On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung; Le Thi Phuong Ngoc; Nguyen Huu Nhan; Nguyen Thanh Long

Mathematica Bohemica (2024)

  • Volume: 149, Issue: 2, page 261-285
  • ISSN: 0862-7959

Abstract

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We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values u ( η 1 , t ) , ⋯ , u ( η q , t ) with 0 ≤ η 1 < η 2 < ⋯ < η 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 ( P q ) of (P) in which the nonlinear term contains the sum S q [ u 2 ] ( t ) = q - 1 ∑ i = 1 q u 2 ( ( i - 1 ) q , t ) . Under suitable conditions, we prove that the solution of ( P q ) converges to the solution of the corresponding problem ( P ∞ ) as q → ∞ (in a certain sense), here ( P ∞ ) is defined by ( P q ) in which S q [ u 2 ] ( t ) is replaced by ∫ 0 1 u 2 ( y , t ) 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.

How to cite

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Dzung, Nguyen Vu, et al. "On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values." Mathematica Bohemica 149.2 (2024): 261-285. <http://eudml.org/doc/299590>.

@article{Dzung2024,
abstract = {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 $0\le \eta _\{1\}<\eta _\{2\}<\cdots <\eta _\{q\}<1.$ 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.},
author = {Dzung, Nguyen Vu, Ngoc, Le Thi Phuong, Nhan, Nguyen Huu, Long, Nguyen Thanh},
journal = {Mathematica Bohemica},
keywords = {Kirchhoff-Carrier equation; Robin-Dirichlet problem; nonlocal term; Faedo-Galerkin method; linearization method},
language = {eng},
number = {2},
pages = {261-285},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values},
url = {http://eudml.org/doc/299590},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Dzung, Nguyen Vu
AU - Ngoc, Le Thi Phuong
AU - Nhan, Nguyen Huu
AU - Long, Nguyen Thanh
TI - On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 2
SP - 261
EP - 285
AB - 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 $0\le \eta _{1}<\eta _{2}<\cdots <\eta _{q}<1.$ 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.
LA - eng
KW - Kirchhoff-Carrier equation; Robin-Dirichlet problem; nonlocal term; Faedo-Galerkin method; linearization method
UR - http://eudml.org/doc/299590
ER -

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