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| This is a joint research with Vincenzo Vespri e Francesco Ragnedda.
We deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with $L^{\infty}$ coefficients, which prototypes are the p-laplacian and the Porous medium equation (in the supercritical range).
Estimates for the p-Laplacian and Porous medium equations (both for the slow diffusion and fast diffusion case) have been considered by several authors. To prove estimates for more general operators, we are forced to use a completely different and more sophisticated approach, based on DiBenedetto's techniques, recent Harnack inequalities and De Giorgi estimates.
Sharp pointwise estimates from above and from below for the fundamental solutions are derived, assuming as initial data the Dirac mass. Our results can be extended to general nonnegative $L^1$ initial data. |
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