Decay estimates for a class of Dunkl wave equations

Let Δ_κ be the Dunkl Laplacian on Rⁿ and φ: R⁺ → R is a smooth function. The aim of this manuscript is twofold. First, we study the decay estimate for a class of dispersive semigroup of the form eitφ(√-Δ_κ).W e overcome the difficulty arising from the non-homogeneousity of φ by frequency localization. As applications, in the next part of the paper, we establish Strichartz estimates for some concrete wave equations associated with the Dunkl Laplacian Δₖ, which corresponds to φ(r)=r, r², r²+r⁴, √1+r², √1+r⁴, and r^μ,0<μ≤ 2, μ≠ 1. More precisely, we unify and simplify all the known dispersive estimates and extend to more general cases. Finally, using the decay estimates, we prove the global-in-time existence of small data Sobolev solutions for the nonlinear Klein-Gordon equation and beam equation with the power type nonlinearities.