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Almost automorphic solutions for hyperbolic semilinear evolution equations. (English) Zbl 1086.34052

Summary: Here, we deal with mild solutions for the semilinear evolution equation \[ \tfrac{d}{dt}x(t) = Ax(t) + f(t,x(t)),\qquad t\in \mathbb R, \] under the sectoriality of \(A\), a linear operator with not necessarily dense domain, in a Banach space \(X\) and \(\sigma(A)\cap i\mathbb R = \emptyset\). We discuss the existence and uniqueness of an almost automorphic solution in an intermediate space \(X_{\alpha}\), when the function \(f \colon \mathbb R \times X_{\alpha} \longrightarrow X\) is almost automorphic. An example illustrating the obtained result is given (a partial differential equation).

MSC:

34G20 Nonlinear differential equations in abstract spaces
47A55 Perturbation theory of linear operators
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