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).
Very weak solutions to hypoelliptic wave equations
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abstract
In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, H\"older, and distributional. For H\"older coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [GR15] and [RT17b]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$-evolution equations for higher order operators on $\mathbb{R}^{n}$ or on groups, the results already being new in all these cases.
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math.AP 1years
2019 1verdicts
REJECT 1representative citing papers
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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).