$L^p$ asymptotics for the heat equation on symmetric spaces for non-symmetric solutions
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915005955309568 |
|---|---|
| author | Papageorgiou, Effie |
| author_facet | Papageorgiou, Effie |
| contents | The main goal of this work is to study the $L^p$-asymptotic behavior of solutions to the heat equation on arbitrary rank Riemannian symmetric spaces of non-compact type $G/K$ for non-bi-$K$ invariant initial data. For initial data $u_0$ compactly supported or in a weighted $L^1(G/K)$ space with a weight depending on $p\in [1, \infty]$, we introduce a mass function $M_p(u_0)(\cdot)$, and prove that if $h_t$ is the heat kernel on $G/K$, then $$\|h_t\|_p^{-1}\,\|u_0\ast h_t \, - \,M_p(u_0)(\cdot)\,h_t\|_p \rightarrow 0 \quad \text{as} \quad t\rightarrow \infty.$$ Interestingly, the $L^p$ heat concentration leads to completely different expressions of the mass function for $1\leq p <2$ and $2\leq p\leq \infty$. If we further assume that the initial data are bi-$K$-invariant, then our mass function boils down to the constant $\int_{G/K}u_0$ in the case $p=1$, and more generally to $\mathcal{H}{u_0}(iρ(2/p-1))$ if $1\leq p<2$, and to $\mathcal{H}{u_0}(0)$ if $2\leq p \leq \infty$. Thus we improve upon results by Vázquez, Anker et al, Naik et al, clarifying the nature of the problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02940 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$ asymptotics for the heat equation on symmetric spaces for non-symmetric solutions Papageorgiou, Effie Analysis of PDEs 22E30, 35B40, 35K05, 58J35 The main goal of this work is to study the $L^p$-asymptotic behavior of solutions to the heat equation on arbitrary rank Riemannian symmetric spaces of non-compact type $G/K$ for non-bi-$K$ invariant initial data. For initial data $u_0$ compactly supported or in a weighted $L^1(G/K)$ space with a weight depending on $p\in [1, \infty]$, we introduce a mass function $M_p(u_0)(\cdot)$, and prove that if $h_t$ is the heat kernel on $G/K$, then $$\|h_t\|_p^{-1}\,\|u_0\ast h_t \, - \,M_p(u_0)(\cdot)\,h_t\|_p \rightarrow 0 \quad \text{as} \quad t\rightarrow \infty.$$ Interestingly, the $L^p$ heat concentration leads to completely different expressions of the mass function for $1\leq p <2$ and $2\leq p\leq \infty$. If we further assume that the initial data are bi-$K$-invariant, then our mass function boils down to the constant $\int_{G/K}u_0$ in the case $p=1$, and more generally to $\mathcal{H}{u_0}(iρ(2/p-1))$ if $1\leq p<2$, and to $\mathcal{H}{u_0}(0)$ if $2\leq p \leq \infty$. Thus we improve upon results by Vázquez, Anker et al, Naik et al, clarifying the nature of the problem. |
| title | $L^p$ asymptotics for the heat equation on symmetric spaces for non-symmetric solutions |
| topic | Analysis of PDEs 22E30, 35B40, 35K05, 58J35 |
| url | https://arxiv.org/abs/2411.02940 |