论文标题
Heisenberg组中具有粗核的非局部过滤方程
Nonlocal Filtration Equations with Rough Kernels in the Heisenberg Group
论文作者
论文摘要
通过对$ \ mathbb {r}^n $的全面差异方程的广泛调查，我们考虑了Heisenberg Group $ \ Mathbb {H}^n $上的粗核的非局部过滤类型方程。我们证明了与合适的初始数据相对应的弱解决方案的存在和独特性。此外，我们获得了解决方案的较大时间行为，以及针对多孔培养基方程（$ M \ geq 1 $）的签名解决方案的均匀Hölder规律性。请注意，两个共形分数运算符$ \ MATHSCR {l} _ {α/2} $和纯电源分数操作员$ \ Mathscr {l}^{l}^{α/2} $上的Heisenberg Group $ \ Mathbb {h}^n $具有合适的kernels。因此，本文中的所有结果都将与运算符$ \ mathscr {l} _ {α/2} $或$ \ Mathscr {l}^{α/2} $保持。
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 ($m\geq 1$). Notice that both conformal fractional operators $\mathscr{L}_{α/2}$ and pure power fractional operators $\mathscr{L}^{α/2}$ 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 $\mathscr{L}_{α/2}$ or $\mathscr{L}^{α/2}$.
