Display Abstract

Title On New Decay Rates for the Plate and Wave Equations with Fractional Damping

Name Ruy Coimbra Charao
Country Brazil
Email ruy.charao@ufsc.br
Co-Author(s) Ryo Ikehata (Hiroshima University) and Cleverson R. da Luz (Federal University of Santa Catarina)
Submit Time 2014-02-25 09:17:06
Special Session 60: Recent advances in evolutionary equations
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].