| Title:
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On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values (English) |
| Author:
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Dzung, Nguyen Vu |
| Author:
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Ngoc, Le Thi Phuong |
| Author:
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Nhan, Nguyen Huu |
| Author:
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Long, Nguyen Thanh |
| Language:
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English |
| Journal:
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Mathematica Bohemica |
| ISSN:
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0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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149 |
| Issue:
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2 |
| Year:
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2024 |
| Pages:
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261-285 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
|
We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u(\eta _{1},t),\cdots ,u(\eta _{q},t)$ with $0\leq \eta _{1}<\eta _{2}<\cdots <\eta _{q}<1.$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $({\rm P}_{q})$ of (P) in which the nonlinear term contains the sum $S_{q}[u^{2}](t)=q^{-1}\sum _{i=1}^{q}u^{2}(\frac{(i-1)}{q},t)$. Under suitable conditions, we prove that the solution of $({\rm P}_{q})$ converges to the solution of the corresponding problem $({\rm P}_{\infty })$ as $q\rightarrow \infty $ (in a certain sense), here $({\rm P}_{\infty })$ is defined by $({\rm P}_{q})$ in which $S_{q}[u^{2}](t)$ is replaced by $ \int _{0}^{1}u^{2}( y,t) {\rm d}y.$ The proof is done by using the compactness lemma of Aubin-Lions and the method of continuity with a priori estimates. We end the paper with remarks related to similar problems. (English) |
| Keyword:
|
Kirchhoff-Carrier equation |
| Keyword:
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Robin-Dirichlet problem |
| Keyword:
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nonlocal term |
| Keyword:
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Faedo-Galerkin method |
| Keyword:
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linearization method |
| MSC:
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35A01 |
| MSC:
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35A02 |
| MSC:
|
35B45 |
| MSC:
|
35L05 |
| MSC:
|
35M11 |
| DOI:
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10.21136/MB.2023.0153-21 |
| . |
| Date available:
|
2024-07-10T15:06:35Z |
| Last updated:
|
2024-07-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152472 |
| . |
| Reference:
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