Well-posedness and exponential stability for a nonlinear wave equation
with acoustic boundary conditions
Abstract
In this paper, we prove the well-posedness of a nonlinear wave equation
coupled with boundary conditions of Dirichlet and acoustic type imposed
on disjoints open boundary subsets. The proposed nonlinear equation
models small vertical vibrations of an elastic medium with weak internal
damping and a general nonlinear term. We also prove the exponential
decay of the energy associated with the problem. Our results extend the
ones obtained by Frota-Goldstein [18] and
Limaco-Clark-Frota-Medeiros [26] to allow weak internal dampings and
removing the dimensional restriction 1≤ n≤4. The method we use is
based on a finite-dimensional approach by combining the Faedo-Galerkin
method with suitable energy estimates and multiplier techniques.