Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation.
Résumé
We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions.
In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution.
What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term.
We also provide counterexamples in order to show the optimality of our results.