Decay estimates for a class of Dunkl wave equations
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| Format: | Preprint |
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2024
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| author | Luo, Cheng Mondal, Shyam Swarup Song, Manli |
| author_facet | Luo, Cheng Mondal, Shyam Swarup Song, Manli |
| contents | Let $Δ_κ$ be the Dunkl Laplacian on $\mathbb{R}^n$ and $ϕ: \mathbb{R}^+ \to \mathbb{R}$ 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 $e^{itϕ(\sqrt{-Δ_κ})}$.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 $Δ_k,$ which corresponds to $ϕ(r)=r, r^2, r^2+r^4, \sqrt{1+r^2}, \sqrt{1+r^4}$, and $r^μ,0<μ\leq 2, μ\neq 1$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06949 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decay estimates for a class of Dunkl wave equations Luo, Cheng Mondal, Shyam Swarup Song, Manli Functional Analysis Primary: 22E25, 33C45, Secondary: 35H20, 35B40 Let $Δ_κ$ be the Dunkl Laplacian on $\mathbb{R}^n$ and $ϕ: \mathbb{R}^+ \to \mathbb{R}$ 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 $e^{itϕ(\sqrt{-Δ_κ})}$.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 $Δ_k,$ which corresponds to $ϕ(r)=r, r^2, r^2+r^4, \sqrt{1+r^2}, \sqrt{1+r^4}$, and $r^μ,0<μ\leq 2, μ\neq 1$. 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. |
| title | Decay estimates for a class of Dunkl wave equations |
| topic | Functional Analysis Primary: 22E25, 33C45, Secondary: 35H20, 35B40 |
| url | https://arxiv.org/abs/2407.06949 |