The Liouville theorem of a torsion system and its application to symmetry group of a porous medium type equation on symmetric spaces

An optimal Liouville theorem for the porous medium equation

Under a sharp asymptotic growth condition at infinity, we prove a Liouville type theorem for the inhomogeneous porous medium equation, provided it stays universally close to the heat equation. Additionally, for the homogeneous equation, we show that for the conclusion to hold, it is enough to assume the sharp asymptotic growth at infinity only in the space variable. The results are optimal, meaning that the growth condition at infinity cannot be weakened.

Gradient estimates for $u_t=ΔF(u)$ on manifolds and some Liouville-type theorems

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $$u_t=ΔF(u),$$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): $$u_t=Δ(u^p),\qquad p>0,$$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of $p$ than that for Aronson-Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for…

f-Laplace非线性多孔介质方程的非线性多孔介质方程：Hamilton-Souplet-Zhang类型的梯度估计及其影响

The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications

本文研究了f-Laplace非线性多孔介质方程的梯度估计，通过多种方法建立了Hamilton-Souplet-Zhang类型的估计，并探讨了其对抛物型Liouville型结果和古老解特征化的影响。

On distributional solutions of local and nonlocal problems of porous medium type

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of $$\partial_tu-\mathfrak{L}^{σ,μ}[φ(u)]=g(x,t)\quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $φ$ is merely continuous and nondecreasing and $\mathfrak{L}^{σ,μ}$ is the generator of a general symmetric Lévy process. This means that $\mathfrak{L}^{σ,μ}$ can have both local and nonlocal parts like e.g. $\mathfrak{L}^{σ,μ}=Δ-(-Δ)^{\frac12}$. New uniqueness results for bounded distributional solutions of this problem and the corresponding elliptic equation are presented…

有界区域中分数阶多孔介质型方程的边界正则性

Boundary regularity for fractional porous medium-type equations in bounded domains

该研究证明了有界光滑区域中分数阶多孔介质型方程的边界Hölder正则性，通过结合比较原理、边界障碍与基于半空间古老解Liouville定理的爆破紧性论证，得到相关解的正则性估计。

CR Sub-Laplacian Comparison and Liouville-type Theorem in a Complete Noncompact Sasakian Manifold

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the CR sub-Laplacian comparison property. It is served as the CR analog of Yau's gradient estimate. As a consequence, we have the natural CR analogue of Liouville-type theorems in a complete noncompact Sasakian manifold of nonnegative pseudohermitian Ricci curvature tensors.