Nonlocal Filtration Equations with Rough Kernels in the Heisenberg Group

TL;DR

This paper studies nonlocal filtration equations with rough kernels on the Heisenberg group, establishing existence, uniqueness, regularity, and long-term behavior of solutions, extending results to fractional operators on the group.

Contribution

It introduces existence, uniqueness, and regularity results for nonlocal filtration equations with rough kernels on the Heisenberg group, including fractional operators.

Findings

Existence and uniqueness of weak solutions.

Solutions exhibit uniform Hölder regularity.

Long-time behavior of solutions is characterized.

Abstract

Motivated by the extensive investigations of integro-differential equations on $\mathbb{R}^n$, we consider nonlocal filtration type equations with rough kernels on the Heisenberg group $\mathbb{H}^n$. We prove the existence and uniqueness of weak solutions corresponding to suitable initial data. Furthermore, we obtain the large time behavior of solutions and the uniform H\"older regularity of sign-changing solutions for the porous medium type equations ($m\geq 1$). Notice that both conformal fractional operators $\mathscr{L}_{\alpha/2}$ and pure power fractional operators $\mathscr{L}^{\alpha/2}$ on the Heisenberg group $\mathbb{H}^n$ have their integral representations with suitable kernels. Therefore, all the results in this paper will hold for these equations with operators $\mathscr{L}_{\alpha/2}$ or $\mathscr{L}^{\alpha/2}$.