In　this　paper,　we　propose　a　scheme　for　generating　multipartite　entanglement　of　distant　four-level　atoms　separately　trapped　in　individual　cavities,　each　of　which　is　coupled　to　a　non-Markovian　reservoir.　The　entanglement　of　atoms　is　generated　by　measuring　the　photons　leaking　from　cavities.　In　non-Markovian　environments,　we　derive　dynamical　evolution　of　the　entanglement　and　obtain　the　condition　of　generating　the　long-living　multipartite　maximally　entangled　state.　When　the　condition　is　not　satisfied,　by　introducing　a　time-varying　coupling　strength,　the　maximal　multipartite　entangled　state　can　also　be　generated.　(C)　2018　Elsevier　B.V.　All　rights　reserved.