  We study a semilinear differential-algebraic equation (DAE) with the focus on
the Lagrange stability (instability). The conditions for the existence and
uniqueness of global solutions (a solution exists on an infinite interval) of
the Cauchy problem, as well as conditions of the boundedness of the global
solutions, are obtained. Furthermore, the obtained conditions for the Lagrange
stability of the semilinear DAE guarantee that every its solution is global and
bounded, and, in contrast to theorems on the Lyapunov stability, allow to prove
the existence and uniqueness of global solutions regardless of the presence and
the number of equilibrium points. We also obtain the conditions of the
existence and uniqueness of solutions with a finite escape time (a solution
exists on a finite interval and is unbounded, i.e., is Lagrange unstable) for
the Cauchy problem. We do not use constraints of a global Lipschitz condition
type, that allows to use the work results efficiently in practical
applications. The mathematical model of a radio engineering filter with
nonlinear elements is studied as an application. The numerical analysis of the
model verifies the results of theoretical investigations.
