Abstract
We consider a locally damped wave equation in a bounded domain. The damping is nonlinear, involves the Laplace operator, and is localized in a suitable open subset of the domain under consideration. First, we discuss the well-posedness and regularity of the solutions of the system by using a combination of the nonlinear semigroup theory and the Faedo–Galerkin scheme. Then, using the energy method combined with the piecewise multipliers method and relying on the localized smoothing property, we derive the exponential decay of the energy when the nonlinear damping grows linearly, the damping coefficient is smooth enough and further satisfies a structural condition; this is in agreement with what is known in the case of a linear damping of Kelvin–Voigt type. The novelty of our approach lies on the fact that: (1) we are using a nonlinear damping, which makes it trickier to study the regularity of solutions as well as the exploitation of the localized smoothness property, in order to prove that the energy decays exponentially; (2) the energy estimates are carried out directly in the time domain, unlike the frequency domain method utilized in all earlier works on the stabilization of the wave equation involving a localized Kelvin–Voigt damping.
Published Version
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