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

Abstract

In this article, we study the quasilinear parabolic problemwhere Ω is a bounded domain in ℝ N , m > 0 and g(u) satisfies |g(u)| ≤ K 1 |u| 1+ν with 0 ≤ ν <m. By the Moser's technique, we prove that if a, b > 1, 0 ≤ p <q, 1 ≤ q <m + 2, p + a <q + b, there exists a weak solutionThe asymptotic behavior of global weak solution u(t) for small initial data u 0 L 2 (Ω) also be established if p + a > max{m + 2, q + b}.