Evolution Equations and Control Theory | 2019
Optimal energy decay rates for some wave equations with double damping terms
Abstract
We consider the Cauchy problem in \\begin{document}$ {\\bf R}^{n} $\\end{document} for some wave equations with double damping terms, that is, one is the frictional damping \\begin{document}$ u_{t}(t, x) $\\end{document} and the other is very strong structural damping expressed as \\begin{document}$ (-\\Delta)^{\\theta}u_{t}(t, x) $\\end{document} with \\begin{document}$ \\theta > 1 $\\end{document} . We will derive optimal decay rates of the total energy and the \\begin{document}$ L^{2} $\\end{document} -norm of solutions as \\begin{document}$ t \\to \\infty $\\end{document} . These results can be obtained in the case when the initial data have a sufficient high regularity in order to guarantee that the corresponding high frequency parts of such energy and \\begin{document}$ L^{2} $\\end{document} -norm of solutions are remainder terms. A strategy to get such results comes from a method recently developed by the first author [ 11 ].