Nonlinear perturbations of the Kirchhoff equation




Miranda, Manuel M.
Louredo, Aldo T.
Medeiros, Luiz A.

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Texas State University, Department of Mathematics


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)∥2)∆u + |u|ρ = ƒ in Ω x (0, T0), u = 0 on Γ0 x]0, T0[, ∂u/∂v + δh(u′) = 0 on Γ1 x]0, T0[, where Ω is a bounded domain of ℝn with its boundary consisting of two disjoint parts Γ0 and Γ1; ρ > 1 is a real number; v(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.



Kirchhoff equation, Nonlinear boundary condition, Existence of solutions


Miranda, M. M., Louredo, A. T., & Medeiros, L. A. (2017). Nonlinear perturbations of the Kirchhoff equation. <i>Electronic Journal of Differential Equations, 2017</i>(77), pp. 1-21.


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