$L_1$–$L_\infty$ estimates of solutions of the Cauchy problem for an anisotropic degenerate parabolic equation with double non-linearity and growing initial data

The Cauchy problem for a degenerate parabolic equation with anisotropic $p$-Laplacian and double non-linearity is considered. For increasing initial data local estimates for the $L_\infty$-norm of a solution are obtained, which yield a precise characterization of the growth of solutions at infinity. An estimate for the order of the length of the time interval on which a solution is defined is found in its dependence on the initial data.
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