Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

Philip Jameson Graber

july, 2011

Publication type:

Paper in peer-reviewed journals

Journal:

Nonlinear Analysis: Theory, Methods & Applications, vol. 74, pp. 3137-3148

External link:

http://dx.doi.org/10.1016/j.na.2011.01.029

http://www.sciencedirect.com/science/article/pii/S0362546X11000460

DOI:

10.1016/j.na.2011.01.029

HAL:

hal-00973539

Keywords :

Wave equation; Acoustic semilinear boundary conditions; Interface; Structural acoustic model; Boundary stabilization; Strong stability; Decay rates

Abstract:

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt–Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.

BibTeX:

@article{Gra-2011,
    author={Philip Jameson Graber },
    title={Strong stability and uniform decay of solutions to a wave 
           equation with semilinear porous acoustic boundary conditions },
    doi={10.1016/j.na.2011.01.029 },
    journal={Nonlinear Analysis: Theory, Methods & Applications },
    year={2011 },
    month={7},
    volume={74 },
    pages={3137--3148},
}