Mathematica Bohemica (Dec 2024)
Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems
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 $\mathbb{R}^n$ $(n\geq1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n\times n$ matrix of variable coefficients depending on $u$ and $M \mathbb{R}\rightarrow\mathbb{R}$, $\phi \Omega\rightarrow\mathbb{R}$, $g \Omega\times(0,T)\times\mathbb{R}\rightarrow\mathbb{R}$ 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.
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