| Summary: | We consider the wave equation with two types of locally distributed damping
mechanisms: a frictional damping and a Kelvin-Voigt type damping.
The location of each damping is such that none of them alone is able
to exponentially stabilize the system; the main obstacle being that there
is a quite big undamped region. Using a combination of the multiplier
techniques and the frequency domain method, we show that a convenient interaction
of the two damping mechanisms is powerful enough for the exponential stability
of the dynamical system, provided that the coefficient of the Kelvin-Voigt
damping is smooth enough and satisfies a structural condition. When the
latter coefficient is only bounded measurable, exponential stability may still
hold provided there is no undamped region, else only polynomial stability is
established. The main features of this contribution are:
(i) the use of the Kelvin-Voigt or short memory damping as opposed to the usual
long memory type damping; this makes the problem more difficult to solve due
to the somewhat singular nature of the Kelvin-Voigt damping,
(ii) allowing the presence of an undamped region unlike all earlier works where
a combination of frictional and viscoelastic damping is used.
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