The Cauchy problem in classes of increasing functions for the equation of filtration with convection

We consider the Cauchy problem with a non-negative continuous initial function for the equation
$$ u_t=(u^m)_{xx}+c(u^n)_x, $$
where $m>1$, $m\geqslant n\geqslant 1$ and $c$ is a positive constant. We prove a number of existence and uniqueness theorems for generalized solutions increasing at infinity for this Cauchy problem; we also investigate the behaviour of these solutions for large values of the time.