Resumen: In this paper we give uniform stable spatial bounds for the resolvent operator fami- lies of the abstract fractional Cauchy problem on R+. Such bounds allow to prove existence and uniqueness of µ-pseudo almost automorphic e-mild regular solutions to the nonlinear fractional Cauchy problem in the whole real line. Finally, we apply our main results to the fractional heat equation with critical nonlinearities. Idioma: Inglés Año: 2018 Publicado en: Topological Methods in Nonlinear Analysis ISSN: 1230-3429 Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2016-77710-P Tipo y forma: Article (PostPrint) Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)