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Decay rates for the shifted wave equation on a symmetric space of noncompact type. (English) Zbl 1408.35095

Summary: We derive \(L^{\infty}-L^{1}\) decay rate estimates for solutions of the shifted wave equation on certain symmetric spaces \((M, g)\). The Cauchy problem for the shifted wave operator on these spaces was studied by Helgason, who obtained a closed form for its solution. Our results extend to this new context the classical estimates for the wave equation in \(\mathbb R^{n}\). Then, following an idea from Klainerman, we introduce a new norm based on Lie derivatives with respect to Killing fields on M and we derive an estimate for the case that \(n = \dim M\) is odd.

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
58J45 Hyperbolic equations on manifolds
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