The Cauchy problem for certain degenerate quasilinear parabolic equations with absorption

The Cauchy problem for the equation
\begin{equation} u_t=\Delta(|u|^{\mu-1}u)-c|u|^{\nu-1} u, \tag{1} \end{equation}
where $\mu>1$, $\nu>1$, and $c>0$, with initial data
\begin{equation} u(0,x)=u_0(x). \tag{2} \end{equation}
is considered in the halfspace $\mathbf{R}_+^{n+1}$. A growth of the initial function at infinity is admitted. For various relations between $\mu$ and $\nu$, existence and uniqueness theorems for (1), (2) in classes of increasing functions are proved. Examples are given which indicate that the results obtained are exact in a sense.