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)
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 $({\rm P}_q)$ of (P) in which the nonlinear term contains the sum $S_q[u^2](t)=q^{-1}\sum_{i=1}^qu^2(\frc{(i-1)}q,t)$. Under suitable conditions, we prove that the solution of $({\rm P}_q)$ converges to the solution of the corresponding problem $({\rm P}_{∞})$ as $q→∞$ (in a certain sense), here $({\rm P}_{∞})$ is defined by $({\rm P}_q)$ in which $S_q[u^2](t)$ is replaced by $ \int_0^1u^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.