On the strongly damped wave equation with constraint

A weak formulation for the so-called semilinear strongly damped wave equation with constraint is introduced and a corresponding notion of solution is defined. The main idea in our approach consists in the use of duality techniques in Sobolev-Bochner spaces, aimed at providing a suitable “relaxation” of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite “physical” energy.