The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source

The Cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form ∂ u ∂ t = d i v ( ρ ( x ) u m − 1 | D u | λ − 1 D u ) + u p is studied. Global in time existence and nonexistence conditions are found for a solution to the Cauchy problem. Exact estimates of a solution are obtained in the case of global solvability. A sharp universal (i.e., independent of the initial function) estimate of a solution near the blow-up time is obtained.