Strong unique continuation for parabolic operators. (English) Zbl 1404.35075
Summary: We prove a unique continuation theorem for functions \(u\) vanishing to infinite order in the space-time variables at \((0,0)\) and satisfying the inequality \(|\varDelta u+\partial_t u|\leq V(x,t)|u|+W(x,t)|\nabla u|\) for some unbounded time-dependent \(V\) and \(W\) on \({\mathbb R}\), and in the radial case on \(\mathbb R^n\).
MSC:
| 35B60 | Continuation and prolongation of solutions to PDEs |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35K05 | Heat equation |
| 35R45 | Partial differential inequalities and systems of partial differential inequalities |
References:
| [1] | T. Carleman, Sur un probl‘eme d’unicit´e pour les syst‘emes d’´equations aux deriv´ees partielles ‘a deux variables ind´ependantes, Ark. Mat. 26 (1939), 1-9. · Zbl 0022.34201 |
| [2] | X. Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann. 311 (1996), 603-630. [E]L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J. 104 (2000), 113-127. [EK+] L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, Decay at infinity of caloric functions within characteristic hyperplanes, Math. Res. Lett. 13 (2006), 441-453. [EV]L. Escauriaza and L. Vega, Carleman inequalities and the heat operator II, Indiana Univ. Math. J. 50 (2001), 1149-1169. [F]F. J. Fernandez, Unique continuation for parabolic operators. II, Comm. Partial Differential Equations 28 (2003), 1597-1604. [JK]D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schr¨odinger operators, Ann. of Math. 121 (1985), 465-488. |
| [3] | G. Karadzhov, Riesz summability of multiple Hermite series in Lpspace, C. R. Acad. Bulgare Sci. 47 (1994), 5-8. · Zbl 0829.40003 |
| [4] | F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math. 42 (1988), 125-136. [P]C. C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations 21 (1996), 521-539. [R]R. Regbaoui, Unique continuation for differential equations of Schr¨odinger’s type, Comm. Anal. Geom. 7 (1999), 303-323. |
| [5] | S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes 42, Princeton Univ. Press, Princeton, 1993. [T2]S. Thangavelu, Hermite and special Hermite expansions revisited, Duke Math. J. 94 (1998), 257-278. · Zbl 0945.42014 |
| [6] | Th. H. Wolff, Recent work on sharp estimates in second order elliptic unique continuation problems, in: Fourier Analysis and Partial Differential Equations (Miraflores de la Sierra, 1992), CRC, Boca Raton, FL, 1995, 99-128. Yueyang Men Institute of Applied Physics and Computational Mathematics Beijing, 100088, P.R. China E-mail: menyuey@amss.ac.cn · Zbl 0871.35019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
