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Local center manifold for parabolic equations with infinite delay. (English) Zbl 0818.45005

This work continues the authors earlier studies [Math. Bohem. 118, No. 2, 175-193 (1993; Zbl 0798.35153)] on invariant manifolds of functional equations. Consider the equation \[ u_ t = Au(t) + Lu_ t + g \bigl( u(t), u_ t \bigr) \] with \(u(0) = x\), \(u(\tau) = \varphi (\tau)\) for \(\tau < 0\), where \(u_ t (\tau) = u(t + \tau)\), \(\tau < 0\); \(A\) generates an analytic semigroup in a Banach space \(X\), \(L\) is a continuous linear operator from a function space \(Y\) into \(X\) and \(g\) is nonlinear, smooth, satisfying \(g(0,0) = 0\), \(Dg (0,0)= 0\). Via the resolvent operator for \(A+L\) the equation is rewritten as \(v'(t) = Bv(t) + f(v(t))\), with \(v(t) = (u(t), u_ t)^ T\) and \(B\) the generator of a specific semigroup.
Using the techniques of interpolation spaces the author obtains (under certain assumptions on the spectrum of \(B)\) strong enough estimates to prove the existence of a center manifold for the equation considered. Finally, an example with \(A = \Delta + bI\) and Neumann boundary conditions is considered.
Reviewer: S.O.Londen (Espoo)

MSC:

45K05 Integro-partial differential equations
45N05 Abstract integral equations, integral equations in abstract spaces

Citations:

Zbl 0798.35153
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