Well-posedness and stability for Kirchhoff equation with non-porous acoustic boundary conditions. (English) Zbl 1483.35137
Summary: In this paper we prove the well-posedness to the wave equation of Kirchhoff type. Under a portion of the boundary, we consider the acoustic boundary conditions. We also prove the exponential stability of the energy associated to the problem. Our result generalize the previous literature where only the Carrier model was considered (when the nonlinearity involves the \(L^2(\Omega)\) norm) or the Kirchhoff model with porous acoustic boundary conditions. The main tool is the Faedo-Galerkin method. Due to the presence of acoustic boundary conditions, we can not use special basis and new estimates are necessary. To prove the stability we use integral estimates.
MSC:
| 35L72 | Second-order quasilinear hyperbolic equations |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35B35 | Stability in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
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