Stability of solutions to abstract differential equations

2010/07/18 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Computer Science · Engineering · Mathematics · #34G20 #447J05 #47J35 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Stability and Controllability of Differential Equations #math.CA #msc:34G20 #msc:447J05 #msc:47J35

paper · pdf · doi:10.48550/arxiv.1007.3001

arxiv created 2010/07/18 · openalex publication_date 2010/07/18 · arxiv updated 2010/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A sufficient condition for asymptotic stability of the zero solution to an abstract nonlinear evolution problem is given. The governing equation is u=A(t)u+F(t,u), where A(t) is a bounded linear operator in Hilbert space H and F(t,u) is a nonlinear operator, ‖F(t,u)‖≤ c₀‖u‖^1+p, p=const >0, c₀=const>0. It is not assumed that the spectrum σ:=σ(A(t)) of A(t) lies in the fixed halfplane Rez≤ -κ, where κ>0 does not depend on t. As t→ ∞ the spectrum of A(t) is allowed to tend to the imaginary axis.

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