Fractional semilinear damped wave equation on the Heisenberg group
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909460785528832 |
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| author | Dasgupta, Aparajita Mondal, Shyam Swarup Tushir, Abhilash |
| author_facet | Dasgupta, Aparajita Mondal, Shyam Swarup Tushir, Abhilash |
| contents | This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian $(-\mathcal{L}_{\mathbb{H}})^α$, $α>0$ on the Heisenberg group $\mathbb{H}^{n}$ with power type non-linearity. With the presence of a positive damping term and nonnegative mass term, we derive $L^2-L^2$ decay estimates for the solution of the homogeneous linear fractional damped wave equation on $\mathbb{H}^{n}$, for its time derivative, and for its space derivatives. We also discuss how these estimates can be improved when we consider additional $L^1$-regularity for the Cauchy data in the absence of the mass term. Also, in the absence of mass term, we prove the global well-posedness for $2\leq p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}}$ $(\text{or }1+\frac{4α}{\mathcal{Q}}<p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}})$ in the case of $L^1\cap L^2$ $(\text{or } L^2)$ Cauchy data, respectively. However, in the presence of the mass term, the global (in time) well-posedness for small data holds for $1<p \leq 1+ \frac{2α}{(\mathcal{Q}-2α)_{+}}$. Finally, as an application of the linear decay estimates, we investigate well-posedness for the Cauchy problem for a weakly coupled system with two semilinear fractional damped wave equations with positive mass term on $\mathbb{H}^{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional semilinear damped wave equation on the Heisenberg group Dasgupta, Aparajita Mondal, Shyam Swarup Tushir, Abhilash Analysis of PDEs Primary 43A80, 35L71, 35A01, Secondary 35B33 This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian $(-\mathcal{L}_{\mathbb{H}})^α$, $α>0$ on the Heisenberg group $\mathbb{H}^{n}$ with power type non-linearity. With the presence of a positive damping term and nonnegative mass term, we derive $L^2-L^2$ decay estimates for the solution of the homogeneous linear fractional damped wave equation on $\mathbb{H}^{n}$, for its time derivative, and for its space derivatives. We also discuss how these estimates can be improved when we consider additional $L^1$-regularity for the Cauchy data in the absence of the mass term. Also, in the absence of mass term, we prove the global well-posedness for $2\leq p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}}$ $(\text{or }1+\frac{4α}{\mathcal{Q}}<p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}})$ in the case of $L^1\cap L^2$ $(\text{or } L^2)$ Cauchy data, respectively. However, in the presence of the mass term, the global (in time) well-posedness for small data holds for $1<p \leq 1+ \frac{2α}{(\mathcal{Q}-2α)_{+}}$. Finally, as an application of the linear decay estimates, we investigate well-posedness for the Cauchy problem for a weakly coupled system with two semilinear fractional damped wave equations with positive mass term on $\mathbb{H}^{n}$. |
| title | Fractional semilinear damped wave equation on the Heisenberg group |
| topic | Analysis of PDEs Primary 43A80, 35L71, 35A01, Secondary 35B33 |
| url | https://arxiv.org/abs/2501.10816 |