On the backward uniqueness property for a class of parabolic operators

Backward uniqueness for general parabolic operators in the whole space

We prove the backward uniqueness for general parabolic operators of second order in the whole space under assumptions that the leading coefficients of the operator are Lipschitz and their gradients satisfy certain decay conditions. This result extends in some ways a classical result of Lions and Malgrange [12] and a recent result of the authors [10].

Global in Time Asymptotic Solutions to Kolmogorov--Feller-Type Parabolic Pseudodifferential Equations with a Small Parameter. Forward and Backward in Time Motion

The goal of the present paper is to present a new approach to the construction of asymptotic (approximating) solutions to parabolic PDE by using the characteristics.

Uniqueness of renormalized solutions to nonlinear parabolic problems with lower order terms

In this paper we prove uniqueness results for renormalized solutions to a class of nonlinear parabolic problems.

时间反向的半线性耦合抛物型系统的稳定性

Stability of backward-in-time semilinear coupled parabolic systems

本文研究有界区域中半线性耦合抛物型系统的反向问题，利用带权函数$e^{λt}$的修正Carleman估计方法，证明了线性与半线性强耦合系统的条件稳定性估计。

Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations

We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of fractional parabolic and elliptic equations.

A Semi-Linear Backward Parabolic cauchy Problem with Unbounded Coefficients of Hamilton-Jacobi-Bellman Type and Applications to optimal control

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton-Jacobi-Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the Value Function of the controlled equation and that the feedback law is verified.

Backward Uniqueness of Kolmogorov Operators

The backward uniqueness of the Kolmogorov operator $L=\sum_{i,k=1}^n\partial_{x_i}(a_{i,k}(x,t)\partial_{x_k})+\sum_{l=1}^m x_l\partial_{y_l}-\partial_t$, was proved in this paper. We obtained a weak Carleman inequality via Littlewood-Paley decomposition for the global backward uniqueness.

Backward uniqueness for parabolic operators with non-Lipschitz coefficients

In this paper we study the backward uniqueness for parabolic equations with non-Lipschitz coefficients in time and space. The result presented here improves an old uniqueness theorem due to Lions and Malgrange [Math. Scand. ${\bf 8}$ (1960), 277--286] and some more recent results of Del Santo and Prizzi [J. Math. Pures Appl. ${\bf 84}$ (2005), 471--491; Ann. Mat. Pura Appl., to appear].

Uniqueness for fractional parabolic and elliptic equations with drift

We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of fractional parabolic and elliptic equations with a drift.

Uniqueness for degenerate parabolic equations in weighted $L^1$ spaces

We study uniqueness of solutions to degenerate parabolic problems, posed in bounded domains, where no boundary conditions are imposed. Under suitable assumptions on the operator, uniqueness is obtained for solutions that satisfy an appropriate integral condition; in particular, such condition holds for possibly unbounded solutions belonging to a suitable weighted $L^1$ space.

Two-phase entropy solutions of forward-backward parabolic problems with unstable phase

This paper study the two--phase problem for the forward-backward parabolic equation with diffusion function of cubic type. Existence and uniqueness for these kind of problems were obtained in literature in the case in which the phases are both stable. Here we consider the situation in which the unstable phase is taken in account, obtaining not trivial solution of the problem. It is interesting to note that such solutions are given by solving generalized Abel's equations.

Short time existence for a general backward-forward parabolic system arising from Mean-Field Games

We study the local in time existence of a regular solution of a nonlinear parabolic backward-forward system arising from the theory of Mean-Field Games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.

A new result on backward uniqueness for parabolic operators

Using Bony's paramultiplication we improve a result obtained in in a previous paper for operators having coefficients non-Lipschitz-continuous with respect to $t$ but ${\mathcal C}^2$ with respect to $x$, showing that the same result is valid when ${\mathcal C}^2$ regularity is replaced by Lipschitz regularity in $x$.

一类一般非强制非线性抛物型方程重整化解的存在性与唯一性

Existence and uniqueness for renormalized solutions to a general noncoercive nonlinear parabolic equation

本文引入包含奇性与无界低阶项的一般非强制非线性抛物问题的重整化解概念，并证明其存在性与唯一性。

Uniqueness of singular self-similar solutions of a semilinear parabolic equation

We study the uniqueness of singular radial (forward and backward) self-similar positive solutions of the equation $u_t-Δu = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ where $p\geq(n+2)/(n-2)_+$.

Quantitative uniqueness of solutions to parabolic equations

We investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth manifolds. Quantitative uniqueness is a quantitative form of strong unique continuation property. We characterize quantitative uniqueness by the rate of vanishing. We can obtain the vanishing order of solutions by $C^{1, 1}$ norm of the potential functions, as well as the $L^\infty$ norm of the coefficient functions. Some quantitative Carleman estimates and three cylinder inequalities are established.

Quasilinear parabolic stochastic partial differential equations: existence, uniqueness

In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone.

Continuous dependence and uniqueness for lateral Cauchy problems for linear integro-differential parabolic equations

Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.

Conditional stability up to the final time for backward-parabolic equations with Log-Lipschitz coefficients

We prove logarithmic conditional stability up to the final time for backward-parabolic operators whose coefficients are Log-Lipschitz continuous in $t$ and Lipschitz continuous in $x$. The result complements previous achievements of Del Santo and Prizzi (2009) and Del Santo, Jaeh and Prizzi (2015), concerning conditional stability (of a type intermediate between Hoelder and logarithmic), arbitrarily closed, but not up to the final time.

Non-uniqueness results for entropy two-phase solutions of forward-backward parabolic problems with unstable phase

This paper study the well--posedness of the entropy formulation given by Plotnikov in [{Differential Equations}, 30 (1994), pp. 614--622] for forward-backward parabolic problem obtained as singular limit of a proper pseudoparabolic approximation. It was proved in [C. Mascia, A. Terracina, and A. Tesei, {Arch.\ Ration.\ Mech.\ Anal.}, 194 (2009), pp. 887--925] that such formulation gives uniqueness when the solution takes values in the stable phases. Here we consider the situation in which unstable phase is taken in account, proving that, in general, uniqueness does not hold.

Hardy-Hénon 抛物方程的无条件唯一性与非唯一性

Unconditional uniqueness and non-uniqueness for Hardy-Hénon parabolic equations

针对 Hardy-Hénon 抛物方程，采用加权 Lorentz 空间作为解空间，证明了无条件唯一性与非唯一性，并建立唯一性准则，推广了 Fujita 方程和 Hardy 方程的已有结果。

抛物方程中Hölder势的定量化唯一性

Quantitative uniqueness for parabolic equations with Hölder potentials

本文研究了具有Hölder零阶项的抛物算子的空间-like定量化唯一性，扩展了Teng等人关于时间不变Schrödinger算子的研究。

Uniqueness for weak solutions of parabolic equations with a fractional time derivative

We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.

On the Backward Uniqueness Property for the Heat Equation in Two-Dimensional Conical Domains

In this article we deal with the backward uniqueness property of the heat equation in conical domains in two spatial dimensions via Carleman inequality techniques. Using a microlocal interpretation of the pseudoconvexity condition, we improve the bounds of Šverák and Li on the minimal angle in which the backward uniqueness property is displayed: We reach angles of slightly less than $95^{\circ}$. Via two-dimensional limiting Carleman weights we obtain the uniqueness of possible controls of the heat equation with lower order perturbations in conical domains with opening angles larger than $90^{…

Conditional stability for backward parabolic equations with $\rm{Log}\rm{Lip}_t \times \rm{Lip}_x$-coefficients

In this paper we present an improvement of [Math. Ann. 345 (2009), 213--243], where the authors proved a result concerning continuous dependence for backward parabolic operators whose coefficients are Log-Lipschitz in $t$ and $C^2$ in $x$. The $C^2$ regularity with respect to $x$ had to be assumed for technical reasons. Here we remove this assumption, replacing it with Lipschitz-continuity. The main tools in the proof are Littlewood-Paley theory and Bony's paraproduct as well as a result of Coifman and Meyer [Astérisque 57, 1978, Th. 35].

Quasilinear parabolic problem with $p(x)$-Laplacian: existence, uniqueness of weak solutions and stabilization

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-Δ_{p(x)}u = f(x,u)$ in $ (0,T)\timesΩ$; $u = 0$ on $(0,T)\times\partialΩ$; $u(0,x)=u_0(x)$ in $Ω$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.

Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems

Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.

On quantitative uniqueness for parabolic equations

We consider the quantitative uniqueness properties for a parabolic type equation $ u_t-Δu = w(x,t) \nabla u + v(x,t) u$, when $v \in L^{p_2}_{t} L^{p_1}_x$ and $w \in L^{q_2}_{t} L^{q_1}_x$, with a suitable range for exponents $p_1$, $p_2$, $q_1$, and $q_2$. We prove a strong unique continuation property and provide a pointwise in time observability estimate.

度量图上椭圆与抛物型方程解的唯一性

Uniqueness of solutions to elliptic and parabolic equations on metric graphs

本文研究了度量图上特定椭圆与抛物型方程解的唯一性，针对带势的线性 Schrödinger 方程与变密度热方程，通过设定与势或密度在无穷远处渐近行为相关的适当解增长条件，证明了其解的唯一性。

关于抛物型偏微分方程解的正则性与唯一性的若干问题

Some questions about the regularity and the uniqueness of solutions of parabolic partial differential equations

本文基于热问题解导出线性抛物型偏微分问题的逐点不动点方程，利用非标准技术研究解的符号，由此推出抛物型问题解的正则性，并证明了三维Navier-Stokes问题解的唯一性。

Existence and uniqueness of solutions to parabolic equations with superlinear Hamiltonians

We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on classical techniques for uniformly parabolic quasilinear equations and on the Lipschitz estimates proved in [S.N. Armstrong and H.V. Tran, Viscosity solutions of general viscous Hamilton-Jacobi equations, Math. Ann., 361 (2015)], as well as on viscosity solution arguments.

A note on the uniqueness from sets of positive measure for time dependent parabolic operators

The purpose of this short note is to show how it is possible to combine existing results in the literature to get the unique continuation from sets of positive measure for time dependent parabolic equations with Lipschitz principal part and bounded lower order terms, result which was known in the case of analytic coefficients in [3].

Backward Uniqueness for a PDE Fluid-Structure Interaction

In this work, we establish the so-called backward uniqiueness property for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a boundary interface with a two-dimensional mechanical plate equation. By way of attaining this result, a certain estimate is obtained for the associated semigroup generator resolvent.

On Lipschitz solutions for some forward-backward parabolic equations

We investigate the existence and properties of Lipschitz solutions for some forward-backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Finally, we study two important cases of forward-backward anisotropic diffusion in which the density condition can be realized and therefo…

A Necessary and Suffcient Condition for Uniqueness of the Trivial Solution in Semilinear Parabolic Equations

In their 1968 paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. In particular they showed that if the underlying ODE has non-unique solutions (as characterised via an Osgood-type condition) {\em and} the nonlinearity $f$ satisfies a concavity condition, then the parabolic PDE also inherits the non-uniqueness property. This concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other, …

非线性前向-后向抛物型方程的自相似解

Self-Similar Solutions to a Nonlinear Forward-Backward Parabolic Equation

本文研究源于边界层分离的混合型方程 u u_x = u_{yy}，通过相似变量将其化为常微分方程组并证明存在性，从而构造出带符号变化的自相似解。

Existence and uniqueness of solutions for a nonlocal parabolic thermistor-type problem

In this paper we prove existence and uniqueness of solutions to a nonlocal parabolic problem which generalizes the electric heating problem of a conducting body.

Backward parabolic Ito equations and second fundamental inequality

Regularity of solutions is studied for backward stochastic parabolic Ito equations. An analog of the second energy inequality and the related existence theorem are obtained for domains with boundary.

二维有界区域中由非局部、非线性乘性噪声驱动的抛物-抛物型Keller-Segel系统的全局存在性和路径唯一性

Global Existence and Pathwise Uniqueness for a Stochastic Parabolic-Parabolic Keller-Segel System

研究二维有界区域中由非局部、非线性乘性噪声驱动的抛物-抛物型Keller-Segel系统，通过引入定制截断系统、构建最大局部强解、引入特殊Lyapunov泛函等方法，建立了任意初始数据的强解的全局存在性和路径唯一性。

From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation

A rigorous proof is given for the convergence of the solutions of a viscous Cahn-Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Non-homogenous Neumann boundary condition are handled for the chemical potential and the subdifferential of a possible non-smooth double-well functional is considered in the equation. An error estimate for the difference of solutions is also proved in a suitable norm and with a specified rate of convergence.

退化抛物方程时间反向问题的稳定性

Stability for backward problems in time for degenerate parabolic equations

本文研究退化抛物方程的时间反向问题，在初值有界假设下建立条件稳定性，基于仅含时间权重的Carleman型加权$L^2$估计证明，并将方法推广至半线性情形。

Uniqueness for $L_{p}$-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms

In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be Hölder continuous in $x$ with exponent slightly less than $1/2$. This case is treated by using stability of maximal and minimal $L_{p}$-viscosity solutions.

Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation

The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane inthe general framework of weak (or "free energy") solutions associated to an initial datum with finite mass $M\textless{} 8π$, finite second log-moment and finite entropy. The aim of the paper is twofold:(1) We prove the uniqueness of the "free energy" solution. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the "optimal regularity" as well as an estimate of the difference of two possible solutions in the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$ vorticit…

非线性各向异性抛物方程解的唯一性、正则性及时间行为

Uniqueness, regularity and behaviour in time of the solutions to nonlinear anisotropic parabolic equations

本文研究有界域上非线性各向异性抛物问题，分析初始数据和强迫项对解的影响，证明解的存在唯一性，并特别关注自治情形下解的时间行为。

A non-smooth regularization of a forward-backward parabolic equation

In this paper we introduce a model describing diffusion of species by a suitable regularization of a "forward-backward" parabolic equation. In particular, we prove existence and uniqueness of solutions, as well as continuous dependence on data, for a system of partial differential equations and inclusion, which may be interpreted, e.g., as evolving equation for physical quantities such as concentration and chemical potential. The model deals with a constant mobility and it is recovered from a possibly non-convex free-energy density. In particular, we render a general viscous regularization via…

Uniqueness of limit flow for a class of quasi-linear parabolic equations

We investigate the issue of uniqueness of the limit flow for a relevant class of quasi-linear parabolic equations defined on the whole space. More precisely, we shall investigate conditions which guarantee that the global solutions decay at infinity uniformly in time and their entire trajectory approaches a single steady state as time goes to infinity. Finally, we obtain a characterization of solutions which blow-up, vanish or converge to a stationary state for initial data of the form $λφ_0$ while $λ>0$ crosses a bifurcation value $λ_0$.

Space-like quantitative uniqueness for parabolic operators

We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a $C{^1}$ potential $V$. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.

Backward uniqueness for parabolic operators with non-Lipscitz coefficients

We investigate the relation between the backward uniqueness and the regularity of the coefficients for a parabolic operator. A necessary and sufficient condition for uniqueness is given in terms of the modulus of continuity of the coefficients.

抛物型非局部1-Laplacian方程弱解的存在性与唯一性

Existence and uniqueness of weak solutions to a parabolic nonlocal 1-Laplacian equation

本文研究抛物型非局部1-Laplacian方程，采用Rothe时间离散化方法证明了其弱解的存在性与唯一性，并得出弱解在时间上保持1/2-Hölder连续性的新结论。

On the Dirichlet Problem for Backward Parabolic Stochastic Partial Differential Equations in General Smooth Domains

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces allowing the derivatives of the solutions to blow up near the boundary.

黎曼流形上抛物薛定谔方程的松弛唯一性条件

Relaxed uniqueness conditions for the parabolic Schrodinger equation on Riemannian manifolds

本文研究了完备非紧黎曼流形上抛物薛定谔方程 Cauchy 问题的解的唯一性，通过利用相关稳态方程正解的衰减性质，显著松弛了所需的积分可积性条件。

非线性抛物问题在孔隙介质中的存在性、唯一性和均质化

Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media

本文研究了非线性抛物问题在孔隙介质中的存在性、唯一性及均质化，推导了无孔域上的新问题并引入了额外项。

Duality and semi-group property for backward parabolic Ito equations

We study existence, uniqueness, semi-group property, and a priori estimates for solutions for backward parabolic Ito equations in domains with boundary. We study also duality between forward and backward equations. The semi-group for backward equations is established in the form of some anti-causality. The novelty is that the semi-group property involves the diffusion term that is a part of the solution.

Second order backward stochastic differential equations and fully non-linear parabolic PDEs

We introduce a class of second order backward stochastic differential equations and show relations to fully non-linear parabolic PDEs. In particular, we provide a stochastic representation result for solutions of such PDEs and discuss Monte Carlo methods for their numerical treatment.