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Stability in nonlinear evolution problems by means of fixed point theorems. (English) Zbl 0891.34065

Summary: The stabilization of solutions to an abstract differential equation is investigated. The initial value problem is considered in the form of an integral equation. The equation is solved by means of the Banach contraction mapping theorem or the Schauder fixed point theorem in the space of functions decreasing to zero at an appropriate rate. Stable manifolds for singular perturbation problems are compared with each other. A possible application is illustrated on an initial-boundary-value problem for a parabolic equation in several space variables.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34C30 Manifolds of solutions of ODE (MSC2000)
35K20 Initial-boundary value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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