{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:04:08Z","timestamp":1760144648489,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,9]],"date-time":"2024-05-09T00:00:00Z","timestamp":1715212800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The study of dynamic singularity formation in spacetime, focusing on scalar field collapse models, is analyzed. We revisit key findings regarding open spatial topologies, concentrating on minimal conditions necessary for singularity and apparent horizon formation. Moreover, we examine the stability of initial data in the dynamical system governed by Einstein\u2019s equations, considering variations in parameters that influence naked singularity formation. We illustrate how these results apply to a family of scalar field models, concluding with a discussion on the concept of genericity in singularity studies.<\/jats:p>","DOI":"10.3390\/sym16050583","type":"journal-article","created":{"date-parts":[[2024,5,9]],"date-time":"2024-05-09T05:16:45Z","timestamp":1715231805000},"page":"583","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Global Models of Collapsing Scalar Field: Endstate"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7575-710X","authenticated-orcid":false,"given":"Dario","family":"Corona","sequence":"first","affiliation":[{"name":"School of Science and Technology, University of Camerino, 62032 Camerino, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7147-134X","authenticated-orcid":false,"given":"Roberto","family":"Giamb\u00f2","sequence":"additional","affiliation":[{"name":"School of Science and Technology, University of Camerino, 62032 Camerino, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sez. di Perugia, 06123 Perugia, Italy"},{"name":"IAPS, Istituto Nazionale di Astrofisica\u2014Tor Vergata, 00133 Roma, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1002\/cpa.3160440305","article-title":"The formation of black holes and singularities in spherically symmetric gravitational collapse","volume":"44","author":"Christodoulou","year":"1991","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1002\/cpa.3160460803","article-title":"Bounded variation solutions of the spherically symmetric einstein-scalar field equations","volume":"46","author":"Christodoulou","year":"1993","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"607","DOI":"10.2307\/2118619","article-title":"Examples of naked singularity formation in the gravitational collapse of a scalar field","volume":"140","author":"Christodoulou","year":"1994","journal-title":"Ann. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"183","DOI":"10.2307\/121023","article-title":"The Instability of Naked Singularities in the Gravitational Collapse of a Scalar Field","volume":"149","author":"Christodoulou","year":"1999","journal-title":"Ann. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"575","DOI":"10.4310\/ATMP.2005.v9.n4.a3","article-title":"A Note on naked singularities and the collapse of selfgravitating Higgs fields","volume":"9","author":"Dafermos","year":"2005","journal-title":"Adv. Theor. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3485","DOI":"10.1088\/0264-9381\/15\/11\/014","article-title":"Scalar field cosmologies and the initial space-time singularity","volume":"15","author":"Foster","year":"1998","journal-title":"Class. Quant. Grav."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2981","DOI":"10.1088\/0264-9381\/20\/14\/301","article-title":"Scalar field cosmologies with an arbitrary potential","volume":"20","author":"Miritzis","year":"2003","journal-title":"Class. Quant. Grav."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3900","DOI":"10.1063\/1.1602161","article-title":"Dynamical system approach to FRW models in higher order gravity theories","volume":"44","author":"Miritzis","year":"2003","journal-title":"J. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"082502","DOI":"10.1063\/1.2009648","article-title":"The Recollapse problem of closed FRW models in higher-order gravity theories","volume":"46","author":"Miritzis","year":"2005","journal-title":"J. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Malafarina, D. (2017). Classical collapse to black holes and quantum bounces: A review. Universe, 3.","DOI":"10.3390\/universe3020048"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"103540","DOI":"10.1103\/PhysRevD.89.103540","article-title":"Coupled quintessence with double exponential potentials","volume":"89","author":"Tzanni","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"035009","DOI":"10.1088\/0264-9381\/32\/3\/035009","article-title":"Negative potentials and collapsing universes","volume":"32","author":"Miritzis","year":"2015","journal-title":"Class. Quant. Grav."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"165017","DOI":"10.1088\/0264-9381\/32\/16\/165017","article-title":"Negative potentials and collapsing universes II","volume":"32","author":"Miritzis","year":"2015","journal-title":"Class. Quant. Grav."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"031302","DOI":"10.1103\/PhysRevLett.96.031302","article-title":"Quantum evaporation of a naked singularity","volume":"96","author":"Goswami","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"135021","DOI":"10.1088\/0264-9381\/32\/13\/135021","article-title":"Scalar field collapse with negative cosmological constant","volume":"32","author":"Baier","year":"2015","journal-title":"Class. Quant. Grav."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"064048","DOI":"10.1103\/PhysRevD.105.064048","article-title":"Singularity resolution in gravitational collapse","volume":"105","author":"Mosani","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"924","DOI":"10.1140\/epjc\/s10052-020-08486-7","article-title":"Strength of the naked singularity in critical collapse","volume":"80","author":"Guo","year":"2020","journal-title":"Eur. Phys. J. C"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"042504","DOI":"10.1063\/1.2907949","article-title":"Genericity of black hole formation in the gravitational collapse of homogeneous self-interacting scalar fields","volume":"49","author":"Giannoni","year":"2008","journal-title":"J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"012501","DOI":"10.1063\/1.3032755","article-title":"Gravitational collapse of homogeneous perfect fluid in HOG theories","volume":"50","year":"2009","journal-title":"J. Math. Phys."},{"key":"ref_20","unstructured":"Khalil, H. (2002). Nonlinear Systems, Pearson Education, Prentice Hall."},{"key":"ref_21","unstructured":"Dumortier, F., Llibre, J., and Art\u00e9s, J.C. (2006). Qualitative Theory of Planar Differential Systems, Springer."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1023\/A:1018819521971","article-title":"Generalized Vaidya solutions","volume":"31","author":"Wang","year":"1999","journal-title":"Gen. Rel. Grav."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2295","DOI":"10.1088\/0264-9381\/22\/11\/023","article-title":"Gravitational collapse of homogeneous scalar fields","volume":"22","year":"2005","journal-title":"Class. Quant. Grav."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Malafarina, D., and Joshi, P.S. (2024). New Frontiers in Gravitational Collapse and Spacetime Singularities, Springer.","DOI":"10.1007\/978-981-97-1172-7"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2784","DOI":"10.1140\/epjc\/s10052-014-2784-8","article-title":"Arbitrary scalar field and quintessence cosmological models","volume":"74","author":"Harko","year":"2014","journal-title":"Eur. Phys. J. C"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"245005","DOI":"10.1088\/1361-6382\/abbd5a","article-title":"Generalized scalar field cosmologies: Theorems on asymptotic behavior","volume":"37","author":"Leon","year":"2020","journal-title":"Class. Quant. Grav."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"015004","DOI":"10.1088\/1361-6382\/abc095","article-title":"Generalized scalar field cosmologies: A global dynamical systems formulation","volume":"38","author":"Leon","year":"2021","journal-title":"Class. Quant. Grav."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1600415","DOI":"10.1002\/andp.201600415","article-title":"The final state of gravitational collapse in Eddington-inspired Born-Infeld theory","volume":"529","author":"Tavakoli","year":"2017","journal-title":"Annalen Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1140\/epjc\/s10052-021-08930-2","article-title":"Collapse of an axion scalar field","volume":"81","author":"Chakrabarti","year":"2021","journal-title":"Eur. Phys. J. C"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1950035","DOI":"10.1142\/S021988781950035X","article-title":"Gravitational collapse in General Relativity and in R2-gravity: A comparative study","volume":"16","author":"Astashenok","year":"2019","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1140\/epjp\/s13360-021-02082-8","article-title":"Non-singular collapse scenario from matter\u2013curvature coupling","volume":"136","author":"Ziaie","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"044009","DOI":"10.1103\/PhysRevD.95.044009","article-title":"Gravitational Collapse of a Homogeneous Scalar Field Coupled Kinematically to Einstein Tensor","volume":"95","author":"Koutsoumbas","year":"2017","journal-title":"Phys. Rev. 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