{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T15:51:45Z","timestamp":1776354705937,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"10","license":[{"start":{"date-parts":[[2014,6,18]],"date-time":"2014-06-18T00:00:00Z","timestamp":1403049600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    We establish local well-posedness results for the Initial Value Problem associated to the Schr\u00f6dinger-Debye system in dimensions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N equals 2 comma 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N=2, 3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for data in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript s Baseline times upper H Superscript script l\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u2113\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H^s\\times H^{\\ell }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n                        <mml:semantics>\n                          <mml:mi>s<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u2113\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    satisfying\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"max left-brace right-brace comma 0 comma minus minus s 1 less-than-or-equal-to script l less-than-or-equal-to min left-brace right-brace comma times times 2 s comma plus plus s 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo movablelimits=\"true\" form=\"prefix\">max<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mo movablelimits=\"true\" form=\"prefix\">min<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\max \\{0, s-1\\} \\le \\ell \\le \\min \\{2s, s+1\\}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In particular, these include the energy space\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1 Baseline times upper L squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1\\times L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Our results improve the previous ones obtained by B. Bid\u00e9garay, and by A.\u00a0J. Corcho and F.\u00a0Linares. Moreover, in the critical case (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N equals 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N=2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) and for initial data in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1 Baseline times upper L squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1\\times L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned by G.\u00a0Fibich and G.\u00a0C. Papanicolau concerning the formation of singularities for these solutions.\n                  <\/p>","DOI":"10.1090\/s0002-9939-2013-11612-6","type":"journal-article","created":{"date-parts":[[2013,6,18]],"date-time":"2013-06-18T15:59:30Z","timestamp":1371571170000},"page":"3485-3499","source":"Crossref","is-referenced-by-count":8,"special_numbering":"652","title":["Local and global well-posedness for the critical Schr\u00f6dinger-Debye system"],"prefix":"10.1090","volume":"141","author":[{"given":"Ad\u00e1n","family":"Corcho","sequence":"first","affiliation":[]},{"given":"Filipe","family":"Oliveira","sequence":"additional","affiliation":[]},{"given":"Jorge","family":"Silva","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,6,18]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1051\/m2an:2001106","article-title":"Numerical study of self-focusing solutions to the Schr\u00f6dinger-Debye system","volume":"35","author":"Besse, Christophe","year":"2001","journal-title":"M2AN Math. 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