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Li Ma

Publications and source records attributed to Li Ma.

At least 217 records · Page 12Linked to original sources
arXiv2010

Boundary value problem for a classical semilinear parabolic equation

In this paper, we study the boundary value problem of the classical semilinear parabolic equations $$ u_t-Δu=|u|^{p-1}u, \ \ in \ \ Ω\times (0,T) $$ and $u=0$ on the boundary $\partialΩ\times [0,T)$ and $u=ϕ$ at $t=0$, where $Ω\subset R^n$ is a compact $C^1$ domain, $1<p\leq p_S$ is a fixed constant, and $ϕ\in C^2_0(Ω)$ is a given smooth function. Introducing new idea, we show that there are two sets $\tilde{W}$ and $\tilde{Z}$ such that for $ϕ\in W$, there is a global positive solution $u(t)\in \tilde{W}$ with $h^1$ omega limit $\{0\}$ and for $ϕ\in \tilde{Z}$, the solution blows up at finite time.

math.AP↗
arXiv2010

Yamabe flow and the Myers-type theorem on complete manifolds

In this paper,we prove the following Myers-type theorem: if $(M^n,g)$, $n\geq 3$, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition $Rc\geq εRg>0$, where $ε>0$ is an uniform constant, then $M^n$ must be compact.

math.DG↗
arXiv2010

Optional Pólya tree and Bayesian inference

We introduce an extension of the Pólya tree approach for constructing distributions on the space of probability measures. By using optional stopping and optional choice of splitting variables, the construction gives rise to random measures that are absolutely continuous with piecewise smooth densities on partitions that can adapt to fit the data. The resulting "optional Pólya tree" distribution has large support in total variation topology and yields posterior distributions that are also optional Pólya trees with computable parameter values.

math.ST↗
arXiv2010

Gradient estimates for a simple nonlinear heat equation on manifolds

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $$ u_t-Δu=au\log u+Vu, \ \ u>0 $$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative Ricci curvature. Here $a\leq 0$ is a constant, $V$ is a smooth function on $M$ with $-ΔV\leq A$ for some positive constant $A$. This heat equation is a basic evolution equation and it can be considered as the negative gradient heat flow to $W$-functional (introduced by G.Perelman), which is the Log-Sobolev inequalities on the Riemannian manifold and $V$ corresponds to the scalar curvature.

math.DG↗
arXiv2010

A proof of Hamilton's conjecture

This paper has been withdrawn by the author due to the version of [A complete proof of Hamilton's conjecture] at arXiv:1008.1576

math.DG↗
arXiv2010

Ricci expanders and type III Ricci flow

In this paper, we study how to get the Ricci expanders from W+-functional through the heat kernel estimate of the conjugate heat equation to the type III singularity of Ricci flow. The Gaussian upper and lower bounds are established for the related heat kernel in accordance to the interesting work of Cao-Zhang for the type I Rici flow.

math.DG↗
arXiv2010

Heat flow method to Lichnerowicz type equation on closed manifolds

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -Δu=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$, $B(x)\geq0$ are given smooth functions. Our analysis is based on the global existence of positive solutions to the following heat equation {ll} u_t-Δu=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad in\quad M with the positive smooth initial data $u_0$.

math.DG↗
arXiv2010

On the generalized Feynman-Kac transformation for nearly symmetric Markov processes

Suppose $X$ is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on $L^{2}(E;m)$. For $u\in D(\mathcal{E})$, we have Fukushima's decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$. In this paper, we investigate the strong continuity of the generalized Feynman-Kac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$. Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$. Denote by $J_1$ the dissymmetric part of the jumping measure $J$ of $(\mathcal{E},D(\mathcal{E}))$. Under the assumption that $J_1$ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant $α_0\ge 0$ such that $\|P^{u}_{t}\|_2\leq e^{α_0 t}$ for every $t>0$. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on $L^{2}(E;m)$. If $X$ is equipped with a differential structure, then this result also holds without assuming that $J_1$ is finite.

math.PR↗
arXiv2009

Eigenvalue estimates and L1 energy on closed manifolds

In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and L1 and L2 energy for drifting heat equation on closed manifolds with weighted measure. In some sense, this study is about the eigenvalue estimate on Ricci solitons.

math.DG↗
arXiv2009

Blow-up and global solutions to L^p norm preserving non-local flows

In this paper, we study global existence and blow up properties to $L^p$ norm preserving non-local heat flows. We first study two kinds of $L^p$ norm preserving non-local flows and prove that these flows have the global solutions. Finally, we give a example to show that one kind of this heat flow may blow up in $L^{\infty}$ norm though its $L^p$ norm is preserved.

math.AP↗
arXiv2009

Curvature tensor under the complete non-compact Ricci Flow

We prove that for a solution $(M^n,g(t))$, $t\in[0,T)$, where $T<\infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on $M^n\times [0,T)$, the curvature tensor stays uniformly bounded on $M^n\times [0,T)$. Some other results are also presented.

math.DG↗