Asymptotic stability of solutions to abstract differential equations

الأصل · EN

An evolution problem for abstract differential equations is studied. The typical problem is: u=A(t)u+F(t,u), t≥ 0; u(0)=u₀; u=du/dt (*) Here A(t) is a linear bounded operator in a Hilbert space H, and F is a nonlinear operator, F(t,u)≤ c₀uᵖ,p>1, c₀, p=const>0. It is assumed that Re(A(t)u,u)≤ -γ(t)u² ∀ u∈ H, where γ(t)>0, and the case when ₜ→ ∞γ(t)=0 is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this estimate uses a nonlinear differential inequality.