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

Benhamoud, Tayeb ; Zaouche, Elmehdi ; Bousselsal, Mahmoud

Mathematica Bohemica, Tome 149 (2024) no. 4, pp. 533-548

Abstract (VO)

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 $\mathbb {R}^{n}$ $(n\geq 1)$, $T>0$ 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.

Détail

DOI : 10.21136/MB.2024.0065-23

Classification : 35D30, 35K55, 35Q92
Keywords: nonlocal nonlinear parabolic problem; Schauder fixed point theorem; weak solution; existence; uniqueness