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
Received October 4, 2021. Published online April 25, 2023.
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 $0\leq\eta_1<\eta_2<\cdots<\eta_q<1.$ 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 $S_q[u^2](t)=q^{-1}\sum_{i=1}^qu^2(\frc{(i-1)}q,t)$. Under suitable conditions, we prove that the solution of $({\rm P}_q)$ converges to the solution of the corresponding problem $({\rm P}_{\infty})$ 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_0^1u^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.
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Affiliations: Nguyen Vu Dzung, Department of Mathematics and Computer Science, University of Science, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam, Cao Thang Technical College, 65 Huynh Thuc Khang Str., Ben Nghe Ward, Dist. 1, Ho Chi Minh City, Vietnam, e-mail: dzungngv@gmail.com; Le Thi Phuong Ngoc, University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam, e-mail: ngoc1966@gmail.com; Nguyen Huu Nhan, Ho Chi Minh University of Foreign Languages and Information Technology, 828 Su Van Hanh Str., Dist. 10, Ho Chi Minh City, Vietnam, e-mail: nhannh1@huflit.edu.vn; Nguyen Thanh Long (corresponding author), Department of Mathematics and Computer Science, University of Science, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam, e-mail: longnt2@gmail.com
