On uniqueness of solution of the second initial-boundary value problem for non-Nеwtonian politropical filtration equation

The uniqueness of non-negative solution of the second initial-boundary value problem for the equation
$$ u_t-\sum_{i=1}^n\frac{\partial}{\partial x_i}|u|^l|u_{x_i}|^{m-2}u_{x_i}=0,\quad l\geqslant1,\ m\geqslant2,\ n\geqslant2, $$
in the class of functions $\{u: u\in L^\infty(Q_T)\cap C([0,T];L^2(\Omega)), (u^{l/m+1})_{x_i}\in L^m(Q_T)\}$ is established.