Nonlinear perturbations of the Kirchhoff equation

Abstract

In this article we study the existence and uniqueness of local solutions for the initial-boundary value problem for the Kirchhoff equation $$\displaylines{ u'' - M(t,\|u(t)\|^{2})\Delta u + |u|^{\rho} =f \quad\text{in } \Omega \times (0, T_0), \cr u=0\quad\text{on }\Gamma_0 \times ]0, T_0[, \cr \frac{\partial u}{\partial \nu} + \delta h(u')=0 \quad\text{on } \Gamma_1 \times ]0, T_0[, }$$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$ with its boundary constiting of two disjoint parts $\Gamma_0$ and $\Gamma_1$; $\rho >1$ is a real number; $\nu(x)$ is the exterior unit normal vector at $x \in \Gamma_1$ and $\delta(x), h(s)$ are real functions defined in $\Gamma_1$ and $\mathbb{R}$, respectively. Our result is obtained using the Galerkin method with a special basis, the Tartar argument, the compactness approach, and a Fixed-Point method.

Keywords

• Kirchhoff equation
• nonlinear boundary condition
• existence of solutions