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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.

##### MSC:
 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
##### Keywords:
nonhomogeneous abstract equation; mild solution