2010/12/13 by A. G. Ramm, Ramm, A. G.
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations #math.CA #math.DS #msc:26D10 #msc:34G20 #msc:37L05 #msc:44J05 #msc:47J35 #msc:70K20
paper · pdf · doi:10.48550/arxiv.1012.2785
arxiv created 2010/12/13 · arxiv updated 2010/12/14
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, 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 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). The basic technical tool in this work is the following nonlinear inequality: g(t)≤ -γ(t)g(t)+α(t,g(t))+β(t), t≥ 0; g(0)=g0.