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 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 $\mathscr{L}^{\alpha/2}$.