2024/04/08 by Marianna Chatzakou, Chatzakou, Marianna, Aidyn Kassymov +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2404.05611
openalex publication_date 2024/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we study the global well-posedeness of the heat equation with variable time-dependent nonlinearity of the form φ(t)f(u) on unimodular Lie groups when the differential operator arises as the sum of squares of Hörmander vector fields. For general unimodular Lie groups, we derive the necessary conditions for the nonexistence of global positive solutions. This gives different conditions in the cases of compact, polynomial, and exponential volume growth groups. In the case of the Heisenberg groups ℍn, we also derive sufficient conditions, which coincide with the necessary ones in the case of ℍ1 (and this is also true for ℝn). In particular, in the case of the Heisenberg group ℍ1 we obtain the necessary and sufficient conditions under which the aforesaid initial value problem with variable nonlinearity has a global positive solution.