| Contents |
| In this report we show recent (almost) optimal decay rates for
the total energy and the $L^2$ norm of solutions associated to the
linear dissipative wave equation and plate equation with effects of
rotational inertia. The damping is given by a fractional damping
term depending on a parameter $\theta \in [0,1]$. We observe that
the dissipative structure of the plate equation with $\theta=0$ is
of the regularity-loss type. This decay structure still remains true
for the plate equation with a power of fractional damping $\theta \in [0,1)$. This means that we can have an optimal decay estimate
of solutions under an additional regularity assumption on the
initial data. The structure of regularity-loss becomes more weak
when $\theta$ increase and does not occur when $\theta$ arrive in
$\theta=1$. The wave equation does not have a such structure of
regularity loss and due to this fact it is not necessary to assume
additional regularity on the initial data when get estimates in the
region of high frequencies in Fourier space. In fact, in that region
the energy decays exponentially. Our results generalized previous
recent. We use a special method based on the energy method in the
Fourier space combined with the Haraux-Komornik lemma. This method
shows to be very effective to study decay properties for several
initial problems in ${\bf R}^n$ [ R. C. Char\~ao, C. R. da Luz and
R. Ikehata, Sharp decay rates for wave equations with a fractional
damping via new method in the Fourier space, JMAA 408 (2013),
247-255; R. C. Char\~ao, C. R. da Luz and R. Ikehata, New decay
rates for a problem of plate dynamics with fractional damping, JHDE
10 (2013), 563-575]. |
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