Behaviour of solutions of certain quasilinear parabolic equations with power-type non-linearities

The Cauchy problem with non-negative continuous initial function for the equation
$$ u_t=\Delta u^m-u^p, \qquad (x,t)\in S=\mathbb R^N\times\mathbb R_+, $$
is considered for $0<p<1$, $p<m$. For generalized solutions of this problem with initial data increasing at infinity several results on their behaviour as $t\to\infty$ are established.