| Title:
|
Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems (English) |
| Author:
|
Benhamoud, Tayeb |
| Author:
|
Zaouche, Elmehdi |
| Author:
|
Bousselsal, Mahmoud |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
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149 |
| Issue:
|
4 |
| Year:
|
2024 |
| Pages:
|
533-548 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
nonlocal nonlinear parabolic problem |
| Keyword:
|
Schauder fixed point theorem |
| Keyword:
|
weak solution |
| Keyword:
|
existence |
| Keyword:
|
uniqueness |
| MSC:
|
35D30 |
| MSC:
|
35K55 |
| MSC:
|
35Q92 |
| DOI:
|
10.21136/MB.2024.0065-23 |
| . |
| Date available:
|
2024-12-13T19:05:58Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152678 |
| . |
| Reference:
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