Differentiability, analyticity and optimal rates of decay for damped wave equations

Abstract

We give necessary and sufficient conditions on the damping term of a wave equation for the corresponding semigroup to be analytic. We characterize damped operators for which the corresponding semigroup is analytic, differentiable, or exponentially stable. Also when the damping operator is not strong enough to have the above properties, we show that the solution decays polynomially, and that the polynomial rate of decay is optimal.

Keywords

• Dissipative systems
• decay rate
• analytic semigroups
• polynomial stability