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On the Theory of Integral Manifolds for Some Delayed Partial Differential Equations with Nondense Domain

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Ukrainian Mathematical Journal Aims and scope

Integral manifolds are very useful in studying the dynamics of nonlinear evolution equations. We consider a nondensely defined partial differential equation

$$ \frac{du}{dt}=\left(A+B(t)\right)u(t)+f\left(t,{u}_t\right),\kern0.72em t\in \mathrm{\mathbb{R}}, $$
(1)

where (A,D(A)) satisfies the Hille–Yosida condition, (B(t))t∈R is a family of operators in L(D(A),X) satisfying certain measurability and boundedness conditions, and the nonlinear forcing term f satisfies the inequality ‖f(t, ϕ) − f(t, ψ)‖≤φ(t)‖ϕ − ψ‖c, where 𝜑 belongs to admissible spaces and 𝜙,ψ ∈ C ≔ C([−r, 0], X). We first present an exponential convergence result between the stable manifold and every mild solution of (1). Then we prove the existence of center-unstable manifolds for these solutions. Our main methods are invoked by the extrapolation theory and the Lyapunov–Perron method based on the properties of admissible functions.

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Authors and Affiliations

  1. University of Sfax, Sfax, Tunisia

    C. Jendoubi

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  1. C. Jendoubi
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Correspondence to C. Jendoubi.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 776–789, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.6020.

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Jendoubi, C. On the Theory of Integral Manifolds for Some Delayed Partial Differential Equations with Nondense Domain. Ukr Math J 72, 900–916 (2020). https://doi.org/10.1007/s11253-020-01831-9

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  • DOI: https://doi.org/10.1007/s11253-020-01831-9

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