quasilinear parabolic problems in the lebsgue-sobolev space with variable exponent and l1 data

In this work, we study the existence of an initial boundary problem of a quasilinear parabolic problem with variable exponent and l 1 -data of the type {(b(u))t − div(|∇u| p(x)−2 ∇u) + λ |u| p(x)−2 u = f(x, t, u) in q = ω×]0, t[, u = 0 on σ = ∂ω×]0, t[, b(u)(t = 0) = b(u0) in ω, where λ > 0 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.