Electron. J. Differential Equations, Vol. 2017 (2017), No. 77, pp. 1-21.

Nonlinear perturbations of the Kirchhoff equation

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

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.

Submitted January 24, 2017. Published March 21, 2017.
Math Subject Classifications: 35L15, 35L20, 35K55, 35L60, 35L70.
Key Words: Kirchhoff equation; nonlinear boundary condition; existence of solutions.

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Manuel Milla Miranda
Universidade Estadual da Paraíba
DM, PB, Brazil
email: milla@im.ufrj.br
Aldo T. Louredo
Universidade Estadual da Paraíba
DM, PB, Brazil
email: aldolouredo@gmail.com
Phone: +55 (83) 3315-3340
Luiz A. Medeiros
Universidade Federal do Rio de Janeiro
IM, RJ, Brazil
email: luizadauto@gmail.com

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