Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. (English) Zbl 1007.35049

This paper is excellent. The authors prove global existence for the following equation of Kirchhoff-Carrier type subject to nonlinear boundary dissipation, that is, \[ \begin{aligned} u_{tt}-M\left( t,\int_{\Omega }\left|\nabla u\right|^{2}dx\right) \Delta u&=0\quad\text{in }\Omega \times (0,\infty), \\ u&=0\quad\text{on }\Gamma _{0}\times (0,\infty), \\ \frac{\partial u}{\partial \upsilon }+g(u_{t})&=0\quad\text{on }\Gamma _{1}\times (0,\infty), \\ ((u(x,0),u_{t}(x,0))&=((u^{0}(x),u^{1}(x)), \end{aligned} \] where \(\Omega \) is a bounded, star-shaped domain of \(\mathbb{R}^{n}\), \(n\geq 1,\) with a smooth boundary \(\Gamma =\Gamma _{0}\cup \Gamma _{1}.\) Here, \(\Gamma _{0}\) and \(\Gamma _{1}\) are closed and disjoint and \(\upsilon \) represents the unit outward normal to \(\Gamma\). Besides, the authors place a natural hypothesis about the function \(M\), and in general problems of Kirchhoff-Carrier equation when the function \(M\) depends on the time the problem becomes more difficult. It is important to mention here, that the authors prove the existence and uniqueness of regular solutions without any smallness on the initial data. Moreover, uniform decay rates are obtained by assuming a nonlinear feedback acting on the boundary.


35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations