ON A KIRCHHOFF-CARRIER EQUATION WITH NONLINEAR TERMS CONTAINING A FINITE NUMBER OF UNKNOWN VALUES.

We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values u(η₁, t), . . ., u(η_q, t) with 0 ⩽ η₁ < η₂ < . . . < 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 Sq[u ² ](t) is replaced by ʃ¹₀ u ² (y, t) dy. 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. [ABSTRACT FROM AUTHOR]