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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

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

Published: 15 February 2012

Abstract

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>

(0.1)

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.

Keywords:
quasilinear parabolic equation; Lestimates; asymptotic behavior of solution