The linearized uniform asymptotic stability of evolution differential equations. (English) Zbl 0599.34081

Neustupa, Jiří

Czech. Math. J. 34(109), 257-284 (1984).

We examine the correctness of the method of linearization in the case of uniform asymptotic stability. Under certain conditions we show that the uniform asymptotic stability of the zero solution of the equation (0,1) implies the same property of the zero solution of the equation (0,2). This main result is proved in Section 2. The restriction to the question of stability of the zero solution does not represent a great loss of generality, because the problem of stability of a generally ”nonzero” solution can be mostly transformed quite simply to a similar problem concerning the zero solution. In Section 1, we list all necessary assumptions about the equations (0.1) and (0.2). Nonetheless, these equations remain very general throughout Sections 1 and 2. Sections 3, 4 and 5 are devoted to applications of the results derived to some more special classes of differential equations, containing as particular cases for instance the Navier-Stokes equations, the wave equation, the equation of oscillations of a beam and the Timoshenko-type equation.

MSC:

34G20	Nonlinear differential equations in abstract spaces
35B35	Stability in context of PDEs
34D20	Stability of solutions to ordinary differential equations

Keywords:

method of linearization; uniform asymptotic stability; Navier-Stokes equations; wave equation; equation of oscillations of a beam; Timoshenko- type equation

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