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C0-semigroups of linear operators on some ultrametric Banach spaces

C 0 -semigroups of linear operators play a crucial role in the solvability of evolution equations in the classical context. This paper is concerned with a brief conceptualization of C 0 -semigroups on (ultrametric) free Banach spaces E . In contrast with the classical setting, the parameter of a given C 0 -semigroup belongs to a clopen ball Ω r of the ground field K . As an illustration, we will discuss the solvability of some homogeneous p -adic differential equations.