Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces
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| Format: | Preprint |
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2024
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| _version_ | 1866917638865682432 |
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| author | Dasgupta, Aparajita Kumar, Vishvesh Mondal, Shyam Swarup Ruzhansky, Michael |
| author_facet | Dasgupta, Aparajita Kumar, Vishvesh Mondal, Shyam Swarup Ruzhansky, Michael |
| contents | Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $ν\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-γ}(\mathbb G), γ>0$. In the framework of Sobolev spaces of negative order, we prove that $p_{\text{Crit}}(Q, γ, ν) :=1+\frac{2ν}{Q+2γ}$ is the new critical exponent for $γ\in (0, \frac{Q}{2})$. More precisely, we show the global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{Crit}}(Q, γ, ν) $ in the energy evolution space $ \mathcal{C}\left([0, T], H^{s}(\mathbb{G})\right), s\in (0, 1]$. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for $1<p<p_{\text{Crit}}(Q, γ, ν)$. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical cases. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces Dasgupta, Aparajita Kumar, Vishvesh Mondal, Shyam Swarup Ruzhansky, Michael Analysis of PDEs Primary 43A80, 35L15, 35L71, 35A01, Secondary 35L15, 35B33, 35B44 Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $ν\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-γ}(\mathbb G), γ>0$. In the framework of Sobolev spaces of negative order, we prove that $p_{\text{Crit}}(Q, γ, ν) :=1+\frac{2ν}{Q+2γ}$ is the new critical exponent for $γ\in (0, \frac{Q}{2})$. More precisely, we show the global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{Crit}}(Q, γ, ν) $ in the energy evolution space $ \mathcal{C}\left([0, T], H^{s}(\mathbb{G})\right), s\in (0, 1]$. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for $1<p<p_{\text{Crit}}(Q, γ, ν)$. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical cases. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups. |
| title | Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces |
| topic | Analysis of PDEs Primary 43A80, 35L15, 35L71, 35A01, Secondary 35L15, 35B33, 35B44 |
| url | https://arxiv.org/abs/2404.08766 |