6533b82ffe1ef96bd1295e35

RESEARCH PRODUCT

The asymptotic behavior of the solutions of the Cauchy problem generated by ϕ-accretive operators

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Abstract The purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in Banach spaces. In particular, we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem { u t − Δ p ( u ) + | u | γ − 1 u = f on  ] 0 , ∞ [ × Ω , − ∂ u ∂ η ∈ β ( u ) on  [ 0 , ∞ [ × ∂ Ω , u ( 0 , x ) = u 0 ( x ) in  Ω , where Ω is a bounded open domain in R n with smooth boundary ∂Ω, f ( t , x ) is a given L 1 -function on ] 0 , ∞ [ × Ω , γ ⩾ 1 and 1 ⩽ p ∞ . Δ p represents the p-Laplacian operator, ∂ ∂ η is the associated Neumann boundary operator and β a maximal monotone graph in R × R with 0 ∈ β ( 0 ) .