Research
L∞ estimates of solutions for the quasilinear parabolic equation with nonlinear gradient term and L1 data
Author affiliations
College of Science, Hohai University, Nanjing 210098, P. R. China
Citation and License
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.
