Positively invariant subset for non-densely defined Cauchy problems

This study develops a generalized notion of sub tangential condition to establish the positive invariance of a closed subset under the semiflow generated by a semi-linear non densely defined Cauchy problem. We also remark that the sufficient condition for the positivity of the semiflow implies our sub tangentiality condition. In other words, our sub tangential condition is more powerful since it can be used to show the positive invariance of a much larger class of closed subset. As an illustration we apply our results to an age-structured equation in L^p space which is only defined on a closed subset of L^p.