The damped wave equation on the Heisenberg group has heat-like L2 decay: with L1 data the solution decays like (1+t)^(-Q/4), the horizontal gradient like (1+t)^(-Q/4-1/2), and the time derivative like (1+t)^(-Q/4-1).
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Decay estimates for the linear damped wave equation on the Heisenberg group
The damped wave equation on the Heisenberg group has heat-like L2 decay: with L1 data the solution decays like (1+t)^(-Q/4), the horizontal gradient like (1+t)^(-Q/4-1/2), and the time derivative like (1+t)^(-Q/4-1).