On solvability of the dissipative Kirchhoff equation with nonlinear boundary damping
Bull. Korean Math. Soc. 2014 Vol. 51, No. 1, 189-206
https://doi.org/10.4134/BKMS.2014.51.1.189
Printed January 1, 2014
Zai-yun Zhang and Jian-hua Huang
Hunan Institute of Science and Technology, National University of Defense Technology
Abstract : In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equation \begin{eqnarray*} \begin{array}{ll} u_{tt}-M(\|\nabla u\|^2)\bigtriangleup u+\alpha u_{t}+f(u)=0 ~\text{in} \,\Omega\times [0,\infty),\\ u(x,t) = 0 \,\ \text{on} \, \Gamma_{1}\times[0,\infty),\\ \frac{\partial u}{\partial \nu}+g(u_t)=0 ~\text{on}\, \Gamma_{0}\times[0,\infty),\\ u(x,0) = u_0, \ u_{t}(x,0) = u_1 ~\text{in}\,\Omega \end{array} \end{eqnarray*} with nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al.~\cite{1}. Furthermore,we overcome some difficulties due to the presence of nonlinear terms $M(\|\nabla u\|^2)$ and $g(u_t)$ by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity.
Keywords : global existence, dissipative Kirchhoff equation, Galerkin approximation, boundary damping
MSC numbers : 35B40, 35L70
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