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Integral manifolds for semilinear evolution equations and admissibility of function spaces

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Ukrainian Mathematical Journal Aims and scope

We prove the existence of integral (stable, unstable, and center) manifolds for the solutions to a semilinear integral equation

$$ u(t)=U\left( {t,s} \right)u(s)+\int\limits_s^t {U\left( {t,\xi } \right)f\left( {\xi, u\left( \xi \right)} \right)} d\xi $$

in the case where the evolution family (U(t, s)) ts has an exponential trichotomy on a half line or on the whole line, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions, i.e.,

$$ \left\| {f\left( {t,x} \right)-f\left( {t,y} \right)} \right\|\leqslant \varphi (t)\left\| {x-y} \right\|, $$

where φ(t) belongs to some classes of admissible function spaces. Our main method is based on the Lyapunov–Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 6, pp. 772–796, June, 2012.

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Huy, N.T., Ha, V.T.N. & Phi, H. Integral manifolds for semilinear evolution equations and admissibility of function spaces. Ukr Math J 64, 881–911 (2012). https://doi.org/10.1007/s11253-012-0686-1

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  • DOI: https://doi.org/10.1007/s11253-012-0686-1

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