{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T13:02:20Z","timestamp":1775480540657,"version":"3.50.1"},"reference-count":4,"publisher":"American Mathematical Society (AMS)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"published-print":{"date-parts":[[1985,8]]},"abstract":"<p>\n                    We find a necessary and sufficient condition for a Goursat problem for a third order partial differential operator with constant coefficients of the form\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C 2 left-parenthesis upper D Subscript x Baseline comma upper D Subscript y Baseline right-parenthesis upper D Subscript t plus upper C 3 left-parenthesis upper D Subscript x Baseline comma upper D Subscript y Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>C<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>D<\/mml:mi>\n                                <mml:mi>x<\/mml:mi>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>D<\/mml:mi>\n                                <mml:mi>y<\/mml:mi>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>D<\/mml:mi>\n                                <mml:mi>t<\/mml:mi>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>C<\/mml:mi>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>D<\/mml:mi>\n                                <mml:mi>x<\/mml:mi>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>D<\/mml:mi>\n                                <mml:mi>y<\/mml:mi>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{C_2}({D_x},{D_y}){D_t} + {C_3}({D_x},{D_y})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    to be\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Superscript normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:msup>\n                              <mml:mi>C<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{C^\\infty }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -well posed, showing at the same time that a necessary and sufficient condition of Hasegawa cannot be extended. The result can be generalised to operators of higher orders but leads to cumbersome conditions; nevertheless, we show that the condition of Hasegawa is also not sufficient in this case.\n                  <\/p>","DOI":"10.1090\/s0002-9939-1985-0792271-9","type":"journal-article","created":{"date-parts":[[2010,6,30]],"date-time":"2010-06-30T15:58:06Z","timestamp":1277913486000},"page":"612-616","source":"Crossref","is-referenced-by-count":0,"title":["On the well-posedness of a \ud835\udc36^{\u221e} Goursat problem for a partial differential operator of order greater than two"],"prefix":"10.1090","volume":"94","author":[{"given":"Jaime","family":"Carvalho e Silva","sequence":"first","affiliation":[]}],"member":"14","reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1215\/kjm\/1250522472","article-title":"On the \ud835\udc36^{\u221e}-problem for equations with constant coefficients","volume":"19","author":"Hasegawa, Yukiko","year":"1979","journal-title":"J. Math. Kyoto Univ.","ISSN":"https:\/\/id.crossref.org\/issn\/0023-608X","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"L. H\u00f6rmander, Linear partial differential operators, Springer-Verlag, Berlin, 1963.","DOI":"10.1007\/978-3-642-46175-0"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1215\/kjm\/1250522329","article-title":"On the \\cal\ud835\udc38-well posedness for the Goursat problem with constant coefficients","volume":"20","author":"Nishitani, Tatsuo","year":"1980","journal-title":"J. Math. Kyoto Univ.","ISSN":"https:\/\/id.crossref.org\/issn\/0023-608X","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/BF02589555","article-title":"Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part","volume":"8","author":"Svensson, S. Leif","year":"1969","journal-title":"Ark. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0004-2080","issn-type":"print"}],"container-title":["Proceedings of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/proc\/1985-094-04\/S0002-9939-1985-0792271-9\/S0002-9939-1985-0792271-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/proc\/1985-094-04\/S0002-9939-1985-0792271-9\/S0002-9939-1985-0792271-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T12:21:36Z","timestamp":1775478096000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/proc\/1985-094-04\/S0002-9939-1985-0792271-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,8]]},"references-count":4,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1985,8]]}},"alternative-id":["S0002-9939-1985-0792271-9"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9939-1985-0792271-9","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6826","0002-9939"],"issn-type":[{"value":"1088-6826","type":"electronic"},{"value":"0002-9939","type":"print"}],"subject":[],"published":{"date-parts":[[1985,8]]}}}