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Existence of \(p\)-almost automorphic mild solution to some abstract differential equations. (English) Zbl 1083.35052
The nonhomogeneous abstract evolution equation in a Banach space \[ u'(t)=Au(t)+f(t) \] is considered with \(A\) being the generator of a semigroup of class \(C_0\) and \(f\) satisfying certain conditions weaker than in the classical setup. Some properties of \(p\)-almost automorphic functions \(f\) introduced by the author are studied. These functions can be written in the form \(f=h+\phi\), where \(h\) is almost automorphic and \(\phi\) is \(p\)-locally integrable. Application of the functions to constructing \(p\)-almost automorphic mild solutions to the equation is given.

35K90 Abstract parabolic equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
47D06 One-parameter semigroups and linear evolution equations