On weak/Strong Attractor for a 3-D Structural-Acoustic Interaction with Kirchhoff–Boussinesq Elastic Wall Subject to Restricted Boundary Dissipation

• Published in: Journal of dynamics and differential equations Vol. 36; no. 3; pp. 2793 - 2825
• Main Authors: Lasiecka, Irena, Rodrigues, José H.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2024; Springer Nature B.V
• Subjects: Acoustics; Applications of Mathematics; Attractors (mathematics); Blowing time; Boussinesq equations; Dissipation; Dynamical systems; Energy methods; Internal forces; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; System theory; Wave equations; Structure-acoustic dynamical system; Kirchhoff-Boussinesq plate and interface; 74K20; Boundary-interface dissipation; 74F99; 35L05; 35B41; 37L30; Global attractors
• DOI: 10.1007/s10884-023-10325-8
• ISSN: 1040-7294

Existence of global attractors for a structural-acoustic system, subject to restricted boundary dissipation, is considered. Dynamics of the acoustic environment is given by a linear 3-D wave equation subject to locally distributed boundary dissipation, while the dynamics on the (flat) structural wall is given by a 2D-Kirchhoff-Boussinesq plate equation, subject to linear dissipation and supercritical nonlinear restoring forces. It is shown that the trajectories of the dynamical system defined on finite energy phase space are attracted asymptotically to a global attractor. The main challenges of the problem are related to: (i) superlinearity of the elastic energy of the structural component, (ii) Boussinesq effects of internal forces potentially leading to a finite time blowing up solutions, (iii) partially-restricted boundary dissipation placed on the interface only. The resulting system lacks dissipativity along with the suitable compactness properties, both corner stones of PDE dynamical system theories [ 1 ]. To contend with the difficulties, a new hybrid approach based on a suitable adaptation of the so called “energy methods” [ 2 , 3 ] and compensated compactness [ 4 ] has been developed. The geometry of the acoustic chamber plays a critical role.