Research
L∞ estimates of solutions for the quasilinear parabolic equation with nonlinear gradient term and L1 data
College of Science, Hohai University, Nanjing 210098, P. R. China
Boundary Value Problems 2012, 2012:19 doi:10.1186/1687-2770-2012-19
Published: 15 February 2012Abstract
In this article, we study the quasilinear parabolic problem
where Ω is a bounded domain in ℝN, m > 0 and g(u) satisfies |g(u)| ≤ K1|u|1+ν with 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 
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.
