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Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations. (English) Zbl 1206.35053
The paper deals with the Harnack inequality for non negative solutions of a class of singular, quasilinear, parabolic equations are established. The prototype example is
$u_t- \text{div} |Du|^{p-1} Du=0, \quad 1<p<2.$
The authors prove that non-negative weak solutions satisfy an intrisic form of Harnack inequality provided that $$p_*=\frac{2N}{N+1}<p<2$$, which is the optimal range for this estimate.
The relevant fact is that the inequality obtained is a forward and backward in time Harnack estimate as well as a Harnack elliptic type.

##### MSC:
 35B45 A priori estimates in context of PDEs 35K65 Degenerate parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35K59 Quasilinear parabolic equations
weak solutions
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