Weak solutions of second-order quasilinear parabolic equations with double non-linearity

The first boundary-value problem for the equation
$$ \beta (u)\frac {\partial u}{\partial t}-\sum _{i=1}^nD_iA_i(t,x,u,Du)+ A_0(t,x,u,Du)=0 $$
is considered in a bounded subdomain of $n$. The function $\beta (u)$ is assumed to be continuous and satisfy the following growth conditions:
$$ c|u|^{r-2}\leqslant \beta (u)\leqslant C\bigl (|u|^{r-2}+1\bigr ),\qquad r\geqslant 2. $$
The other coefficients satisfy the standard conditions of the theory of monotone operators. An existence theorem for a global weak solution is proved.