Fractional semilinear damped wave equation on the Heisenberg group

2025/01/18 by Aparajita Dasgupta, Shyam Swarup Mondal, Dasgupta, Aparajita +3
Engineering · Mathematics · #35A01 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Primary 43A80 #Secondary 35B33 #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2501.10816

openalex publication_date 2025/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian (-L_ℍ)^α, α>0 on the Heisenberg group ℍⁿ with power type non-linearity. With the presence of a positive damping term and nonnegative mass term, we derive L²-L² decay estimates for the solution of the homogeneous linear fractional damped wave equation on ℍⁿ, for its time derivative, and for its space derivatives. We also discuss how these estimates can be improved when we consider additional L¹-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≤ p≤ 1+\frac2α(Q-2α)₊ (or 1+(4α)/(Q)

Related

• Decay estimates for a class of wave equations on the Heisenberg group
• Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order
• Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups
• Existence of solutions to the semilinear damped wave equation with non-L² slowly decaying data : polynomial nonlinearity case
• Nonlinear fractional damped wave equation on compact Lie groups