2011, 2011(Special): 117-125. doi: 10.3934/proc.2011.2011.117

Existence results to a quasilinear and singular parabolic equation

1. 

Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex

2. 

LMAP (UMR 5142), Bat. IPRA, Université de Pau et des Pays de l'Adour, Avenue de l'Université, 64013 cedex Pau, France, France

Received  July 2010 Revised  January 2011 Published  October 2011

We investigate the following quasilinear parabolic and singular equation, $ u_t-\Delta_p u &=\frac{1}{u^\delta}+f(t,x)\;\text{ in }\,Q_T=(0,T]\times\Omega\\ u>0 \text{ in }\, Q_T\; , u &=0\,\text{ on} \;\Gamma=[0,T]\times\partial\Omega,\\ u(0,x) &=u_0(x)\;\text{ in }\Omega $ where $\Omega$ is an open bounded domain with smooth boundary in ${\rm R}^N$, $1 < p< \infty$ and $0<\delta$, $T>0$, $f\in L^\infty(Q_T)$ and $u_0\in L^\infty(\Omega)\cap W^{1,p}_0(\Omega)$. For any $\delta\in (0,2+\frac{1}{p-1})$, $u_0$ satisfying a cone condition defined below and any $T>0$, In this paper we prove the existence and the uniqueness of a weak solution $u$ to $({\rm P_t})$.
Citation: Mehdi Badra, Kaushik Bal, Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Conference Publications, 2011, 2011 (Special) : 117-125. doi: 10.3934/proc.2011.2011.117
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