Existence of positive solutions for a class of parabolic equations with natural growth terms and L1 data

We study a class of nonlinear parabolic equations of the type: ∂b(u)∂t−div (a(x,t,u)∇u)+g(u)|∇u|2=f,where the right hand side belongs to L1(Q), b is a strictly increasing C1-function and −div (a(x,t,u)∇u)−div (a(x,t,u)∇u) is a Leray-Lions operator. The function g is just assumed to be continuous on ℝ and to satisfy a sign condition. Without any additional growth assumption on u, we prove the existence of a renormalized solution.