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

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 {ut−Δp(u)+|u|γ−1u=fon ]0,∞[×Ω,−∂u∂η∈β(u)on [0,∞[×∂Ω,u(0,x)=u0(x)in Ω, where Ω is a bounded open domain in Rn with smooth boundary ∂Ω, f(t,x) is a given L1-function on ]0,∞[×Ω, γ⩾1 and 1⩽p<∞. Δp 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).