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}\displaystyle\frac{\partial b(x,u)}{\partial t}-\mbox{div}\>\mathcal{A}(x,t,u,\nabla u)-\mbox{div}\>\Phi(x,t,u)=f &\mbox{ in }\Omega\times (۰,T)\\b(x,u)(t=۰)=b(x,u_۰) & \mbox{ in } \Omega\\u=۰ &\mbox{ on } \partial\Omega\times (۰,T).\end{array}\right.is given in the non reflexive Musielak spaces, where b(x,\cdot) is a strictly increasing C^۱-function for every x\in\Omega with b(x,۰)=۰, 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.

کلیدواژه ها:

Parabolic problems ، Musielak spaces ، Renormalized solutions ، natural growth