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On stabilization of a solution of the Cauchy problem for a first-order differential equation in a Banach space. (English. Russian original) Zbl 0796.34040
Russ. Acad. Sci., Dokl., Math. 46, No. 2, 245-247 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 2, 224-226 (1992).
The authors consider the initial value problem $$v(o)= v_ 0$$ for the differential equation $$dv/dt= Av$$, $$t>0$$, where $$v$$ belongs to a Banach space $$E$$ and $$A$$ is the generator of a strongly continuous cosine function. The necessary and sufficient conditions for existence of the limit $$\lim_{t\to+\infty} v(t)$$ are formulated in this note.

##### MSC:
 34G10 Linear differential equations in abstract spaces 34C11 Growth and boundedness of solutions to ordinary differential equations 35K99 Parabolic equations and parabolic systems 35Q05 Euler-Poisson-Darboux equations