Fujita type theorems for quasilinear parabolic equations with initial data slowly decaying to zero

This work deals with the Cauchy problem for a parabolic equation with a double non-linearity of the following type:
$$ u_t=\operatorname{div}(u^\alpha|Du|^{m-1}Du)+u^p, $$
where $0<m+\alpha \leqslant1$. Existence and non-existence results for global solutions of this problem with initial conditions that slowly decay to zero are established.