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We study the problem ∂b(x,u)/∂t - div(a(x,t,u,Du)) + H(x,t,u,Du) = μ in Q = Ω×(0,T), ${b(x,u)|}_{t=0}=b(x,u₀)$ in Ω, u = 0 in ∂Ω × (0,T). The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on H(x,t,u,Du). The critical growth condition on H is only with respect to Du and not with respect to u. The datum μ is assumed to be in $L¹\left(Q\right)+L^{p^{\text{'}}}(0,T;W^{-1,p^{\text{'}}}\left(\Omega \right))$ and b(x,u₀) ∈ L¹(Ω).