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Stable manifolds for some partial neutral functional differential equations with non-dense domain. (English) Zbl 1429.35190

Summary: We investigate the asymptotic solution behavior for the partial neutral functional differential equation \[ \begin{cases} \frac{d}{dt}\mathcal{D}u_t=(A+B(t))\mathcal{D}u_t+f(t,u_t), \quad t\geq s\geq 0, \\ u_s=\phi \in \mathcal{C}:=C([-r,0],X), \end{cases} \] where the linear operator \(A\) is not necessarily densely defined and satisfies the Hille-Yosida condition, and the delayed part \(f\) is assumed to satisfy the \(\varphi \)-Lipschitz condition, i.e., \(\|f(t,\phi)-f(t,\psi)\|\leq \varphi (t)\|\phi -\psi \|_{\mathcal{C}} \). Here \(\varphi\) belongs to some admissible spaces and \(\phi,\psi \in \mathcal{C}:=C([-r,0],X)\).
More precisely, we prove the existence of stable (respectively, center stable) manifolds when the linear part generates an evolution family having an exponential dichotomy (respectively, trichotomy) on the half positive line. Furthermore, we show that such stable manifold attracts all mild solutions of the considered neutral equation. An example is given to assimilate our theory.

MSC:

35R10 Partial functional-differential equations
34D35 Stability of manifolds of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
34K40 Neutral functional-differential equations
35B40 Asymptotic behavior of solutions to PDEs
37D10 Invariant manifold theory for dynamical systems
47D06 One-parameter semigroups and linear evolution equations
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References:

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