Decay at infinity of caloric functions within characteristic hyperplanes
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In this article, it was shown that a function u satisfying, |�u+@tu| ≤ M (|u| + |∇u|), |u(x, t)| ≤ Me M|x| 2 in R n × (0, T) and |u (x,0)| ≤ Cke k|x | 2 in r n and for allAbstract:
It is shown that a function u satisfying, |�u+@tu| ≤ M (|u| + |∇u|), |u(x, t)| ≤ Me M|x| 2 in R n × (0, T) and |u(x,0)| ≤ Cke k|x| 2 in R n and for allread more
Citations
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On Uniqueness Properties of Solutions of Schrödinger Equations
TL;DR: In this article, the uniqueness properties of solutions of Schrodinger equations of the form are studied and sufficient conditions on the asymptotic behavior of the difference of two solutio...
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Backwards Uniqueness for the Ricci Flow
TL;DR: In this article, the authors prove a backwards uniqueness theorem for solutions to the Ricci flow and prove that the isometry group of a solution cannot expand within the lifetime of the solution.
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Uniqueness of self-similar shrinkers with asymptotically conical ends
TL;DR: In this article, the uniqueness of smooth embedded selfshrinkers asymptotic to generalized cylinders of infinite order was shown and non-rotationally symmetric self-shrinking ends were constructed with rate as fast as any given polynomial.
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Uniqueness properties of solutions to Schrödinger equations
TL;DR: The strong unique continuation property for operators whose coefficients are not necessarily real analytic was established by Carleman et al. as discussed by the authors, who showed that the uniqueness of the Cauchy problem can be established for nonlinear operators with real analytic coefficients.
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A short proof of backward uniqueness for some geometric evolution equations
TL;DR: In this article, a simple direct proof of the backward uniqueness of solutions to a class of second-order geometric evolution equations which includes Ricci and cross-curvature flows is presented.
References
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Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Book
Linear and Quasilinear Equations of Parabolic Type
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
Journal ArticleDOI
Unique continuation for some evolution equations
Jean-Claude Saut,Bruno Scheurer +1 more
TL;DR: In this article, the authors prove a unique continuation result for a second order parabolic operator with smooth coefficients. But their proof is based on the derivation of a Carleman estimate which is reminiscent of the classical Carlemann inequality for second order elliptic operators.