Nonlinear Perturbations of the Kirchhoff Equation

In this article we study the existence and uniqueness of local solutions for the initial-boundary value problem for the Kirchhoff equation u′′ −M(t, ‖u(t)‖)∆u+ |u| = f in Ω× (0, T0), u = 0 on Γ0×]0, T0[, ∂u ∂ν + δh(u′) = 0 on Γ1×]0, T0[, where Ω is a bounded domain of Rn with its boundary constiting of two disjoint parts Γ0 and Γ1; ρ > 1 is a real number; ν(x) is the exterior unit normal vector at x ∈ Γ1 and δ(x), h(s) are real functions defined in Γ1 and R, respectively. Our result is obtained using the Galerkin method with a special basis, the Tartar argument, the compactness approach, and a Fixed-Point method.