2012/08/31 by А. Г. Рамм, Ramm, A. G.
Computer Science · Engineering · Medicine · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical and Theoretical Epidemiology and Ecology Models #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1208.6462
openalex publication_date 2012/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: u=A(t)u+F(t,u)+b(t), t≥ 0; u(0)=u0. (*) Here u:=\frac dudt, u=u(t)∈ H, H is a Hilbert space, t∈ \R+:=[0,∞), A(t) is a linear dissipative operator: Re(A(t)u,u)≤ -γ(t)(u,u), %γ(t)≥ 0, F(t,u) is a nonlinear operator, |F(t,u)|≤ c0|u|p, p>1, c0,p are positive constants, |b(t)|≤ β(t), β(t)≥ 0 is a continuous function. Sufficient conditions are given for the solution u(t) to problem (*) to exist for all t≥0, to be bounded uniformly on \R+, and a bound on |u(t)| is given. This bound implies the relation limt→ ∞|u(t)|=0 under suitable conditions on γ(t) and β(t).