ACTA issues

Bounded solutions of finite lifetime to differential equations in Banach spaces

Rafael Dahmen, Helge Glöckner

Acta Sci. Math. (Szeged) 81:3-4(2015), 457-468

Abstract. Consider a smooth vector field $f\colon{\msbm R} ^n\to{\msbm R} ^n$ and a maximal solution $\gamma\colon ]a,b[ \to{\msbm R} ^n$ to the ordinary differential equation $x'=f(x)$. It is a well-known fact that, if $\gamma $ is bounded, then $\gamma $ is a global solution, i.e., $ ]a,b[ ={\msbm R} $. We show by example that this conclusion becomes invalid if ${\msbm R} ^n$ is replaced with an infinite-dimensional Banach space.

DOI: 10.14232/actasm-014-271-7

AMS Subject Classification (1991): 34C11; 26E20, 34A12, 34G20, 37C10, 34--01

Keyword(s): ordinary differential equation, smooth dynamical system, autonomous system, Banach space, finite life time, maximal solution, bounded solution, relatively compact set, tubular neighborhood, nearest point

Received March 25, 2014. (Registered under 21/2014.)