Some existence and regularity results for porous media and fast diffusion equations with a gradient term
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- by Boumediene Abdellaoui, Ireneo Peral and Magdalena Walias PDF
- Trans. Amer. Math. Soc. 367 (2015), 4757-4791 Request permission
Abstract:
In this article we consider the problem \begin{equation}\tag {\textit {P}} \left \{\begin {array}{rclll} u_t-\Delta u^m&=&|\nabla u|^q + f(x,t),\quad u\ge 0 &\hbox { in } \Omega _T\equiv \Omega \times (0,T),\\ u(x,t)&=&0 &\quad \hbox { on } \partial \Omega \times (0,T),\\ u(x,0)&=&u_0(x)&\quad \hbox { in } \Omega , \end{array} \right . \end{equation} 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
- 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
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 4757-4791
- MSC (2010): Primary 35K10, 35K59, 35K61, 35K65, 35K67
- DOI: https://doi.org/10.1090/S0002-9947-2015-06125-X
- MathSciNet review: 3335400
Dedicated: To the memory of Juan Antonio Aguilar, our dearest friend