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The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source. (English) Zbl 1195.35068

Summary: The Cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form

$\frac{\partial u}{\partial t}=div\left(\rho \left(x\right){u}^{m-1}{|Du|}^{\lambda -1}Du\right)+{u}^{p}$

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.

##### MSC:
 35B44 Blow-up (PDE) 35K65 Parabolic equations of degenerate type 35K15 Second order parabolic equations, initial value problems 35K59 Quasilinear parabolic equations
##### Keywords:
existence and nonexistence theorems