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Estimates for the linear viscoelastic damped wave equation on the Heisenberg group. (English) Zbl 1461.35146

Summary: In the paper, we investigate the linear wave equation with viscoelastic damping on the Heisenberg group \(\mathbf{H}_n\). Basing on the group Fourier transform on \(\mathbf{H}_n\) and on the properties of the Hermite functions, we derive some \(L^2( \mathbf{H}_n) - L^2( \mathbf{H}_n)\) estimates with additional \(L^1( \mathbf{H}_n)\) regularity on initial data for the solution and its higher-order horizontal gradients.

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35B45 A priori estimates in context of PDEs
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
58J45 Hyperbolic equations on manifolds
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