Abstract: This paper dwells upon the asymptotic behavior, as t→∞, of the integral solution u(t) to the nonhaear evolution equation u'(t) ∈A(t)u(t) + g(t),t ≥ s,u(s)=x₀∈D(A(a)),where {A(t)}_{t ≥ s} is a family m-disipative operator in a Hlilbert space H,and g∈L_loc(0,∞;H). We prove that σ(t,h)=1/t-x∫_x^tu(θ+h)dθ converges weakly, as t→∞, uniforluly in h ≥ 0, which applies that u(t) is weak convergence if and only if u(t) is weakly asymptotically regular i.e., u(t + h) -u(t) →0 for h ≥ 0.

Key words: Nonlinear evolution equation, dissipative operator, integral solution, asymptotic behavior