Longtime behavior of the hyperbolic equations with an arbitrary internal damping (Q1938457)

For damped hyperbolic equations, IBV-problems with Neumann boundary conditions are considered. Assuming sharp regularity conditions, i.e. coefficients and boundary of class \(C^1\), logarithmic decay rate for solutions is proved. Results for \(C^2\) coefficients and domains with \(C^2\) boundary have been published by the author in a previous paper [Commun. Partial Differ. Equations 34, No. 9, 957--975 (2009; Zbl 1180.35104)]. The decay rate result is a consequence of a suitable resolvent estimate, which is proved by means of a global Carleman estimate and an interpolation inequality for elliptic equations with Neumann boundary conditions.