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Some existence and regularity results for porous media and fast diffusion equations with a gradient term


Authors: Boumediene Abdellaoui, Ireneo Peral and Magdalena Walias
Journal: Trans. Amer. Math. Soc. 367 (2015), 4757-4791
MSC (2010): Primary 35K10, 35K59, 35K61, 35K65, 35K67
Published electronically: March 2, 2015
MathSciNet review: 3335400
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Abstract: In this article we consider the problem

$\displaystyle \left \{\begin {array}{rclll} u_t-\Delta u^m&=&\vert\nabla u\vert... ...times (0,T),\\ u(x,0)&=&u_0(x)&\quad \hbox { in } \Omega , \end{array} \right .$ (P)

where $ \Omega \subset \mathbb{R}^N$ is a bounded regular domain, $ N\ge 1$, $ 1<q\le 2$, and $ f\ge 0$, $ u_0\ge 0$ are in a suitable class of measurable functions.

We obtain some results for the so-called elliptic-parabolic problems with measure data related to problem $ (P)$ that we use to study the existence of solutions to problem $ (P)$ according with the values of the parameters $ q$ and $ m$.


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Additional Information

Boumediene Abdellaoui
Affiliation: Département de Mathématiques, Université Aboubekr Belkaïd, Tlemcen, Tlemcen 13000, Algeria
Email: boumediene.abdellaoui@uam.es

Ireneo Peral
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: ireneo.peral@uam.es

Magdalena Walias
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: magdalena.walias@uam.es

DOI: https://doi.org/10.1090/S0002-9947-2015-06125-X
Keywords: Elliptic-parabolic equations, porous media equation, fast diffusion equation, dependence on the gradient
Received by editor(s): October 18, 2012
Received by editor(s) in revised form: March 13, 2013
Published electronically: March 2, 2015
Additional Notes: The first author was partially supported by a grant of ICTP, Trieste, Italy. This work was partially supported by project MTM2010-18128, MICINN, Spain
Dedicated: To the memory of Juan Antonio Aguilar, our dearest friend
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.