(a). Using time analyticity result, we address a basic question for a nonhomogeneous backward heat equation (exact control problem) in the setting of smooth domains and compact manifolds, namely: when is essentially time independent control possible? i.e. The control function is 0 on one time interval and stationary on the other. For general $L^2$ initial values, the answer is: if and only if the full space domain is used for the control function. Also an explicit formula for the control function is found in the form of an infinite series involving the heat kernel, which converges rapidly.
(b). A formal exact formula for a time dependent control function supported in a proper subdomain is also obtained via eigenfunctions of the Laplacian. The formula is rigorous on any finite dimensional space spanned by the eigenfunctions and there is no smoothness assumption on the whole domain, making partial progress on a problem on p74 \cite{Zu:1}.
(c). A byproduct is an inversion formula for the heat kernel.
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…
In this paper, we study the local backward problem of a linear heat equation with time-dependent coefficients under the Dirichlet boundary condition. Precisely, we recover the initial data from the observation on a subdomain at some later time. Thanks to the "optimal filtering" method of Seidman, we can solve the global backward problem, which determines the solution at initial time from the known data on the whole domain. Then, by using a result of controllability at one point of time, we can connect local and global backward problem.
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^{…
In this paper, we regularize the nonlinear inverse time heat problem in the unbounded region by Fourier method. Some new convergence rates are obtained. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. Especially, the optimal convergence of the approximate solution at t = 0 is also proved. This work extends to many earlier results in (f2,f3, hao1,Quan,tau1, tau2, Trong3,x1).
It is known that a bounded solution of the heat equation in a half-space which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the half-space is replaced by cones with opening angle smaller than 90 degrees. Here we show the result remains true for cones with opening angle larger than 110 degrees. The proof covers heat equations having lower-order terms with bounded measurable coefficients.
In this paper, we are concerned with backward solvabilities of heat equations, in an abstract framework. We show that semigroups $T_t$ in Banach spaces $X$, generated by heat operators, are extendable to groups in an extended space $E$, which is obtained by considering a sequence of wider Banach spaces containing $X$, i.e. $X$$/subset$$X_t$$/subset$$X_s$... $(t<s)$, under the following two conditions. One is the density assumption on a subset $D$ of $X$, the set of initial values $x$ from which $T_{-t}x$ exists for all $t>0$. Another is the backward uniqueness of the semigroup $T_t$. For examp…
The inclination or $λ$-Lemma is a fundamental tool in finite dimensional hyperbolic dynamics. In contrast to finite dimension, we consider the forward semi-flow on the loop space of a closed Riemannian manifold $M$ provided by the heat flow. The main result is a backward $λ$-Lemma for the heat flow near a hyperbolic fixed point $x$. There are the following novelties. Firstly, infinite versus finite dimension. Secondly, semi-flow versus flow. Thirdly, suitable adaption provides a new proof in the finite dimensional case. Fourthly and a priori most surprisingly, our $λ$-Lemma moves the given dis…
In this paper we proposed two new quasi-boundary value methods for regularizing the ill-posed backward heat conduction problems. With a standard finite difference discretization in space and time, the obtained all-at-once nonsymmetric sparse linear systems have the desired block $ω$-circulant structure, which can be utilized to design an efficient parallel-in-time (PinT) direct solver that built upon an explicit FFT-based diagonalization of the time discretization matrix. Convergence analysis is presented to justify the optimal choice of the regularization parameter. Numerical examples are rep…
This paper addresses a backward heat conduction problem with fractional Laplacian and time-dependent coefficient in an unbounded domain. The problem models generalized diffusion processes and is well-known to be severely ill-posed. We investigate a simple and powerful variational regularization technique based on mollification. Under classical Sobolev smoothness conditions, we derive order-optimal convergence rates between the exact solution and regularized approximation in the practical case where both the data and the operator are noisy. Moreover, we propose an order-optimal a-posteriori par…
We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by the J-self-adjoint linear operator L depending on a small parameter. The problem has been originated from the lubrication approximation of a viscous fluid film on the inner surface of the rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on the numerical evidence, that the set of eigenvectors of the operator $L$ does not form a Riesz basis in $Ł^2 (-π,π)$. Our method can be applied to a wide range of the evolutional problems given by $PT-$symm…
We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $\varepsilon>0$ but divided by a factor of $\sqrt{\log\varepsilon^{-1}}$. We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the "weak noise" regime. We show that, as $\varepsilon\downarrow0$, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the li…
Consider the system $|\partial_tu+Δu|\leq M(|u|+|\nabla u|)$, $|u(x,t)|\leq Me^{M|x|^2}$ in $\mathcal{C}_θ\times[0,T]$ and $u(x,0)=0$ in $\mathcal{C}_θ$, where $\mathcal{C}_θ$ is a cone with opening angle $θ$. L. Escauriaza constructed an example to show that such system has a nonzero bounded solution when $θ<90^\circ$, and it's conjectured that the system has only zero solution for $θ>90^\circ$. Recently Lu Li and V. Šverák \cite{LlS} proved that the claim is true for $θ>109.5^\circ$. Here we improve their result and prove that only zero solution exists for this system when $θ>99^\circ$ by ex…
Application of the cut-off projection to solve a backward heat conduction problem in a two-slab composite system
Institute for Computational Science and Technology（计算科学与技术研究所）；Gran Sasso Science Institute（格兰萨索科学研究院）；University of Goettingen（哥廷根大学）；Hankyong National University（汉城国立大学）；University of Hamburg（汉堡大学）；University of Helsinki（赫尔辛基大学）
In this paper, we focus on the backward heat problem of finding the function $θ(x,y)=u(x,y,0)$ such that \[ {l l l} u_t - a(t)(u_{xx} + u_{yy}) & = f(x,y,t), & \qquad (x,y,t) \in Ω\times (0,T), u(x,y,T) & = h(x,y), & \qquad (x,y) \in\barΩ. \] where $Ω= (0,π) \times (0,π)$ and the heat transfer coefficient $a(t)$ is known. In our problem, the source $f = f(x,y,t)$ and the final data $h(x,y)$ are unknown. We only know random noise data $g_{ij}(t)$ and $d_{ij}$ satisfying the regression models g_{ij}(t) &=& f(x_i,y_j,t) + \varthetaξ_{ij}(t), d_{ij} &=& h(x_i,y_j) + σ_{ij}ε_{ij}, where $ξ_{ij}(t)$…
This second version of the manuscript includes, in the appendices, an erratum that points out an error on the published version and offers alternative results for the Lipschitz stability analysis of the backward heat propagation problem and its applications to light sheet fluorescence microscopy. Abstract of the first version: In this work, we study a Lipschitz stability result in the reconstruction of a compactly supported initial temperature for the heat equation in $\mathbb{R}^n$, from measurements along a positive time interval and over an open set containing its support. We take advantage…
In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat equation. Based on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem, which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time. Using the backward pseudolocality theorem…
In this note we obtain local derivative estimates of Shi-type for the heat equation coupled to the Ricci flow. As applications, in part combining with Kuang's work, we extend some results of Zhang and Bamler-Zhang including distance distortion estimates and a backward pseudolocality theorem for Ricci flow on compact manifolds to the noncompact case.
It is well known that generic solutions of the heat equation are not analytic in time in general. Here it is proven that ancient solutions with exponential growth are analytic in time in ${\M} \times (-\infty, 0]$. Here $\M=\R^n$ or is a manifold with Ricci curvature bounded from below. Consequently a necessary and sufficient condition is found on the solvability of backward heat equation in the class of functions with exponential growth.
We construct a radially smooth positive ancient solution for energy critical semi-linear heat equation in $\mathbb{R}^n$, $n\geq 7$. It blows up at the origin with the profile of multiple Talenti bubbles in the backward time infinity.
In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound.
We prove Fatou type theorems for solutions of the heat equation in sub- Riemannian spaces. The doubling property of L-caloric measure, the Dahlberg estimate, the local comparison theorem, among other results, are established here. A backward Harnack inequality is proved for non-negative solutions vanishing in the lateral boundary.
We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our main result yields the equivalence of (i) dynamic convexity of the Boltzmann entropy on the (time-dependent) $L^2$-Wasserstein space; (ii) monotonicity of $L^2$-Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time); (iii) gradi…
This paper defines a parabolic frequency for solutions of the heat equation on a Ricci flow and proves it's monotonicity along the flow. Frequency monotonicity is known to have many useful consequences; here it is shown to provide a simple proof of backwards uniqueness. For solutions of more general parabolic equations on a Ricci flow, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.
An inverse source problem for the heat equation is considered. Extraction formulae for information about the time and location when and where the unknown source of the equation firstly appeared are given from a single lateral boundary measurement. New roles of the plane progressive wave solutions or their complex versions for the backward heat equation are given.
We establish a unique continuation property for stochastic heat equations evolving in a bounded domain $G$. Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of $G$ at any given positive time constant. Further, when $G$ is convex and bounded, we also give a quantitative version of the unique continuation property. As applications, we get an observability estimate for stochastic heat equations, an approximate result and a null controllability result for a backward stochastic heat equation.
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.
Eternal solutions of parabolic equations (those which are defined for all time) are typically rather rare. For example, the heat equation has exactly one eternal solution -- the trivial solution. While solutions to the heat equation exist for all forward time, they cannot be extended backwards in time. Nonlinearities exasperate the situation somewhat, in that solutions may form singularities in both backward and forward time. However, semilinear parabolic equations can also support nontrivial eternal solutions. This article shows how nontrivial eternal solutions can be constructed for a semili…
We give a sufficient condition for existence of an exponential dichotomy for a general linear dynamical system (not necessarily invertible) in a Banach space, in discrete or continuous time. We provide applications to the backward heat equation with a potential varying in time and to the heat equation with a finite number of slowly moving potentials. We also consider the Klein-Gordon equation with a finite number of potentials whose centres move at sub-light speed with small accelerations.
We derive a non-linear version of the Feynman-Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2.
This paper is concerned with the quasi-linear reflected backward stochastic partial differential equation (RBSPDE for short). Basing on the theory of backward stochastic partial differential equation and the parabolic capacity and potential, we first associate the RBSPDE to a variational problem, and via the penalization method, we prove the existence and uniqueness of the solution for linear RBSPDE with Lapalacian leading coefficients. With the continuity approach, we further obtain the well-posedness of general quasi-linear RBSPDEs. Related results, including Itô formulas for backward stocha…
We study solutions of the focusing energy-critical nonlinear heat equation $u_t = Δu - |u|^2u$ in $\mathbb{R}^4.$ We show that solutions emanating from initial data with energy and $\dot{H}^1-$norm below those of the stationary solution $W$ are global and decay to zero, via the "concentration-compactness plus rigidity" strategy of Kenig-Merle. First, such global solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza, Seregin and Sverak in an argu…
We consider the energy super critical semilinear heat equation $$\partial_t u=Δu+u^{p}, \ \ x\in \mathbb R^3, \ \ p>5.$$ We first revisit the construction of radially symmetric backward self similar solutions and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws …
We prove an equivariant Lefschetz formula for elliptic complexes over a compact manifold carrying the action of a compact Lie group of isometries via heat equation methods.
For high order (than two) in time operator-splitting methods applied to dissipative systems, a folklore issue is the appearance of negative-time/backward-in-time linear evolution operators such as backward heat operators interwoven with nonlinear evolutions. The stability of such methods has remained an ensuing difficult open problem. In this work we consider a fourth order operator splitting discretization for the Allen-Cahn equation which is a prototypical high order splitting method with negative time-stepping, i.e. backward in time integration for the linear parabolic part. We introduce a …
In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time depend…
The Navier-Stokes equation on Rd (d greater or equal to 3) formulated on Besov spaces is considered. Using a stochastic forward-backward differential system, the local existence of a unique solution in B_ r, with r > 1 + d is obtained. We also show p,p p the convergence to solutions of the Euler equation when the viscosity tends to zero. Moreover, we prove the local existence of a unique solution in B_ pr,q, with p > 1, 1 greater or equal to q greater or equal to infinity, r > max(1, d); here the maximal time interval depends on p the viscosity.
By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer's forward formulations in \cite{Co-Iy}. Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is present…
