On the study of some quasilinear non-coercive p(x)-parabolic problem with L1-data

Abstract

In this paper, we consider the following quasilinear and non-coercive $p(.)-$parabolic problem  :

d u/dt   -   div a(x,t,u,\nabla u) =  f(x,t) -   div  F(x,t,u)     in    Q_{T}\\
u = 0                                                                             on    \Sigma_{T}
 u(x,0)=u_{0}                                                               in  \Omega
 
with   f\in L^{1}( Q_{T})   and   u_{0}\in L^{1}(\Omega). We study the existence of a entropy solutions for this problem in the parabolic Sobolev space with variable exponent  V.