{"@context":{"@vocab":"https://cir.nii.ac.jp/schema/1.0/","rdfs":"http://www.w3.org/2000/01/rdf-schema#","dc":"http://purl.org/dc/elements/1.1/","dcterms":"http://purl.org/dc/terms/","foaf":"http://xmlns.com/foaf/0.1/","prism":"http://prismstandard.org/namespaces/basic/2.0/","cinii":"http://ci.nii.ac.jp/ns/1.0/","datacite":"https://schema.datacite.org/meta/kernel-4/","ndl":"http://ndl.go.jp/dcndl/terms/","jpcoar":"https://github.com/JPCOAR/schema/blob/master/2.0/"},"@id":"https://cir.nii.ac.jp/crid/1362825896131598592.json","@type":"Article","productIdentifier":[{"identifier":{"@type":"DOI","@value":"10.5802/aif.2688"}},{"identifier":{"@type":"URI","@value":"https://aif.centre-mersenne.org/item/10.5802/aif.2688.pdf"}}],"dc:title":[{"@value":"Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations"}],"description":[{"type":"abstract","notation":[{"@value":"<jats:p>\n                    In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mrow>\n                        <mml:mrow>\n                          <mml:mo>(</mml:mo>\n                          <mml:mi>t</mml:mi>\n                          <mml:mo>,</mml:mo>\n                          <mml:mi>x</mml:mi>\n                          <mml:mo>)</mml:mo>\n                        </mml:mrow>\n                        <mml:mo>=</mml:mo>\n                        <mml:mrow>\n                          <mml:mo>(</mml:mo>\n                          <mml:mn>0</mml:mn>\n                          <mml:mo>,</mml:mo>\n                          <mml:mn>0</mml:mn>\n                          <mml:mo>)</mml:mo>\n                        </mml:mrow>\n                        <mml:mo>∈</mml:mo>\n                        <mml:msup>\n                          <mml:mi mathvariant=\"bold\">C</mml:mi>\n                          <mml:mn>2</mml:mn>\n                        </mml:msup>\n                      </mml:mrow>\n                    </mml:math>\n                    . Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mi>k</mml:mi>\n                    </mml:math>\n                    -summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.\n                  </jats:p>"}]}],"creator":[{"@id":"https://cir.nii.ac.jp/crid/1382825896131598593","@type":"Researcher","foaf:name":[{"@value":"Zhuangchu Luo"}]},{"@id":"https://cir.nii.ac.jp/crid/1382825896131598594","@type":"Researcher","foaf:name":[{"@value":"Hua Chen"}]},{"@id":"https://cir.nii.ac.jp/crid/1382825896131598592","@type":"Researcher","foaf:name":[{"@value":"Changgui Zhang"}]}],"publication":{"publicationIdentifier":[{"@type":"EISSN","@value":"17775310"}],"prism:publicationName":[{"@value":"Annales de l'Institut Fourier"}],"dc:publisher":[{"@value":"MathDoc/Centre Mersenne"}],"prism:publicationDate":"2012","prism:volume":"62","prism:number":"2","prism:startingPage":"571","prism:endingPage":"618"},"reviewed":"false","url":[{"@id":"https://aif.centre-mersenne.org/item/10.5802/aif.2688.pdf"}],"createdAt":"2012-07-12","modifiedAt":"2025-10-10","relatedProduct":[{"@id":"https://cir.nii.ac.jp/crid/1360002216005232256","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations"}]},{"@id":"https://cir.nii.ac.jp/crid/1360568465740337792","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Maillet type theorem for nonlinear totally characteristic partial differential equations"}]},{"@id":"https://cir.nii.ac.jp/crid/1390282680091920000","@type":"Article","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"Unique solvability of some nonlinear partial differential equations with Fuchsian and irregular singularities"}]},{"@id":"https://cir.nii.ac.jp/crid/1391693801399816320","@type":"Article","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"Uniqueness of the solution of nonlinear singular first order partial differential equations"}]}],"dataSourceIdentifier":[{"@type":"CROSSREF","@value":"10.5802/aif.2688"},{"@type":"CROSSREF","@value":"10.2969/jmsj/06631017_references_DOI_WCS2LTyrBMrFbYdeW1L7xLdr18v"},{"@type":"CROSSREF","@value":"10.1007/s00208-019-01864-x_references_DOI_WCS2LTyrBMrFbYdeW1L7xLdr18v"},{"@type":"CROSSREF","@value":"10.2969/jmsj/82788278_references_DOI_WCS2LTyrBMrFbYdeW1L7xLdr18v"},{"@type":"CROSSREF","@value":"10.1016/j.jde.2013.07.061_references_DOI_WCS2LTyrBMrFbYdeW1L7xLdr18v"}]}