Geodesic
On a sharp Liouville theorem for solutions of a parabolic equation on a characteristic
Sbornik. Mathematics, Tome 26 (1975) no. 3, pp. 349-364 
A. L. Gusarov. On a sharp Liouville theorem for solutions of a parabolic equation on a characteristic. Sbornik. Mathematics, Tome 26 (1975) no. 3, pp. 349-364. http://geodesic.mathdoc.fr/item/SM_1975_26_3_a3/
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     title = {On~a~sharp {Liouville} theorem for solutions of~a~parabolic equation on~a~characteristic},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The equation $u_t=Lu+c(x)$ is considered in the strip $0. The operator $L=\sum_{i,j=1}^n\frac\partial{\partial x_i}\bigl(a_{ij}(x)\frac\partial{\partial x_j}\bigr)$ is a selfadjoint uniformly elliptic operator of second order, $a_{ij}\in C^2(\mathbf R^n)$, $c\in C^1(\mathbf R^n)$, $|D^\beta a_{ij}(x)|=o(|x|^{-|\beta|})$, $|\beta|=1,2$, and $|c(x)|=o(|x|^{-2})$. For a solution $u$ of this equation the following assertions are proved: if $|u(t,x)|=O(\exp\varphi(|x|))$ ($\varphi(r)\geqslant r^{2+\varepsilon}$ is an arbitrary increasing function of one variable) uniformly in $t$ and if in some cone on the characteristic $t=T$ we have $|u(T,x)|=O(\exp(-C\varphi(C'|x|)))$ ($C$ and $C'$ are constants which depend on the equation and the vertex angle of the cone), then $u(T,x)\equiv0$; if $u(T,x)|=O(\exp K|x|^2)$ and if in the cone we have $|u(T,x)|=O(\exp(-C(K+1/T)|x|^2))$ then $u(t,x)\equiv0$. Bibliography: 11 titles.

[1] E. M. Landis, O. A. Oleinik, “Obobschennaya analitichnost i svyazannye s nei svoistva reshenii ellipticheskikh i parabolicheskikh uravnenii”, Uspekhi matem. nauk, XXIX:2(176) (1974), 190–206 | MR

[2] A. L. Gusarov, “O teoreme tipa Liuvillya dlya parabolicheskikh uravnenii”, Vestnik MGU, seriya matem. i mekh., 1975, no. 3, 62–70 | MR | Zbl

[3] R. Kurant, D. Gilbert, Metody matematicheskoi fiziki, Gostekhizdat, Moskva–Leningrad, 1951

[4] S. Ito, H. Yamabe, “A unique continuation theorem for solutions of a parabolic differential equation”, J. Math. Soc. Japan, 10:3 (1958), 314–321 | MR | Zbl

[5] E. M. Landis, “Ob odnoi teoreme tipa Fragmena-Lindelefa dlya reshenii ellipticheskikh uravnenii”, Uspekhi matem. nauk, XXX:5 (185) (1975), 213 | MR | Zbl

[6] M. M. Elborai, “O korrektnosti zadachi Koshi”, Vestnik MGU, seriya matem. i mekh., 1968, no. 4, 15–21 | MR

[7] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, izd-vo «Nauka», Moskva, 1964 | MR

[8] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, izd-vo «Nauka», Moskva, 1967

[9] A. Fridman, Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, izd-vo «Mir», Moskva, 1968

[10] K. Miranda, Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, IL, Moskva, 1957

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