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Local existence and linearized stability for partial functional differential equations. (English) Zbl 0921.35181

In an infinite-dimensional Banach space \((E,| \cdot |)\), the authors consider the Cauchy problem \[ du(t)/dt= Au(t)+F(u_t), \quad t\geq 0,\;u_0=\varphi\in C_E, \tag{1} \] where \(C_E:=C ([-\tau, 0],E)\), \(\tau>0\), is the space of continuous functions from \([-\tau,0]\) into \(E\), with the uniform convergence topology and \(A:D(A)\subseteq E\to E\) is a linear operator. For \(u\in C([-\tau, b],E)\), \(b>0\), and \(t\in[0,b]\), \(u_t\) denotes the element of \(C_E\) defined by \(u_t(\theta)= u(t+\theta)\), \(\theta\in[-\tau, 0]\) and \(F\) is a function which satisfies a local Lipschitz condition.
If the operator \(A\) (with \(D(A)\) non-dense in \(E)\) satisfies a Hille-Yosida condition, then the problem (1) has local solutions. In the case when \(F\) is globally Lipschitz continuous, the problem of linearized stability near an equilibrium point is studied.

MSC:

35R10 Partial functional-differential equations
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
35B35 Stability in context of PDEs
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