On a class of nonlinear parabolic equations with natural growth in non-reflexive Musielak spaces

Abstract

An existence result of renormalized solutions for nonlinear parabolic Cauchy-Dirichlet problems whose model $$left{begin{array}{ll} displaystylefrac{partial b(x,u)}{partial t} -mbox{div}>mathcal{A}(x,t,u,nabla u)-mbox{div}> Phi(x,t,u)= f &mbox{ in }Omegatimes (0,T) b(x,u)(t=0)=b(x,u_0) & mbox{ in } Omega u=0 &mbox{ on } partialOmegatimes (0,T). end{array}right. $$ 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.