Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity

We prove the existence and uniqueness of weak solutions of the Dirichlet problem for the nonlinear degenerate parabolic equations
$$ u_{t}=\operatorname{div}(a|u|^{\gamma(x,t)}\nabla u)+\mathbf{b}|u|^{\gamma(x,t)/2}\nabla u-c|u|^{\sigma (x,t)-2}u+d, $$
where $a$, $\mathbf{b}$, $c$, and $d$ are given functions of the arguments $x$, $t$, and $u(x,t)$, and the exponents of nonlinearity $\gamma(x,t)$ and $\sigma(x,t)$ are known measurable and bounded functions of their arguments.