Nonlinear semigroups and the existence and stability of solutions to semilinear nonautonomous evolution equations. (English) Zbl 0934.34051

Aulbach, Bernd; Nguyen Van Minh

Abstr. Appl. Anal. 1, No. 4, 351-380 (1996).

Summary: This paper is concerned with existence and stability of solutions to a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the so-called evolution semigroup of (possibly nonlinear) operators. Sufficient conditions for existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding infinitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes, the authors obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional-differential equations.

MSC:

34G20	Nonlinear differential equations in abstract spaces
47H20	Semigroups of nonlinear operators
35R10	Partial functional-differential equations
35K90	Abstract parabolic equations

Keywords:

existence; stability; solutions; semilinear nonautonomous evolution equations; evolution semigroup; periodic oscillations; accretiveness; infinitesimal generator