We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse H"older inequality in suitable intrinsic cylinders. Relying on an intrinsic Calder'on-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for $minleft(rac{(n-2)_+}{n+2},1 ight)$. Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for $mgeq 1$ (see cite{GiaSch16} in the list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.
Self-improving property of the fast diffusion equation
Gianazza, Ugo;
2019-01-01
Abstract
We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse H"older inequality in suitable intrinsic cylinders. Relying on an intrinsic Calder'on-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for $minleft(rac{(n-2)_+}{n+2},1 ight)$. Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for $mgeq 1$ (see cite{GiaSch16} in the list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
