Nonlocal heat equations in the Heisenberg group

2017/03/27 by Raúl E. Vidal, Vidal, Raúl Emilio
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1703.09323

openalex publication_date 2017/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the following nonlocal diffusion equation in the Heisenberg group ℍ_n, u_t(z,s,t)=J∗ u(z,s,t)-u(z,s,t), where ∗ denote convolution product and J satisfies appropriated hypothesis. For the Cauchy problem we obtain that the asymptotic behavior of the solutions is the same form that the one for the heat equation in the Heisenberg group. To obtain this result we use the spherical transform related to the pair (U(n),ℍ_n). Finally we prove that solutions of properly rescaled nonlocal Dirichlet problem converge uniformly to the solution of the corresponding Dirichlet problem for the classical heat equation in the Heisenberg group.

Citations

• Reformulation of elasticity theory for discontinuities and long-range forces

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