Geodesic
On the zeros of the solutions of a class of analytic systems
Sbornik. Mathematics, Tome 35 (1979) no. 2, pp. 207-217 
V. I. Ruderfer. On the zeros of the solutions of a class of analytic systems. Sbornik. Mathematics, Tome 35 (1979) no. 2, pp. 207-217. http://geodesic.mathdoc.fr/item/SM_1979_35_2_a3/
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In this paper estimates are obtained for the solution of the Cauchy problem with data on a timelike surface in the case of the nonelliptic analytic system of second order $$ A_{ij}u_{x_ix_j}+B_iu_{x_i}+Cu=0 $$ ($i,j=0,1,\dots,n$; $A_{ij},B_i,C$ are $N\times N$ matrices) in which the operator $\sum_{i,j=0}^nA_{ij}\partial^2/\partial x_i\partial x_j$ is elliptic and the dependence of the estimate on the dimension of the support is given explicitly. On the basis of these estimates the author solves the problem of the rate of decay of the solution and its first derivatives in the neighborhood of a zero on the boundary of the domain of the solution that suffices for the solution to vanish in some region. Bibliography: 7 titles.

[1] E. M. Landis, “O nekotorykh svoistvakh reshenii ellipticheskikh uravnenii”, DAN SSSR, 107:5 (1956), 640–643 | MR | Zbl

[2] M. M. Lavrentev, “O zadache Koshi dlya lineinykh ellipticheskikh uravnenii vtorogo poryadka”, DAN SSSR, 112:2 (1957), 195–197 | MR

[3] E. M. Landis, “Nekotorye voprosy kachestvennoi teorii ellipticheskikh i parabolicheskikh uravnenii”, Uspekhi matem. nauk, XIV:1(85) (1959), 21–85 | MR

[4] B. K. Amonov, S. P. Shishatskii, “Apriornye otsenki resheniya zadachi Koshi dlya parabolicheskogo uravneniya vtorogo poryadka s dannymi na vremenipodobnoi poverkhnosti”, DAN SSSR, 206:1 (1972), 11–13 | MR | Zbl

[5] O. A. Ladyzhenskaya, “O razreshimosti osnovnykh kraevykh zadach dlya uravnenii parabolicheskogo i giperbolicheskogo tipov”, DAN SSSR, 97:3 (1954) | Zbl

[6] M. I. Vishik, O. A. Ladyzhenskaya, “Kraevye zadachi dlya uravnenii v chastnykh proizvodnykh i nekotorykh klassov operatornykh uravnenii”, Uspekhi matem. nauk, XI:6(72) (1956), 41–98

[7] F. John, “Continuous dependence on data for solutions of partial differential equations, with a prescribed bound”, Comm. Pure Appl. Math., 13:4 (1960), 551–585 | DOI | MR | Zbl