M. R. DeDeo, Elinor Velasquez
A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting.
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A note on nonlinear diffusive equations on Poincaré half plane
本文利用广义分离变量法求得庞加莱半平面上非线性扩散方程的精确解，并在不变子空间方法框架下解释其意义。
Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications
本文定义非局部半球梯度、散度、旋度算子，研究有界域上的函数空间，证明非局部Poincaré不等式，并应用于对流扩散、线弹性及Helmholtz分解。
Poincaré compactification for semiflows of reaction-diffusion equations with large diffusion and convection heating at the boundary
本文对具有大扩散和边界对流加热的耦合PDE-ODE系统极限半流进行庞加莱紧化，利用不变流形定理降维，并证明紧化向量场在C^1范数下接近。
Sub-Gaussian heat kernel estimates are typical of fractal graphs. We show that sub-Gaussian estimates on graphs follow from a Poincaré inequality, capacity upper bound, and a slow volume growth condition. An important feature of this work is that we do not assume elliptic Harnack inequality, cutoff Sobolev inequality, or exit time bounds.
The existence and local uniqueness of the eigenfunctions of the non-linear operator $ Δ_H u^{n}$ in the hyperbolic Poincaré half-plane
本文研究了双曲庞加莱半平面中非线性算子Δ_H u^n的特征函数的局部存在性和唯一性，证明了其系数满足特定代数递推规则，为非精确解提供了新的方法。
Poincare inequality and the uniqueness of solutions for the heat equation associated with subelliptic diffusion operators
本文研究了一类满足F.Baudoin和N. Garofalo引入的广义曲率维数不等式的子黎曼流形中球上的整体Poincare不等式.作为推论，我们证明了次椭圆型热方程解的唯一性。我们的结果特别适用于CR Sasakian流形与Tanaka-Webster-里奇曲率有界从下面和卡诺群的步骤二。
Gaussian Poincaré inequalities on the half-space with singular weights
本文在半空间上带奇异权重的高斯测度下证明了 Rellich-Kondrachov 型定理和加权庞加莱不等式，并进一步研究了有界区域中测度为 $y^cdz$ 的局部情形。
Sharp $L^2$-Caffarelli--Kohn--Nirenberg and weighted Poincaré inequalities on half-spaces and orthants and their stability
该研究针对半空间与卦限，显式计算尖锐$L^2$-CKN不等式及加权Poincaré不等式的最优常数、极值函数，建立稳定性估计，推广了经典高斯Poincaré不等式的结果。
Heat kernel estimates of fractional Schrödinger operators with Hardy potential on half-line
本文研究半直线上带哈代位势的分数阶薛定谔算子，建立了其热核的精确双边估计。
In this paper we study the Heat Flow on Metric Random Walk Spaces, which unifies into a broad framework the heat flow on locally finite weighted connected graphs, the heat flow determined by finite Markov chains and some nonlocal evolution problems. We give different characterizations of the ergodicity and prove that a metric random walk space with positive Ollivier-Ricci curvature is ergodic. Furthermore, we prove a Cheeger inequality and, as a consequence, we show that a Poincaré inequality holds if, and only if, an isoperimetric inequality holds. We also study the Bakry-Émery curvature-dime…
Doubling measures, Poincaré inequalities and parabolic Harnack inequalities for a doubly nonlinear equation
通过体积双倍和Poincaré不等式刻画满足双非线性方程抛物型Harnack不等式的度量测度空间，采用纯分析方法扩展线性结果。
Reverse Poincare inequalities, Isoperimetry, and Riesz transforms in Carnot groups
证明了任意阶Carnot群上的次Laplacian生成的热半群的一个最优逆Poincaré不等式。作为应用，我们给出了等周不等式和里斯变换在卡诺群上的有界性的新证明.
We consider the inhomogeneous heat equation on the half-space $\mathbb R_{+}^{d}$ with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol in [10].
We consider the nonlinear half laplacian heat equation $$ u_t+(-Δ)^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad \mathbb{R}^n\times (0, T). $$ We prove that all blows-up are type I, provided that $n \leq 4$ and $ 1<p<p_{*} (n)$ where $ p_{*} (n)$ is an explicit exponent which is below $\frac{n+1}{n-1}$, the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension …
Stationary flows for viscous heat-conductive fluid in a perturbed half-space
本文研究扰动半空间中满足出流与超声速条件的非等熵可压缩Navier-Stokes方程，证明了多方向定常解的唯一存在性及其渐近稳定性。
Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data
研究半空间上具动态边界条件的热方程，扩展了初始数据的可解性条件，证明加权 Lebesgue 空间的无界数据同样保证问题有解。
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.
Asymptotic stability of wave patterns to compressible viscous and heat-conducting gases in the half space
研究了一维半空间中理想多方粘性导热气体可压缩Navier-斯托克斯方程解的大时间行为。对于具有大的初始扰动和一般绝热指数的外流问题，证明了稀疏波及其与非退化定态解的叠加是渐近稳定的。
Well-posedness of half-harmonic map heat flows for rough initial data
本文基于Koch-Tataru理论和Carleson测度估计，为半调和映射热流构建了尺度临界的低正则性框架，引入了比传统空间更广泛的新初始数据类，并证明了其存在性、唯一性与连续依赖性。
Fundamental solution and diffusion limits for the heat equation in a half-space with a diffusive dynamical boundary condition
本文研究半空间热方程在扩散动力边界条件下的基本解，推导出显式表示并建立精确衰减估计，分析解的定性性质及扩散极限行为。
In this note, we give an elementary proof of the lack of null controllability for the heat equation on the half line by employing the machinery inherited by the unified transform, known also as the Fokas method. This approach also extends in a uniform way to higher dimensions and different initial-boundary value problems governed by the heat equation, suggesting a novel methodology for studying problems related to controllability.
We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.
Heat kernel estimates for the Bessel differential operator in half-line
本文考虑了Bessel微分算子L^（μ）= \dfrac{d^2}{dx^2} + \dfrac{2μ+1}{x} \dfrac{d}{dx}在射线（a，\infty）上的Dirichlet热核p_a^（μ）（t，x，y）.对于μ=0，通过结合分析和概率方法，我们给出了空间参数x，y>a和t>0的整个范围内热核的精确双边估计，这是对文献[1]中考虑μ \neq 0的结果的补充.
Global solvability for the heat equations in two half spaces and an interface
本文研究了两个半空间及界面上热方程的时间全局强解存在性，通过引入特定函数空间并结合极大$L^p$-正则性与能量等式，证明了系统在大初值下存在唯一全局强解。
Control of the half-heat equation
本文研究单位圆子弧控制下半热方程的零可控初始状态，推导投影半热方程零可控的近乎充要条件，进而处理原半热方程，并借助全纯函数奇点分离结论证明零可控函数空间与时间无关。
This paper is concerned with the impermeable wall problem for an ideal polytropic model of non-viscous and heat-conductive gas in one-dimensional half space. It is shown that the 3-rarefaction wave is stable under some smallness conditions. The proof is given by an elementary energy method and the key point is to control the boundary terms due to the less dissipativity of the system.
Plateau flow or the heat flow for half-harmonic maps
本文通过把半 Laplace 算子解释为球上 Dirichlet-to-Neumann 算子，建立半调和映射热流的经典存在性理论，并指出其与盘型极小曲面 Plateau 问题梯度流的联系。
The heat equation with inverse square potential on both half-lines of $\mathbb{R}$ is discussed in the presence of \emph{bridging} boundary conditions at the origin. The problem is the lowest energy (zero-momentum) mode of the transmission of the heat flow across a Grushin-type cylinder, a generalisation of an almost Riemannian structure with compact singularity set. This and related models are reviewed, and the issue is posed of the analysis of the dispersive properties for the heat kernel generated by the underlying positive self-adjoint operator. Numerical integration is shown that provides…
Boundary behaviour of the solution of the heat equation on the half line via the Fokas unified transform method
本文利用Fokas统一变换方法分析半直线上带Dirichlet数据的热方程解在时空边界附近的行为，并给出解可光滑延拓至半无限边界的条件。
Thresholding Scheme for the Half-Harmonic Map Heat Flow
本文提出一种基于泊松核卷积与逐点归一化的阈值化格式求解半调和映射热流，证明其收敛到满足能量耗散的弱解，并建立了弱-强唯一性。
Characterizations of sets of finite perimeter using heat kernels in metric spaces
本文的首要目标是在度量测度空间的设置中将有限周长集的概念（与$N^{1,1}$ -空间相关的概念）和热半群理论（与$N^{1,2}$ -空间相关的概念）联系起来，度量测度空间的测度是加倍的，并且支持$1$ -Poincaré不等式。利用热半群的短时性态，证明了在这种度量空间中有限周长集的一个特征。我们还给出了一个新的表征${\rm BV}$功能方面的近对角能量在这个一般的设置。
Stationary waves to viscous heat-conductive gases in half space: existence, stability and convergence rate
本文研究一维半空间中可压缩粘性气体理想多方模型外流问题解的大时间性态，以及解向相应定常解的收敛速度.利用中心流形理论证明了在边界数据很小的条件下，平稳解的存在性。在适当的小性假设下，利用能量方法研究了定常解的时间渐近稳定性，并估计了与初始扰动空间衰减率一致的收敛速度.证明主要是基于先验估计，这是由时间和空间加权能量方法。
In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}ρ\left(kw_x\right)_x+γw$, $\left.\left(\sqrt{\frac{k}ρ}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $ρ$, $k$, $γ$ is applied. The results are illustrated by examples.
Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times $t\in(-\infty,\infty)$) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains. We prove an optimal Liouville theorem for the linear equation in the halfspace complemented by the nonlinear boundary condition $\partial u/\partialν=u^q$, $q>1$.
Pointwise wave behavior of the Navier-Stokes equations in half space
本文研究了半空间中具有混合边界条件的可压缩Navier-斯托克斯方程解的逐点性态。结果表明，半空间中线性系的格林函数的首阶是以声速沿两个相反方向传播的热核和以正声速传播的反射热核（由于边界效应）。在强波相互作用下，非线性分析显示出丰富的波结构：扩散波相互作用，解以代数速率衰减。
Backward uniqueness for the heat equation in cones
已知半空间中的热方程的有界解在某个时刻变为零，即使没有对解的边界值作任何假设，也必定恒为零。在最近的一个例子中，路易斯·埃斯库里亚扎（Luis Escauriaza）证明，如果半空间被开角小于小于90度的圆锥体所取代，这种说法就失败了。在这里，我们表明，结果仍然适用于锥与开口角大于110度。证明涵盖热方程具有较低的可测系数有界项。
Lipschitz continuity of solutions of Poisson equations in metric measure spaces
设$(X,d)$是具有Ahlfors $Q$ -正则测度$μ$，$Q\in[1,\infty)$的路径连通度量空间.假设$(X,d,μ)$对于相应的“Gaussian测度”支持一个2-Poincaré不等式和一个Sobolev-Poincaré型不等式.利用热方程研究了Poisson方程$Δu=f$解的Lipschitz正则性，其中$f\in L^p_\loc$ .当$p>Q$时，$u$的局部Lipschitz连续性成立。
Schottky invariant diffusion on the transcendent p-adic upper half plane
本文证明 Drinfeld p-adic 上半平面的超越部分是 Polish 空间，利用 Schottky 群不变正则微分 1-形式构造自伴扩散算子，显式计算其谱并求解热方程的 Cauchy 问题与边值问题，确定具有 cadlag 路径的 Markov 过程。
Fundamental solution to the heat equation with a dynamical boundary condition
本文给出了${\mathbb R}^N$半空间上具齐次动力学边界条件热方程基本解的显式表示及上下界估计，借此获得了精确的解衰减估计，并应用其确定了半线性热方程的Fujita指数。
In the paper, problems of controllability, approximate controllability, reachability and approximate reachability are studied for the control system $w_t=w_{xx}$, $w(0,\cdot)=u$, $x>0$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. It is proved that each end state of this system is approximately reachable in a given time $T$, and each its initial state is approximately controllable in a given time $T$. A necessary and sufficient condition for reachability in a given time $T$ is obtained in terms of solvability a Markov power moment problem. It is also shown that there is no initial sta…
Zero viscosity and thermal diffusivity limit of the linearized compressible Navier-Stokes-Fourier equations in the half plane
研究了半平面中可压缩粘性导热流体的线性化Navier-Stokes-Fourier方程初边值问题的零粘性和导热极限。我们考虑的情况下，粘度和热扩散率收敛到零，在相同的顺序。本文首先利用多尺度分析方法，形式化地分析了带内、边界展开项的线性化Navier-Stokes-Fourier方程的近似解。然后用能量方法得到了该方程的近似解的误差项的逐点估计，从而建立了线性化Navier-Stokes-Fourier方程在零粘性和零导热极限下的一致稳定性.本文基于\cite{xy}，将等熵情况下的结果推广到具有热扩散效应的一般可压缩流体。在\cite{xy}中，除了粘性层外，还出现了热层，热层与粘性层呈线性耦合。
Harnack inequality and the relevant theorems on Finsler metric measure manifolds
针对加权Ricci曲率有下界的Finsler度量测度流形，本文建立体积比较、加权Poincaré、Sobolev及均值不等式，并据此推出热方程正解的局部梯度估计与Harnack不等式。
Gradient estimates for heat kernels and harmonic functions
设$(X,d,μ)$是一个具有Dirichlet形式$\E$的二重度量测度空间，它是由“carré du champ”导出的.假设$(X,d,μ,\E)$支持尺度不变的$L^2$-Poincaré不等式。本文研究了$p\in (2,\infty]$的调和函数、热核和Riesz变换的下列性质：（i）$(G_p)$：$L^p$ -伴随热半群梯度的估计;（ii）$(RH_p)$：$L^p$ -调和函数梯度的逆Hölder不等式;（iii）$(R_p)$：$L^p$ -Riesz变换的有界性（$p<\infty$）;（iv）$(GBE)$：广义Bakry-Émery条件。证明了对于$p\in (2,\infty)$，（i），（ii），（iii）是等价的，而对于$p=\infty$，（i），（ii），（iv）是等价的.此外，其中一些等价性在比$L^2$ -Poincaré不等式更弱的条件下仍然成立。我们的结果给出了$p=\infty$的Li-Yau的热核梯度估计的特征，而对于$p\in (2,\infty)$，它是对Auscher-Coulhon-Duong-Hofmann [7]和Auscher-Coulhon [6]的早期结果的实质性改进和推广.给出了等周不等式和Sobolev不等式的应用。我们的结果适用于黎曼和次黎曼流形以及非光滑空间，并退化椭圆/抛物方程在这些设置。
Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
该研究针对满足体积约束的局部能量最小化集合，通过证明庞加莱型不等式，分类了单位球、半空间及楔形域中的此类集合，并证实其相对边界具有光滑性。
Asymptotic behaviour of the Bessel heat kernels
考虑半直线$(a,\infty)$，$a>0$中Bessel微分算子$L^{(μ)}=\frac{d^2}{dx^2}+\frac{2μ+1}{2x}$，$μ\in\mathbb{R}$的Dirichlet热核$p_a^{(μ)}(t,x,y)$，并给出了$xy/t\rightarrow\infty$的渐近展开式.
Solvability of the heat equation with a nonlinear boundary condition
得到了具有非线性边界条件的热方程在半空间${\bf R}^N$中可解的必要条件和充分条件。此外，我们还研究了解的寿命与初始函数的行为之间的关系。
New maximal regularity results for the heat equation in exterior domains, and applications
本文证明了在半空间、有界区域或外部区域上的热方程在Besov空间中的新的极大正则性结果。我们在L^1（0，T;X）型正则空间中寻求与时间无关的先验估计，其中X代表某个齐次Besov空间。在有界区域的情况下，我们得到的结果类似于全空间或半空间的结果。对于外部区域，我们需要使用混合Besov范数以获得对低频的控制。这些估计是至关重要的证明全球的时间结果的非线性热方程在临界函数框架。
A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space
本文研究移动半直线上带 Robin 边界条件的热方程，边界以 b(t)∼ct^β 向内移动形成“活塞”效应，利用熵方法和 Duhamel 原理证明解向自相似剖面定量收敛，并揭示 β=1/2 时剖面从高斯型切换为指数型、β=1 时达到稳态平衡。
Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains
在光滑有界区域Ω \subset R^N上，我们考虑薛定谔算子-Δ-V，其中V是Dirichlet边界条件下的临界边界势V（x）=（N-2）^2/4 |x| ^{-2}或V（x）=（1/4）dist（x，\partial Ω）^{-2}。在这项工作中，我们得到尖锐的双边估计相应的热核。为此，我们将Scrödinger算子转化为合适的退化算子，为此我们证明了一个新的抛物Harnack不等式的边界。为了得到Harnack不等式，我们建立了一系列新的不等式，如改进的Hardy不等式，对数HardySobolev不等式，Hardy-Moser不等式和加权Poincaré不等式.作为我们技术的一个副产品，我们能够肯定地回答E. B. Davies的一个猜想。
We use some results from the theory of Reproducing Kernel Hilbert Spaces to show that the reachable space of the heat equation for a finite rod with either one or two Dirichlet boundary controls is a RKHS of analytic functions on a square, and we compute its reproducing kernel. We also show that the null reachable space of the heat equation for the half line with Dirichlet boundary data is a RKHS of analytic functions on a sector, whose reproducing kernel is (essentially) the sum of pullbacks of the Bergman and Hardy kernels on the half plane $\mathbb{C}^+$. We also consider the case with Neum…
We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.
Let $V$ be a finite set, $E \subset 2^{V} $ be a set of hyperedges, and $w : E \to (0, \infty)$ be an edge weight. On the (wighted) hypergraph $G = (V ,E ,w )$, we can define a multivalued nonlinear operator $L_{G,p}$ ($p \in [1 ,\infty )$) as the subdifferential of a convex function on $\mathbb{R} ^V $, which is called "hypergraph $p$-Laplacian." In this article, we first introduce an inequality for this operator $L_{G,p}$ which resembles the Poincaré-Wirtinger inequality in PDEs. Next we consider an ordinary differential equation on $\mathbb{R} ^V $ governed by $L_{G,p}$, which is referred a…
In this paper, we consider solutions of the backward heat equation with Ricci flow on manifolds as a type of infinite dimensional limit of solutions of a wave equation on a larger manifold with an analysis of wavefront set. Specifically, the projection of the solution of the wave equation $ \left(\frac{2t}{N} \cdot \frac{\partial^{2}}{\partial t^{2}} + \frac{tR(t, x)}{N} \frac{\partial}{\partial t} - Δ_{\tilde{g}^{(N)}(t)} \right) u = R(t, x) $ outside its wavefront set onto $ \mathbb{R}_{t} \times M_{x, g(t)} $ solves the backward heat equation $ \partial_{t} u + Δ_{x, g(t)} u = -R(t, x) $ wi…
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