L^p asymptotics for the heat equation on symmetric spaces for non-symmetric solutions

2024/11/05 by Effie Papageorgiou, Papageorgiou, Effie · 1 citation
Computer Science · Mathematics · #22E30 #35B40 #35K05 #58J35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.02940

openalex publication_date 2024/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 u₀ compactly supported or in a weighted L¹(G/K) space with a weight depending on p∈ [1, ∞], we introduce a mass function M_p(u₀)(⋅), and prove that if h_t is the heat kernel on G/K, then ‖h_t‖_p^-1 ‖u₀∗ h_t - M_p(u₀)(⋅) h_t‖_p → 0 as t→ ∞. Interestingly, the L^p heat concentration leads to completely different expressions of the mass function for 1≤ p <2 and 2≤ p≤ ∞. If we further assume that the initial data are bi-K-invariant, then our mass function boils down to the constant ∫_G/Ku₀ in the case p=1, and more generally to Hu₀(iρ(2/p-1)) if 1≤ p<2, and to Hu₀(0) if 2≤ p ≤ ∞. Thus we improve upon results by Vázquez, Anker et al, Naik et al, clarifying the nature of the problem.

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