Séminaire d'Analyse

Damped wave equation with unbounded damping

by Petr Siegl (TU Graz)

Amphi Schwartz

Amphi Schwartz

We present main ideas in the spectral and pseudospectral analysis of the damped wave equation $u_{tt} + 2a(x) u_t = \Delta_x u$ with an unbounded damping coefficient $a$, e.g.~$a(x) = x^2$, $x \in \mathbb R$. The key step is the study of the associated quadratic operator function $T(\lambda) = -\Delta + 2 \lambda a(x) + \lambda^2$, $\lambda \in \mathbb C$. Due to the form of this function, some of the recently developed techniques of the spectral theory for Schr\"odinger operator with complex potential can be generalized; in particular the pseudomode construction and upper resolvent norm estimates. Moreover, the latter and recent results in semigroup theory provide an estimate on the decay of solutions as $t \to + \infty$.

The talk is based mainly on joint works with A. Arnal.