Backward Uniqueness for the Heat Equation in Cones

Abstract

It was shown in [4 Escauriaza , L. , Seregin , G. , Šverák , V. ( 2003 ). L 3, ∞-solutions of the Navier–Stokes equations and backward uniqueness . Uspekhi Mat. Nauk 58:3–44 (in Russian); translation in Russian Math. Surveys 58 : 211 – 250 .[Crossref], [Web of Science ®] , [Google Scholar], 14 Seregin , G. , Šverák , V. ( 2002 ). The Navier–Stokes equations and backward uniqueness. Nonlinear problems in mathematical physics and related topics, II . In : Int. Math. Ser. (N.Y.) . Vol. 2 . New York : Kluwer/Plenum , pp. 353 – 366 . [Google Scholar]] 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°. Here we show that the result remains true for cones with opening angle larger than 110°.

Locations

• Communications in Partial Differential Equations - View
• arXiv (Cornell University) - View - PDF