Local behaviour of solutions to doubly nonlinear parabolic equations. (English) Zbl 1114.35035
Nonnegative weak solutions to the doubly nonlinear parabolic equation
\[
\text{div} (| Du| ^{p-2}Du)=\partial_t (u^{p-1}),
\]
\(p\in (1, \infty),\) are considered. The main purpose of the article is to give a relatively simple and transparent proof for Harnack’s inequality using the approach of Moser. To show that the proof is based on a general principle the Lebesgue measure is replaced with an arbitrary doubling Borel measure which supports a Poincaré inequality.
The arguments can be applied to more general equations with replacement of \(p\)-Laplacian by divergent operator \(\operatorname{div} A(x,t,u,Du)\) with standard structure conditions.
The arguments can be applied to more general equations with replacement of \(p\)-Laplacian by divergent operator \(\operatorname{div} A(x,t,u,Du)\) with standard structure conditions.
Reviewer: Evgeniy A. Kalita (Donetsk)
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
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