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In this article, we study the quasilinear parabolic problem 0.1ut-div(∇um∇u)+uuβ-2∇uq=uuα-2∇up+g(u),x∈Ω,t>0,u(x,0)=u0(x),x∈Ω;u(x,t)=0,x∈∂Ω,t≥0, where Ω is a bounded domain in ℝN, m > 0 and g(u) satisfies |g(u)| ≤ K1|u|1+νwith 0 ≤ ν < m. By the Moser's technique, we prove that if α, β > 1, 0 ≤ p < q, 1 ≤ q < m + 2, p + α < q + β, there exists a weak solution u(t)∈L∞([0,∞),L1)∩Lloc∞((0,∞)W01,m+2) for all u0 ∈ L1(Ω). Furthermore, if 2q ≤ m + 2, we derive the L∞ estimate for ∇u(t). The asymptotic behavior of global weak solution u(t) for small initial data u0 ∈ L2(Ω) also be established if p + α > max{m + 2, q + β}.2000 Mathematics Subject Classification: 35K20; 35K59; 35K65.

L∞ estimates of solutions for the quasilinear parabolic equation with nonlinear gradient term and L1 data