On a sharp Liouville theorem for solutions of a parabolic equation on a characteristic

The equation $u_t=Lu+c(x)$ is considered in the strip $0<t\leqslant T$. 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 $u(t,x)\equiv0$.
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