Abstract.
It is shown that a function u satisfying |∂ t +Δu|≦M(|u|+|∇u|), |u(x, t)|≦Me M|x| 2 in (ℝn \ (B R ) × [0, T] and u(x, 0) = 0 for xℝn \ B R must vanish identically in ℝn \ B R ×[0, T].
References
Ahlfors, L.V.: Complex Analysis. McGraw-Hill, 1966
Chen, X.Y.: A strong unique continuation theorem for parabolic equations. Math. Ann. 311, 603–630 (1996)
Escauriaza, L.: Carleman inequalities and the heat operator. Duke Math. J. 104, 113–127 (2000)
Escauriaza, L., Vega, L.: Carleman inequalities and the heat operator II. Indiana U. Math. J. 50, 1149–1169 (2001)
Escauriaza, L., Fernández, F.J.: Unique continuation for parabolic operators. (to appear)
Fernández, F.J.: Unique continuation for parabolic operators II. (to appear)
Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1950)
Hörmander, L.: Linear Partial Differential Operators. Springer, 1963
Hörmander, L.: Uniqueness theorems for second order elliptic differential equations. Communications in PDE 8, 21–64 (1983)
Jones, B.F.: A fundamental solution of the heat equation which is supported in a strip. J. Math. Anal. Appl. 60, 314–324 (1977)
Kato, T.: Strong L p-solutions of the Navier-Stokes equations in ℝm with applications to weak solutions. Math. Zeit. 187, 471–480 (1984)
Ladyzhenskaya, O.A.: Mathematical problems of the dynamics of viscous incompressible fluids. Gordon and Breach, 1969
Ladyženskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Amer. Math. Soc., 1968
Lin, F.H.: A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math. 42, 125–136 (1988)
Littman, W.: Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients. Annali Scuola Norm. Sup. Pisa Serie IV 3, 567–580 (1978)
Micu, S., Zuazua, E.: On the lack of null-controllability of the heat equation on the half space. Portugaliae Mathematica 58, 1–24 (2001)
Poon, C.C.: Unique continuation for parabolic equations. Comm. Partial Differential Equations 21, 521–539 (1996)
Saut, J.C., Scheurer, E.: Unique continuation for evolution equations. J. Differential Equations 66, 118–137 (1987)
Seregin, G., Šverák, V.: The Navier-Stokes equations and backward uniqueness. (to appear)
Sogge, C.D.: A unique continuation theorem for second order parabolic differential operators. Ark. Mat. 28, 159–182 (1990)
Treves, F.: Linear Partial Differential Equations. Gordon and Breach, 1970
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Escauriaza, L., Seregin, G. & Šverák, V. Backward Uniqueness for Parabolic Equations. Arch. Rational Mech. Anal. 169, 147–157 (2003). https://doi.org/10.1007/s00205-003-0263-8
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DOI: https://doi.org/10.1007/s00205-003-0263-8