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Center manifold and stability in critical cases for some partial functional differential equations. (English) Zbl 1121.37063
Summary: We prove the existence of a center manifold for some partial functional differential equations, whose linear part is not necessarily densely defined but satisfies the Hille-Yosida condition. The attractiveness of the center manifold is also shown when the unstable space is reduced to zero. We prove that the flow on the center manifold is completely determined by an ordinary differential equation in a finite-dimensional space. In some critical cases, when the exponential stability is not possible, we prove that the uniform asymptotic stability of the equilibrium is completely determined by the uniform asymptotic stability of the reduced system on the center manifold.
MSC:
37L10Normal forms, center manifold theory, bifurcation theory
34K30Functional-differential equations in abstract spaces
34G20Nonlinear ODE in abstract spaces
35R10Partial functional-differential equations
37K20Relations of infinite-dimensional systems with algebraic geometry, etc.
47D06One-parameter semigroups and linear evolution equations