Vedansh Arya, Agnid Banerjee, Donatella Danielli, Nicola Garofalo
In this paper we establish the \emph{space-like} strong unique continuation for nonlocal equations of the type $(\partial_t - Δ)^s u= Vu$, for $0<s <1$. The proof of our main result, Theorem 1.1, is achieved via a conditional elliptic type doubling property for solutions to the appropriate extension problem, followed by a blowup analysis.
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We establish a strong unique continuation property for stochastic parabolic equations. Our method is based on a suitable stochastic version of Carleman estimate. As far as we know, this is the first result for strong unique continuation property of stochastic partial differential equations.
Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients
本文研究了具有非光滑系数的二阶抛物型方程的强唯一延拓问题。导言和参考书目已经修订。
In this paper, we establish space like strong unique continuation property (sucp) for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established by Ruland in [Ru] for analogous elliptic sublinear equations. Similar to that in [Ru], this is accomplished via a new $L^{2}-L^{2}$ type Carleman estimate for a class of sublinear parabolic operators.
It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is global in time and we give an example when it never becomes a classical solution again.
In this article, for a fourth-order parabolic equation which is closely related for example to the Cahn-Hilliard equation, we study an inverse source problem by interior data and the continuation of solution from lateral Cauchy data. Our method relies on a Carleman estimate and proves conditional stability for both problems.
In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality \[ |\operatorname{div} (A(x,t) \nabla u) - u_t| \leq \frac{M}{|x|^{2}} |u|,\ \ \ \ \] where the coefficient matrix $A$ is Lipschitz continuous in $x$ and $t$. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends a recent result of one of us with Garofalo and Manna for the heat operator.
We address the quantitative uniqueness properties of the solutions of the parabolic equation $ \partial_t u - Δu = w_j (x,t) \partial_j u + v(x,t) u $ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of vanishing is bounded by $C(\Vert v\Vert_{L^\infty}^{2/3}+\Vert w\Vert_{L^\infty}^2)$ matching the upper bound previously established in the elliptic case. in the elliptic case.
Carleman inequalities for fractional Laplacians and unique continuation
我们得到了Morrey空间中具有位势的分数阶Schr dinger算子的唯一延拓结果。这是基于分数拉普拉斯算子的Carleman不等式。
Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates
本文综述了由抛物型方程的柯西数据确定热传导体未知边界部分的稳定性问题的主要结果。我们给出了详细和完备的证明。我们证明了这样的问题是严重不适定的意义下，根据先验的规则性假设的未知边界，任何有穷级的可微性，连续依赖的未知边界从测量数据是，充其量，对数型。
In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lamé operator $\mathbb{H}$ and we then study the extension problem associated to such non-local operators. We also study the various regularity properties of solutions to such an extension problem via a transformation which reduces the extension problem for the parabolic Lamé operator to another system that resembles the extension problem of the fractional heat operator. Finally in the case when $s \geq 1/2$, by proving a conditional doubling property for solutions to the corresponding reduced…
This article contains a new discussion for the generalized fractional Cauchy-type problem involving Hilfer-Katugampola-type fractional derivative. We study an existence and continuation of its solution. Firstly, we establish a new theorems of local existence. Then, we deduce a continuation theorems for a general fractional differential equations. By applying continuation theorems deduced in this article, we present several global existence results. Moreover, the examples are given to illustrate our main results.
We establish the strong unique continuation property of fractional orders of linear elliptic equations with Lipschitz coefficients by establishing monotonicity of some Almgren-type frequency functional via an extension procedure.
In this paper, we first establish Hopf's lemmas for parabolic fractional equations and parabolic fractional $p$-equations. Then we derive an asymptotic Hopf's lemma for antisymmetric solutions to parabolic fractional equations. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.
Unique Continuation for Fractional Schrödinger Equations with Rough Potential
利用Carleman估计，研究了具有标度临界势和粗糙势的分数阶薛定谔方程的弱唯一延拓原理和强唯一延拓原理.我们的方法允许应用的结果变系数版本的分数薛定谔方程。
In this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth order equations and higher order fractional problems.
We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in general. While discretization thus counteracts the strong rigidity properties of the continuum fractional Laplacian, by discussing quantitative forms of unique continuation, we illustrate that these properties can be recovered if exponentially small (in the lattice size) correctio…
The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation
本文通过解析延拓构造卡诺群中亚拉普拉斯算子的分数次幂，刻画海森堡群中该算子的幂次并计算热核k阶矩，为相关分析提供新方法与结果。
Unique Continuation for Stochastic Parabolic Equations
本文研究随机抛物型方程的唯一连续性。由于随机情形下解的适应性，经典的处理确定性方程唯一延拓问题的方法不起作用。我们的方法是基于一个合适的部分Holmgren坐标变换和随机版本的Carleman型估计。
Unique continuation for fractional Schrödinger operators in three and higher dimensions
证明了微分不等式$|(-Δ)^{α/2}u|\leq|V(x)u|$的唯一连续性，其中$0<α<n$和$V\in L_{\textrm{loc}}^{n/α,\infty}(\mathbb{R}^n)$，$n\geq3$ .
This paper deals with the unique continuation of solutions for a one-dimensional anomalous diffusion equation with Caputo derivative of order $α\in(0,1)$. Firstly, the uniqueness of solutions to a lateral Cauchy problem for the anomalous diffusion equation is given via the Theta function method, from which we further verify the unique continuation principle.
In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrödinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calderón type problems. As our m…
In this paper, we obtain the sharp uniqueness for an inverse $x$-source problem for a one-dimensional time-fractional diffusion equation with a zeroth-order term by the minimum possible lateral Cauchy data. The key ingredient is the unique continuation which holds for weak solutions.
Unique continuation property and local asymptotics of solutions to fractional elliptic equations
研究了具有Hardy型位势的分数阶椭圆型方程解的渐近性。利用Almgren型单调性公式、分离变量和blow-up参数，描述了具有与分数Hardy不等式相关的齐次奇异位势的线性和半线性分数阶椭圆型方程解在奇异点附近的精确行为.从而得到了分数阶椭圆型方程的唯一延拓性质。
On unique continuation for Schrödinger operators of fractional and higher orders
本文研究了$|(-Δ)^{α/2}u|\leq|Vu|$的解的唯一延拓性质，其中$V$属于包含$\bigcup_{p>n/α}L^p(\mathbb{R}^n)$的函数类，其中$n-1\leqα<n$为$\bigcup_{p>n/α}L^p(\mathbb{R}^n)$.特别是，当$n=2$时，我们的结果给出了分数（$1<α<2$）薛定谔算子$(-Δ)^{α/2}+V(x)$在整个$α$值范围内的唯一延拓定理。
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
In this paper we prove strong unique continuation principle and unique continuation from sets of positive measure for solutions of a higher order fractional Laplace equation in an open domain. Our proofs are based on the Caffarelli-Silvestre extension method combined with an Almgren type monotonicity formula. The corresponding extended problem is formulated as a systems of two second order equations with singular or degenerate weights in a half-space, for which asymptotics estimates are derived by a blow-up analysis.
In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-Δ)^s+Q$ with $0<s<1$ in any space dimension. We uniquely determine the unknown bounded potential $Q$ from infinitely many exterior Dirichlet-to-Neumann type measurements. This relies on Runge approximation and the dual global weak unique continuation properties of the equation under consideration. In discussing weak unique continuation of our operator, a main feature of our argument relies on a Carleman estimate for the associated fractional parabolic Caffarelli-Silvestre extension. Furthe…
Strong Unique Continuation for Fractional Schrödinger Operators
该研究证明了0<s<1时，具尺度临界位势的分数阶薛定谔方程的强唯一延拓性质，是经典薛定谔算子Jerison-Kenig结果的非局部对应。
We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulae and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points.
This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain. This system is a simplified version of the chemotaxis model introduced by Keller and Segel. With the aid of priori L^infty-estimates (for solutions of the system) built up in this paper, we treat the semi-linear parabolic equation in the system as a linear parabolic equation, and then use the frequency function method and the localization technique to build up two unique continuation inequalities for the system. As a consequence of the above-mentioned two inequalitie…
A strong-type unique continuation principle for the fractional $p$-Laplacian
本文通过简单直接的方法证明了分数 $p$-Laplacian 的强型唯一持续原理，并将其推广到分数非线性薛定谔方程的强解中。
We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Pa…
On unique continuation in measure for fractional heat equations
本文研究非局部分数阶热方程的唯一延拓问题，证明 $0<s<1$ 时方程的测度唯一延拓定理，将局部情形 $s=1$ 的结果推广到非局部框架。
Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations
在适当的加权Lebesgue空间中，研究了一类分数阶抛物型和椭圆型方程解的唯一性。
In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated from a recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.
A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order $α\,\,(0<α<2)$ and general time dependent second order strongly elliptic time elliptic operator for the diffusion. The estimate is derived via some subelliptic estimate for an operator associated to this equation using calculus of pseudo-differential operators. A special Holmgren type transformation which is linear with respect to time is used to show the local unique continuation of solutions. We developed a new argument to derive the global unique continuatio…
We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of fractional parabolic and elliptic equations with a drift.
We introduce new lower bounds for the fractional Fisher information. Equipped with these bounds we study a hyperbolic-parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes.
We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Holder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative approaches 1.
A Parabolic Problem with a Fractional-Time Derivative
研究了一类具有分数阶时间导数和空间分数阶扩散的抛物型方程的正则性。我们的主要结果是散度型方程解的De Giorgi-Nash-Moser Holder正则性定理。我们还证明了结果的存在性，唯一性和更高的规律性的时间。
Domains of dependence for subelliptic wave equations and unique continuation for fractional powers of Hörmander's operators
本文证明了满足Hörmander括号条件的次椭圆波动方程的精确依赖区域性质，并在附加解析性条件下推导出次椭圆Laplace算子平方根及更一般分数次幂的唯一延拓性质。
We consider time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and variable-coefficient variable-order multiterm time-fractional parabolic equations.
Symmetry and monotonicity of solutions to elliptic and parabolic fractional $p$-equations
该研究建立了抛物型分数阶$p$-方程的反对称函数最大值原理，以此给出含梯度项的椭圆型分数阶$p$-方程解对称性与单调性的新证法，并得到全空间与有界域上非线性抛物型分数阶$p$-方程正解在有限时刻的对称性与单调性，相关结果可推广至含非线性非局部算子的椭圆、抛物问题。
In this paper, we establish the existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces for a large class of initial data including distributions other than Radon measures. We also obtain sufficient conditions for the existence of solutions to viscous Hamilton-Jacobi equations.
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.
We study a class of fractional parabolic equations involving a time-dependent magnetic potential and formulate the corresponding inverse problem. We determine both the magnetic potential and the electric potential from the exterior partial measurements of the Dirichlet-to-Neumann map.
Regularity of solutions to the parabolic fractional obstacle problem
在本文中，我们研究了一个抛物型版本的分数障碍问题，证明了几乎最优的正则性的解决方案。这个问题是由Menton提出的美式期权模型所激发的，该模型将股票价格动态中的不连续路径引入期权评估理论。
This article is concerned with the unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $Ω$ prescribed with some regularity and growth conditions. Our result shows that the value of the solutions can be determined uniquely by its value on an arbitrary open subset $ω$ in $Ω$ at any given positive time $T$. We also derive the quantitative nature of this unique continuation, that is, the estimate of a $L^2(Ω)$ norm of the initial data on $Ω$, which is majorized by that of solution on the bounded open subset $ω$ at termin…
In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an opti…
We study the strong unique continuation property backwards in time for the nonlocal equation in $\mathbb{R}^{n} \times \mathbb{R}$ \begin{equation}\label{one} (\partial_t - Δ)^{s} u = V(x,t)u \end{equation} for $s \in (0,1)$. Our main result Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained by Poon for the case when $s=1$. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation above. With such theory in hands we establish: i) a basic monotonicity result for an adjusted frequency function which plays a central role…
Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients
研究了一类在无穷远（空间）满足点态条件的分数阶抛物型方程解的存在唯一性，该方程可以是不定常的。此外，我们还研究了这类解的渐近性态。我们还考虑了在无穷远处满足适当条件的椭圆型方程的解。
Unique continuation from conical boundary points for fractional equations
本文研究分数阶椭圆方程在锥形边界点附近的解渐近行为，通过光滑化、Pohozaev型不等式及Almgren单调性等方法，建立了强唯一延拓原理。
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.
Nonlinear parabolic problem with time fractional derivative
本文研究了带有双重奇异性硬y型势的p-拉普拉斯时间分数抛物型问题，证明了弱解的比较原理和先验估计，并探讨了全局弱解的存在性和有限时间爆破依赖于最优Hardy常数。
A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren-Poon monotonicity formula combined with a blow-up analysis.
We establish the $L_p$-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend a recent result in [6] to a large extent.
On fractional parabolic $\text{BMO}$ and $\text{Lip}_α$ caloric capacities
本文通过引入分数阶热容量并证明其与抛物型 Hausdorff 容量可比，刻画了分数阶热方程在抛物型 BMO 与 Lipα 规范化条件下解的可去集。
In this paper we introduce a natural function class and prove the existence and uniqueness of both nonnegative renormalized solutions and entropy solutions for the fractional p-Laplacian parabolic problem with L^1 data. And moreover, we obtain the equivalence of renormalized solutions and entropy solutions and establish a comparison result.
Parabolic Frequency on Gaussian Spaces and Unique Continuation
本文在高斯空间中研究抛物线频率，通过加权 $L^2$ 框架推导强唯一持续性原理，扩展了 Poon 的结果并补充了 Colding 和 Minicozzi 的几何推广。
We prove two maximal regularity results in spaces of continuous and Hölder continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^α$. This derivative is intended in the sense of Caputo and $α$ is taken in $(0, 2)$. In case $α= 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.
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