The Damped Waves Equation and generalized Cosine and Sine families on Banach spaces

We study the abstract damped wave equation on a Banach space, allowing the damping coefficient to be unbounded.

By recasting the equation as a first-order system and identifying conditions under which the associated block operator generates a $C_0$-group, we construct generalized cosine and sine families that represent mild and classical solutions, extending the classical undamped theory.

We establish existence, uniqueness, regularity, invariant subspaces, growth rate, and trigonometric type identities for these families.

Our setup applies to a broad class of damped wave, Klein--Gordon, and higher-order PDE examples, including cases where damping restores well-posedness that fails in the undamped equation.