Quasilinear parabolic problems in the Lebsgue-Sobolev space with variable exponent and L^۱ data

In this work, we study the existence of an initial boundary problem of a quasilinear parabolic problem with variable exponent  and   L ^{۱} -data of the type\begin{equation*}\left\{\begin{array}{ll}(b(u))_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-۲}\nabla u)+\lambda\left\vert u\right\vert ^{p(x)-۲}u=f(x,t,u) \text{ } &\text{in}\hspace{۰.۵cm}Q=\Omega \times ]۰,T[, \\u=۰ & \text{on}\hspace{۰.۵cm}\Sigma =\partial \Omega \times ]۰,T[, \\b(u)(t=۰)=b(u_{۰}) & \text{in}\hspace{۰.۵cm}\Omega , \end{array}\right.\end{equation*}where \lambda>۰ and T is positive constant. The main contribution of our work is to prove the existence of a renormalized solution. The functional setting involves Lebesgue– Sobolev spaces with variable exponents.

Keywords:

Quasilinear parabolic problems , Variable exponent , truncations , Renormalized solutions , L۱ data