)×\mathbb{R}→\mathbb{R}$ 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.