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

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.

Keywords

• kirchhoff-carrier equation
• robin-dirichlet problem
• nonlocal term
• faedo-galerkin method
• linearization method