SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Open Badges Research

Lestimates of solutions for the quasilinear parabolic equation with nonlinear gradient term and L1 data

Caisheng Chen*, Fei Yang and Zunfu Yang

Author affiliations

College of Science, Hohai University, Nanjing 210098, P. R. China

For all author emails, please log on.

Citation and License

Boundary Value Problems 2012, 2012:19  doi:10.1186/1687-2770-2012-19

Published: 15 February 2012


In this article, we study the quasilinear parabolic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/19/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/19/mathml/M1">View MathML</a>


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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/19/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/19/mathml/M2">View MathML</a> for all u0 L1(Ω). Furthermore, if 2q m + 2, we derive the Lestimate 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.

quasilinear parabolic equation; Lestimates; asymptotic behavior of solution