Geodesic
On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values
Mathematica Bohemica, Tome 149 (2024) no. 2, pp. 261-285
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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.
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.
DOI : 10.21136/MB.2023.0153-21
Classification : 35A01, 35A02, 35B45, 35L05, 35M11
Keywords: Kirchhoff-Carrier equation; Robin-Dirichlet problem; nonlocal term; Faedo-Galerkin method; linearization method 
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Dzung, Nguyen Vu; Ngoc, Le Thi Phuong; Nhan, Nguyen Huu; Long, Nguyen Thanh. On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values. Mathematica Bohemica, Tome 149 (2024) no. 2, pp. 261-285. doi: 10.21136/MB.2023.0153-21

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