Nonlocal Filtration Equations with Rough Kernels in the Heisenberg Group

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 Hölder regularity of sign-changing solutions for the porous medium type equations ($m\ge 1$). Notice that both conformal fractional operators ${\mathcal{L}}_{\alpha /2}$ and pure power fractional operators ${\mathcal{L}}^{\alpha /2}$ 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 ${\mathcal{L}}_{\alpha /2}$ or ${\mathcal{L}}^{\alpha /2}$.

arXiv comment44 pages. Typos are corrected