Yongjin Lu, University of Virginia


Well-posedness, stability and asymptotic behavior of a nonlinear coupled PDE system arising in structural acoustic interactions


Abstract: We studied the well-posedness, stability and asymptotic behavior of a nonlinear coupled PDE system, which comprises of wave and plate equations with an interface. The system serves as a benchmark for structural acoustic interaction that models pressure in an acoustic chamber.

Our aim is to study long time behavior and related stability issues. In this context, the distinct mathematical feature of the system is that the resolvent is not compact. As is well known, compactness of the resolvent is a critical assumption required by the established asymptotic stability theories. While this requirement can be sometimes circumvented in linear theory (by exploiting spectral-tauberian type of theorems based on the analysis of the spectrum), the inherent nonlinearity of the model prevents this approach from being applicable.

The aim of this work is to present an abstract approach that does not require compactness and is applicable to nonlinear structures. We shall also show that this approach can be applied to the structural interaction considered in this work.


Advisor: Dr. Irena Lasiecka (University of Virginia)