Peter Bella, Alberto Chiarini, Benjamin Fehrman
A first-order Liouville theorem is obtained for random ensembles of uniformly parabolic systems under the mere qualitative assumptions of stationarity and ergodicity. Furthermore, the paper establishes, almost surely, an intrinsic large-scale $C^{1,α}$-regularity estimate for caloric functions.
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We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel [Arxiv preprint 1410.4483, 2014] on the coefficient field $a$ and its inverse, we prove an intrinsic large-scale $C^{1,α}$-regularity estimate for $a$-harmonic functions and obtain a first-order Liouville theorem for subquadratic $a$-harmonic functions.
Liouville theorems and universal estimates for superlinear parabolic problems without scale invariance
本文针对无尺度不变性的超线性抛物问题，利用椭圆刘维尔定理、能量估计和分量比例化约化方法建立全空间与半空间中的刘维尔型定理，并将其应用于普适奇异性与衰减估计。
We consider the Hardy-Hénon parabolic equation $u_t-Δu =|x|^a |u|^{p-1}u$ with $p>1$ and $a\in {\mathbb R}$. We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial boundary value problem.
We consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t σ_k (D^2u) =1$ in $\mathbb{R}^n\times (-\infty, 0]$. We derive that any \textbf{$k+1$-convex-monotone} solution to $-u_t σ_k (D^2u) =1$ when $u(x,0)$ satisfies a quadratic growth and $0<m_1\le -u_t\le m_2$ must be a linear function of $t$ plus a quadratic polynomial of $x$.
Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces
本文在光滑度量测度空间上，针对含Witten Laplace算子的非线性抛物方程，建立Souplet-Zhang、Hamilton与Li-Yau型梯度估计，并由此导出Harnack不等式及Liouville型常值性结果。
In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tσ_k(λ(D^2u))=ψ(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tσ_k(λ(D^2u))=1$ in $\mathbb{R}^n\times(-\infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u \to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and obtain some new …
Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data
本文证明了抛物Monge-Ampère方程周期数据下古老解的Liouville定理，建立了光滑周期解存在性的充要条件，扩展了Caffarelli-Li的结果。
In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm Liouville problem in a general star graph with mixed Dirichlet and Ne…
We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.
Liouville type theorems and periodic solutions for the nonhomogeneous parabolic systems
本文研究了具非齐次非线性项的$χ^{(2)}$型抛物型系统，通过精细积分估计和修改尺度变换与爆破论证，推导出刘维尔型定理、周期解存在性及通用界与衰减估计，是该领域的首个结果。
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. In the case of the nonlinear heat equation $$u_t-Δu=u^p\quad\hbox{in}\quad {R}^n\times{R}, \qquad p>1, $$ the nonexistence of positive classical solutions in the subcritical range $p(n-2)<n+2$ has been conjectured for a long time, but all known results require either a more restrictive assumption on $p$ or deal with a special cl…
We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for ergodic functions in bounded domains related to degenerate/singular operators, and, as a further consequence, we deduce the uniqueness of the ergodic functions.
We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive classical solutions) for the following model problems: the scalar nonlinear heat equation $$ u_t-Δu=u^p \qquad\hbox{in}\ {\mathbb R}^n\times{\mathbb R}, $$ its vector-valued generalization with a $p$-homogeneous nonlinearity and the linear heat equation in ${\mathbb R}^n_+\times{\mathbb R}$ complemented by nonlinear boundary conditions of the form $\partial u/\par…
We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $λβe^{βu }$, forced by an additive space-time white noise. We prove local and global well-posedness of these equations, depending on the sign of $λ$ and the size of $β^2 > 0$, and invariance of the associated Gibbs measures. See the abstract of the paper for a more precise abstract. (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here.)
A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source
本文研究含非线性梯度源的半线性热方程，通过逐点梯度估计、Gidas-Spruck 积分法和 doubling lemma 证明 Liouville 型定理并导出普适先验估计。
We are concerned with a space-time fractional parabolic initial-boundary value problem of Sturm Liouville type in a general star graph with mixed Dirichlet and Neumann boundary controls. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. Using the notion of no-regret control introduced by Lions, we prove the existence, uniqueness, and characterize the low regret control of a quadratic boundary optimal control problem, then we prove that this low regret control converges to the no-regret control and we provide the associated optimality systems an…
We study homogenization for fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media from the qualitative and quantitative perspective. We show that under suitable hypotheses, solutions to fully nonlinear uniformly parabolic equations in spatio-temporal media homogenize almost surely. In addition, we obtain a rate of convergence for this homogenization in measure, assuming that the environment is strongly mixing with a prescribed rate. A general methodology to study the stochastic homogenization of uniformly elliptic equations was introduced by Caffarelli, Soug…
We consider positive solutions of cooperative parabolic Lotka-Volterra systems with equal diffusion coefficients, in bounded and unbounded domains. The systems are complemented by the Dirichlet or Neumann boundary conditions. Under suitable assumptions on the coefficients of the reaction terms, these problems possess both global solutions and solutions which blow up in finite time. We show that any solution $(u,v)$ defined on the time interval $(0,T)$ satisfies a universal estimate of the form $$u(x,t)+v(x,t)\leq C(1+t^{-1}+(T-t)^{-1}),$$ where $C$ does not depend on $x,t,u,v,T$. In particular…
An analog of the Tricomi problem for a mixed type equation with Riemann-Liouville fractional derivative
本文研究混合域内抛物-双曲型方程的Tricomi问题，该方程抛物部分含Riemann-Liouville分数阶导数，双曲部分为第二类退化双曲方程，拓展了分数阶混合型方程的边值问题研究。
We prove a Liouville-type theorem for semilinear parabolic systems of the form $${\partial_t u_i}-Δu_i =\sum_{j=1}^{m}β_{ij} u_i^ru_j^{r+1}, \quad i=1,2,...,m$$ in the whole space ${\mathbb R}^N\times {\mathbb R}$. Very recently, Quittner [{\em Math. Ann.}, DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for $m=2$ in dimension $N\leq 2$, and partial results in higher dimensions in the range $p< N/(N-2)$. By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions $N\geq 3$. In particul…
A Liouville theorem for ancient solutions of the parabolic Monge-Ampère equation with periodic data
本文研究了抛物 Monge-Ampère 方程的古典型解，证明其形式为 -τt + p(x) + v(x,t)，扩展了 Caffarelli-Li 和 Zhang-Bao 的研究成果。
Liouville-type theorems for fully nonlinear elliptic and parabolic equations with boundary degeneracy
本文研究完全非线性边界退化椭圆与抛物型方程，证明算子退化生成隐式边界条件，且除平凡解外不存在有界解，方法借鉴 Hamilton-Jacobi 方程黏性解唯一性论证。
Ergodicity for stochastic T-monotone parabolic obstacle problems
本文研究了由Sobolev空间上T-单调算子控制、受乘性噪声驱动的障碍问题，证明了其解定义了Markov-Feller半群，并在适当假设下确立了遍历不变测度的存在性与唯一性。
Gradient estimate and Liouville theorem for a semilinear parabolic equation with variable coefficient
本文针对完备黎曼流形上带变系数的半线性抛物方程，通过Nash-Moser迭代建立正解的Li-Yau型梯度估计，进而推导古老解与永恒解的刘维尔定理，推广了相关经典结果。
We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*) $$ in the half-space ${\mathbb S} = {\mathbb R}^1_+ \times \mathbb R^n$. Here $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i} [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}],$$ where $a_{ij}(t,x)$, $i,j=1,...,n$, are functions defined, measurable and locally bounded in $\mathbb S$, and such that $a_{ij}(t,x)=a_{ji}(t,x)$ …
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.
In this paper, we give a more physical proof of Liouville's theorem for a class generalized harmonic functions by the method of parabolic equation.
We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far from power like. In this enlarged framework, by adapting the doubling-rescaling method from [37, 38], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale invariant nonlinearities. This leads to a number of ne…
We obtain a new Liouville comparison principle for weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ in the whole space ${\mathbb E} = {\mathbb R} \times \mathbb R^n$. Here, $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i}\left [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}\right],$$ where $a_{ij}(t,x)$, $i,j=1,\dots,n$, are functions that are defined, measurable and locally bounded in $\mathbb E$, and such that $a_{ij}(t,x)=a_…
One-side Liouville Theorem for hypoelliptic Ornstein--Uhlenbeck operators having drifts with imaginary spectrum
本文证明了线性漂移项具虚谱的次椭圆Ornstein-Uhlenbeck方程非负解的刘维尔定理，回答了Priola和Zabczyk的问题，方法基于Kolmogorov方程解在t=-∞处的刘维尔性质，而该性质源于新抛物型Harnack型不等式。
In this paper, we establish a local gradient estimate for a $p$-Lpalacian equation with a fast growing gradient nonlinearity. With this estimate, we can prove a parabolic Liouville theorem for ancient solutions satisfying some growth restriction near infinity.
We consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions, and on the entire real line, with a general nonlinearity $g(t,x,u,u_x)$ either not depending on $t$, or periodic in $t$. While the topological and geometric structure of their attractors has been investigated in depth, we focus here on ergodic-theoretical properties. The main result is that the union of supports of all the invariant measures projects one-to-one to $\mathbb{R}^2$. We rely on a novel application of the zero-number techniques with respect to evolution of measures …
Liouville theorems for parabolic systems with homogeneous nonlinearities and gradient structure
该研究修改既有证明思路，去掉正性假设证明了带梯度结构的齐次非线性抛物方程组的Liouville定理，将解类拓展至变号解，还证明了半空间Dirichlet边界情形的定理，并用于推导特定抛物方程组的通用估计。
We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, we obtain the Liouville-type theorem in the case when the coefficients may grow at the spacial infinity. We also study a corresponding parabolic problem in the $n$-dimensional whole space and discuss the energy identity for solutions in $L^q$. As a co…
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 $\partial u/\partialν=u^q$, $q>1$.
We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was proved in [24] and [12] in this case the homogenized operator is deterministic. The paper focuses on non-diffusive scaling, when the oscillation in spatial variables is faster than that in temporal variable. Our goal is to study the asymptotic behaviour of the normalized difference between solutions of the original and the homogenized problems.
A Liouville Theorem and Radial Symmetry for dual fractional parabolic equations
本文研究对偶分数阶抛物型方程，证明了正有界解的径向对称性，并通过引入新扰动技术建立了全空间齐次方程及Marchaud分数阶方程的Liouville定理。
Liouville theorems and removable sets for bounded $p$-harmonic and quasiharmonic functions on metric spaces under local assumptions
本文针对满足局部p-庞加莱不等式等条件的度量空间，刻画了有界p-调和函数与拟调和函数的可去紧集，揭示可去性与刘维尔型定理等价，确定了相关关键性质并给出刘维尔定理的初等证明，结果适用于流形等空间。
In this paper, we establish blow-up rates for higher-order semilinear parabolic equations with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions.
We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure some qualitative properties of solution such as strong maximum principle, Harnack's inequality, Liouville's theorem. The answers are given in terms of the Lebesgue spaces and the Morrey spaces.
Asymptotic decomposition of solutions to parabolic equations with a random microstructure
研究具有空间周期性和时间随机平稳遍历系数的散度型二阶抛物算子的柯西问题，得到解的渐近展开首项（确定性函数），并证明适当重正化的解与首项之差收敛于某个SPDE的解。
We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.
Liouville theorems for mixed local and nonlocal indefinite equations
本文研究了混合局部-非局部方程正解的定性性质，通过建立最大原理和严格单调性，结合平滑第一特征函数和合适子解，推导了混合算子(-Δ)^s - Δ的非存在性结果，并扩展到抛物情形，揭示了双重非局部性下的新定性特征。
Gradient estimates and Liouville theorems for the \(Φ\)-Laplacian equations on Riemannian manifolds
本文通过非线性 Φ-Bochner 公式和 Nash-Moser 迭代，在 Ricci 曲率下界和 Φ、G 的适当条件下，建立了 Φ-Laplacian 方程的局部梯度界，进而得到全局解的 Liouville 定理；对于抛物情形，利用最大值原理在紧流形上导出梯度估计并获得 Liouville 型结果。
In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.
This article establishes an algebraic error estimate for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. The approach is similar to that of Armstrong and Smart in the study of quantitative stochastic homogenization of uniformly elliptic equations.
On nonnegative solutions of the parabolic differential inequality with $(p,q)$-Laplace on Riemannian manifolds
本文研究了Riemannian流形上带有$(p,q)$-拉普拉斯算子的抛物型微分不等式中的Liouville型定理，通过测试函数方法，在合适的加权体积增长假设下得出非存在性结果，扩展了抛物型不等式的非存在性结论。
We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in \cite{bcr}. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive prob…
A discretization for the nonlinear parabolic evolution equation of fractional order in space
本文提出分数阶空间非线性抛物型方程的数值离散化方法，采用泛函分析设定定义分数导数，通过数值实验验证方法有效性并提出猜想。
We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^p$-Liouville theorems in terms of the lower bound of Bakry-Émery Ricci curvature and the bound of function $f$, which generalize the classical Ricci curvature case and the $N$-Bakry-Émery Ricci curvature case.
Entire solutions to the parabolic Monge--Ampère equation with unbounded nonlinear growth in time
本文研究抛物型蒙日-安培方程在时间导数无界情形下的整解，证明了整抛物凸光滑解的存在性并分析其无穷远渐近行为。
We consider a random conductance model on the $d$-dimensional lattice, $d\in[2,\infty)\cap\mathbb{N}$, where the conductances take values in $(0,\infty)$ and are however not assumed to be bounded from above and below. We assume that the law of the random conductances is stationary and ergodic with respect to translations on $\mathbb{Z}^d$ and invariant with respect to reflections on $\mathbb{Z}^d$ and satisfies a similar moment bound as that by Andres, Deuschel, and Slowik (2015), under which a quenched FCLT holds. We prove a first-order Liouville theorem. In the proof we construct the subline…
This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environmen…
Convergence of The Relative Value Iteration for the Ergodic Control Problem of Nondegenerate Diffusions under Near-Monotone Costs
针对漂移控制的非退化扩散过程，在近单调运行成本结构下，研究相对值迭代算法（拟线性抛物型柯西初值问题）的解对任意有界初始条件收敛到遍历控制问题对应的Hamilton-Jacobi-Bellman方程的解。
Blow up analysis for a parabolic MEMS problem, I: Hölder estimate
本文针对抛物型 MEMS 方程的淬灭现象，运用爆破方法与 Liouville 定理建立解的最优 Hölder 估计及一致连续解序列的收敛理论，并据此证明破裂集分层定理。
We consider a boundary value problem in the half-space for a linear parabolic equation of fourth order with a degeneration on the boundary of the half-space. The equation under consideration is substantially a linearized thin film equation. We prove that, if the right hand side of the equation and the boundary condition are polynomials in the tangential variables and time, the same property has any solution of a power growth. It is shown also that the specified property does not apply to normal variable. As an application, we present a theorem of uniqueness for the problem in the class of func…
A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations
本文证明半空间中超二次扩散Hamilton-Jacobi方程的刘维尔型分类定理，断言解必为一维，并以此研究一般有界区域抛物问题的边界梯度爆破行为，揭示法向普适爆破剖面、切向更奇异行为及粘性解延拓后边界条件的丢失。
Liouville type theorems for dual nonlocal evolution equations involving Marchaud derivatives
本文通过傅里叶分析方法，建立了双非局部演算方程中涉及Marchaud导数的Liouville型定理，证明了在特定条件下所有分布解必须为常数，并提供了关于衰减率的最优估计。
Let $σ>1$ and let $M$ be a complete Riemannian manifold. In a recent work [9], Grigor$^{\prime}$yan and Sun proved that a pointwise upper bound of volume growth is sufficient for uniqueness of nonnegative solutions of elliptic inequality $$(*)\quad\qquad\qquad\qquad Δu(x)+u^σ(x)\leq 0,\qquad x\in M.\quad\qquad \qquad\qquad $$ In this note, we improve their result to that an \emph{integral condition} on volume growth implies the same uniqueness of ($*$). It is inspired by the well-known Varopoulos-Grigor$^{\prime}$yan's criterion for parabolicity of $M$.
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