Some qualitative properties of solutions to the Cauchy problem of the degenerate parabolic equations with a drift term

We consider the Cauchy problem for the weighted doubly degenerate parabolic equation with a source terms, which in fact reduces to the corresponding unweighted equation but with the suitable drift and source terms. Exponentially and power-like growing at infinity classes of weighted functions are considered. For the solution of the Cauchy problem without source term, we analyze the structure of temporal behavior of the solution. While, for the solution of the problem with a source, we prove the Fujita type theorems. Our results show what is the influence of weights on the qualitative behavior of solutions of the problems under consideration.