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A Harnack inequality for a degenerate parabolic equation. (English) Zbl 1112.35113
The degenerate parabolic equation \[ u_t - \sum_{i=0}^n \Big( | u_{x_i}| ^{p-2} u_{x_i}\Big)_{x_i}=0, \qquad p>2, \] is considered. The approach, already developed for the equation \(u_t- \text{div} (| Du| ^{p-2}Du)=0\), fails by no explicit expression for the fundamental solution.
The Harnack inequality is established. The main ingredients of proof are: 1) \(L_\infty\) estimates; 2) measure theoretic estimates; 3) non increasing nature of suitable Rayleigh quotient \(E(t)\); 4) comparison principle.

35K65 Degenerate parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K55 Nonlinear parabolic equations
35B45 A priori estimates in context of PDEs
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