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Existence and linearized stability for partial neutral functional differential equations with nondense domains. (English) Zbl 0983.34075

Summary: The authors consider a class of nonlinear partial neutral functional-differential equations of the form \[ \frac{d}{dt}[{\mathcal D}x_t-G(t,x_t)] =A_0[{\mathcal D}x_t-G(t,x_t)]+F(t,x_t),\;t\geq 0,\;x_0 =\varphi\in{\mathcal C}([-r,0],X), \] with a nondensely defined Hille-Yosida operator \(A_0\). They first give some sufficient conditions ensuring the existence and uniqueness of solutions. In the linear autonomous case, the solutions are shown to generate a locally Lipschitz continuous integrated semigroup on \({\mathcal C}([-r,0],X)\). They give a principle of linearized stability in the nonlinear autonomous case. The results are a natural generalization of their recent work [Dyn. Syst. Appl. 7, No. 3, 389-403 (1998; Zbl 0921.35181), J. Differ. Equations 147, No. 2, 285-332 (1998; Zbl 0915.35109) and Appl. Math. Lett. 12, No. 1, 107-112 (1999; Zbl 0941.34075)], recent work by J. K. Hale [Resen. Inst. Mat. Estat. Univ. São Paulo 1, No. 4, 441-457 (1994; Zbl 0857.35127) and Rev. Roum. Math. Pures Appl. 39, No. 4, 339-344 (1994; Zbl 0817.35119)] and by J. Wu and H. Xia [J. Differ. Equations 124, No. 1, 247-278 (1994; Zbl 0840.34080) and Rotating waves in neutral partial functional-differential equations (preprint)]. The method used here is based on the integrated semigroup theory.

MSC:

34K30 Functional-differential equations in abstract spaces
34K40 Neutral functional-differential equations
34K20 Stability theory of functional-differential equations
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