Boundary stabilization for a nonlinear beam on elastic bearings (Q2731609)

The author considers the equation NEWLINE\[NEWLINEu_{tt}+ u_{xxxx}- M \int^t_0(|u_x|^2 dx) u_{xx}= 0,\quad (x,t)\in [0,L]\times \mathbb{R}^+NEWLINE\]NEWLINE under nonlinear boundary conditions which model the vibrations of a beam clamped at \(x= 0\) and supported by a nonlinear bearing at \(x= L\). The author proved the existence of a global solution and exponential decay of the energy by adding only one damping mechanism at \(x= L\). This results are useful for the existence and decay of another Kirchhoff-type beam equations with nonlinear boundary conditions.