Convergence of solutions of von Karman evolution equations to equilibria

Abstract

We consider von Karman evolution equations with nonlinear interior dissipation and with clamped boundary conditions. Under some conditions we prove that every energy solution converges to a stationary solution and establish a rate of convergence. Earlier this result was known in the case when the set of equilibria was finite and hyperbolic. In our argument we use the fact that the von Karman nonlinearity is analytic on an appropriate space and apply the Lojasiewicz–Simon method in the form suggested by A. Haraux and M. Jendoubi.

Keywords:

• von Karman evolution equations
• nonlinear dissipation
• stabilization to equilibrium
• Lojasiewicz–Simon inequality

AMS Subject Classifications:

• Primary 35Q74
• Secondary 37L15
• 74B20
• 74K20

Acknowledgements

In conclusion, we would like to thank an anonymous referee for pointing out the references 9,15 and for his valuable observation concerning possibility to apply an idea from 9 in the study of the von Karman model.

Notes

1. It is known 2 that the problem may have several solutions.