on a class of nonlinear parabolic equations with natural growth in non-reflexive musielak spaces

An existence result of renormalized solutions for nonlinear parabolic cauchy-dirichlet problems whose model$$left{begin{array}{ll}displaystylefrac{partial b(x,u)}{partial t}-mbox{div}>mathcal{a}(x,t,u,nabla u)-mbox{div}>phi(x,t,u)=f mbox{ in }omegatimes (0,t)b(x,u)(t=0)=b(x,u_0) mbox{ in } omegau=0 mbox{ on } partialomegatimes (0,t).end{array}right.$$is given in the non reflexive musielak spaces, where $b(x,cdot)$ is a strictly increasing $c^1$-function for every $xinomega$ with $b(x,0)=0$, the lower order term $phi$ is a non coercive carath’{e}odory function satisfying only a natural growth condition described by the appropriate musielak function $varphi$ and $f$ is an integrable data.