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Existence results to a quasilinear and singular parabolic equation

Pages: 117 - 125, Issue Special, September 2011

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Mehdi Badra - Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex, France (email)
Kaushik Bal - LMAP (UMR 5142), Bat. IPRA, Université de Pau et des Pays de l'Adour, Avenue de l'Université, 64013 cedex Pau, France (email)
Jacques Giacomoni - LMAP (UMR 5142), Bat. IPRA, Université de Pau et des Pays de l'Adour, Avenue de l'Université, 64013 cedex Pau, France (email)

Abstract: 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 $({\rm P_t})$.

Keywords:  quasilinear parabolic equation, singular nonlinearity, sub and supersolutions
Mathematics Subject Classification:  Primary 35J65, 35J20; Secondary 35J70

Received: July 2010;      Revised: January 2011;      Published: October 2011.