Strong stability of nonlinear semigroups with weak dissipation and non-compact resolvent–applications to structural acoustics

Abstract

We consider asymptotic stability, in the strong topology, of a nonlinear coupled system of partial differential equations (PDEs) arising in structural–acoustic interactions. The coupling involves parabolic and hyperbolic dynamics with interaction on an interface–a manifold of lower dimension. The distinctive feature of the model is that the resolvent associated with the generator governing the evolution is not compact and the dissipation considered is ‘weak’. Thus, strong stability is not to be generally expected. In linear problems this difficulty is circumvented by the use of Taubrien theorems and spectral analysis [W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semi-groups, Trans. Amer. Math. Soc. 306(8) (1988), pp. 837–852, Y.I. Lyubich and V.Q. Phong, Asymptotic stability of linear differential equations ain Banach spaces, Studia Math., LXXXXVII, (1988), pp. 37–42, G.M. Sklyar, On the maximal asymptotica for linear equations in Banach spaces, 2009]. However these methods are not applicable to nonlinear dynamics.

In this article, we present an approach to strong stability that is applicable to nonlinear semigroups governed by multivalued generators with non-compact resolvents. The method relies on a suitable relaxation of Lasalle invariance principle [J.P. LaSalle, Stability theory and invariance principles, in Dynamical Systems, Vol. 1, L. Cerasir, J.K. Hale, J.P. LaSalle, eds., Academic Press, New York, 1976, pp. 211–222] which then requires appropriate unique continuation theorems along with a string of a-priori PDE estimates specific to parabolic–hyperbolic coupled systems.

Keywords:

• strong stability
• relaxed LaSalle invariance principle
• structural-acoustic interactions
• parabolic–hyperbolic coupled systems with an interface
• unique continuation across the boundary
• nonlinear semigroup
• compact resolvent

AMS Subject Classifications::

• 35Q35
• 35Q30

Acknowledgement

This research has been partially supported by the NSF Grant DMS-0606682 and AFOSR Grant FA9550-09-1-0459.