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Abstract wave equations and associated Dirac-type operators. (English) Zbl 1262.35174

The authors consider an abstract version of the damped wave equation \[ \ddot{u}(t) + R\dot{u}(t) + A^*Au(t) = 0, \quad u(0) = f_0, \quad \dot{u}(0) = f_1, \quad t \geq 0, \] where \(A\) is a densely defined closed operator in a separable Hilbert space \(H\) and \(R\) is a perturbation of \(A^*A\). Let \(G_{A,R}\) be the formal generator of the associated first-order system. The authors show that a realization of \(G_{A,R}\) is unitarily equivalent to a realization of \(Q_{A,R}(\mathrm{Id} \oplus [\mathrm{Id} - P_{\mathrm{Ker}(A^*)} ]),\) where \(Q_{A,R}\) is a perturbed supersymmetric Dirac-type operator on \(H \oplus H\). The general equivalence is obtained by considering four cases:
1) case \(A^*A \geq \epsilon I_{\mathcal H}\) for some \(\epsilon > 0\) in absence of damping;
2) case \(\inf(\sigma(A^*A)) = 0\) in absence of damping;
3) case \(A^*A \geq \epsilon I_{\mathcal H}\) for some \(\epsilon > 0\) with damping;
4) case \(\inf(\sigma(A^*A)) = 0\) with damping.
The equivalence is then used to determine properties of the damped wave equation, e.g. well-posedness of the equation and the sizes of the semigroup growth bounds for certain damping terms \(R\), from the properties of the perturbed Dirac operator. For readers with no prior contact to mathematical physics the article contains an appendix which summarizes the used results on supersymmetric Dirac-type operators.

MSC:

35L90 Abstract hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
47D06 One-parameter semigroups and linear evolution equations
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