Lipschitz stability in the determination of the principal part of a parabolic equation

Ganghua Yuan; Masahiro Yamamoto

Displaying similar documents to “Lipschitz stability in the determination of the principal part of a parabolic equation”

Three cylinder inequalities and unique continuation properties for parabolic equations

Sergio Vessella (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We prove the following unique continuation property. Let $u$ be a solution of a second order linear parabolic equation and $S$ a segment parallel to the $t$ -axis. If $u$ has a zero of order faster than any non constant and time independent polynomial at each point of $S$ then $u$ vanishes in each point, $(x, t^{'})$ , such that the plane $t = t^{'}$ has a non empty intersection with $S$ .

Controllability of a parabolic system with a diffuse interface

Jérôme Le Rousseau, Matthieu Léautaud, Luc Robbiano (2013)

Journal of the European Mathematical Society

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We consider a linear parabolic transmission problem across an interface of codimension one in a bounded domain or on a Riemannian manifold, where the transmission conditions involve an additional parabolic operator on the interface. This system is an idealization of a three-layer model in which the central layer has a small thickness $δ$ . We prove a Carleman estimate in the neighborhood of the interface for an associated elliptic operator by means of partial estimates in several microlocal...

$L^{\infty}$ estimates of solution for $m$ -Laplacian parabolic equation with a nonlocal term

Pulun Hou, Caisheng Chen (2011)

Czechoslovak Mathematical Journal

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In this paper, we consider the global existence, uniqueness and $L^{\infty}$ estimates of weak solutions to quasilinear parabolic equation of $m$ -Laplacian type $u_{t} - {div (| \nabla u |}^{m - 2} {\nabla u) = u | u |}^{β - 1} \int_{Ω} {| u |}^{α} d x$ in $Ω \times (0, \infty)$ with zero Dirichlet boundary condition in $\partial Ω$ . Further, we obtain the $L^{\infty}$ estimate of the solution $u (t)$ and $\nabla u (t)$ for $t > 0$ with the initial data $u_{0} \in L^{q} (Ω)$ $(q > 1)$ , and the case $α + β < m - 1$ .

Application of Rothe's method to a parabolic inverse problem with nonlocal boundary condition

Yong-Hyok Jo, Myong-Hwan Ri (2022)

Applications of Mathematics

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We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value $u_{0} \in H^{1} (Ω)$ is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when $u_{0} \in L^{2} (Ω)$ and the integral kernel in the nonlocal boundary condition is symmetric. ...

Estimates of Green functions and their applications for parabolic operators with singular potentials

Lotfi Riahi (2003)

Colloquium Mathematicae

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We prove global pointwise estimates for the Green function of a parabolic operator with potential in the parabolic Kato class on a $C^{1, 1}$ cylindrical domain Ω. We apply these estimates to obtain a new and shorter proof of the Harnack inequality [16], and to study the boundary behavior of nonnegative solutions.

Boundary estimates for certain degenerate and singular parabolic equations

Benny Avelin, Ugo Gianazza, Sandro Salsa (2016)

Journal of the European Mathematical Society

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We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$ -Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_{T}$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of...

Determination of the unknown source term in a linear parabolic problem from the measured data at the final time

Müjdat Kaya (2014)

Applications of Mathematics

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The problem of determining the source term $F (x, t)$ in the linear parabolic equation $u_{t} = {(k (x) u_{x} (x, t))}_{x} + F (x, t)$ from the measured data at the final time $u (x, T) = μ (x)$ is formulated. It is proved that the Fréchet derivative of the cost functional $J (F) = ∥ μ_{T} {(x) - u (x, T) ∥}_{0}^{2}$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence...

Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

Alessandra Lunardi (1998)

Studia Mathematica

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We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in $ℝ^{n}$ . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on $ℝ^{N}$

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove the existence of global attractors for the following semilinear degenerate parabolic equation on $ℝ^{N}$ : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

Static hedging of barrier options with a smile : an inverse problem

Claude Bardos, Raphaël Douady, Andrei Fursikov (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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Let $L$ be a parabolic second order differential operator on the domain $\bar{Π} = [0, T] \times ℝ .$ Given a function $\hat{u} : ℝ \to R$ and $\hat{x} > 0$ such that the support of $\hat{u}$ is contained in $(- \infty, - \hat{x}]$ , we let $\hat{y} : \bar{Π} \to ℝ$ be the solution to the equation: $L \hat{y} = 0, \hat{y} |_{{0} \times ℝ} = \hat{u} .$ Given positive bounds $0 < x_{0} < x_{1},$ we seek a function $u$ with support in $[x_{0}, x_{1}]$ such that the corresponding solution $y$ satisfies: $y (t, 0) = \hat{y} (t, 0) \forall t \in [0, T] .$ We prove in this article that, under some regularity conditions on the coefficients of $L,$ continuous solutions are unique and dense in the...

Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales

Pernilla Johnsen, Tatiana Lobkova (2018)

Applications of Mathematics

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This paper is devoted to the study of the linear parabolic problem $ε \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x / ε, t / ε^{3}) \nabla u_{ε} (x, t)) = f (x, t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $ε$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic...