{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T19:47:45Z","timestamp":1754164065583,"version":"3.41.2"},"reference-count":6,"publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"1","license":[{"start":{"date-parts":[[1993,2,1]],"date-time":"1993-02-01T00:00:00Z","timestamp":728524800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire"],"published-print":{"date-parts":[[1993,2]]},"abstract":"\n In this paper we consider a nonlinear version of the simplified Wheeler\u2013DeWitt equation which describes the minisuperspace model for the wave function \n \n \u03c8<\/jats:tex-math>\n <\/jats:inline-formula>\n of a closed universe (cf. [3], [2], [1]). Following [1], where the linear case has been solved, we study this equation as an evolution equation in the scalar field \n \n y \\in \\mathbf R<\/jats:tex-math>\n <\/jats:inline-formula>\n with a scale factor \n \n x\\in ]0, R[<\/jats:tex-math>\n <\/jats:inline-formula>\n . We solve the Cauchy problem for the initial data \n \n \u03c8(x, 0)<\/jats:tex-math>\n <\/jats:inline-formula>\n and \n \n \\frac{\\mathrm{\u2202}\\mathrm{\\psi }}{\\mathrm{\u2202}y}(x,0)<\/jats:tex-math>\n <\/jats:inline-formula>\n and we study some decay properties and blow-up situations. A particular nonlinear version has been proposed in [6].\n <\/jats:p>\n \n R\u00e9sum\u00e9<\/jats:title>\n \n Dans cet article nous consid\u00e9rons une version non lin\u00e9aire de l\u2019\u00e9quation de Wheeler-DeWitt simplifi\u00e9e qui d\u00e9crit le mod\u00e8le de mini-superespace pour la fonction d\u2019onde \u03c8 d\u2019un univers ferm\u00e9 (cf. [3], [2], [1]). Dans l\u2019esprit de [1], o\u00f9 nous avons r\u00e9solu le cas lin\u00e9aire, nous \u00e9tudions cette \u00e9quation comme une \u00e9quation d\u2019\u00e9volution dans le champ scalaire \n \n y \\in \\mathbf R<\/jats:tex-math>\n <\/jats:inline-formula>\n avec le facteur d\u2019\u00e9chelle \n \n x\\in ]0, R[<\/jats:tex-math>\n <\/jats:inline-formula>\n . Nous r\u00e9solvons le probl\u00e8me de Cauchy pour des donn\u00e9es initiales \n \n \u03c8(x, 0)<\/jats:tex-math>\n <\/jats:inline-formula>\n et \n \n \\frac{\\mathrm{\u2202}\\mathrm{\\psi }}{\\mathrm{\u2202}y}(x,0)<\/jats:tex-math>\n <\/jats:inline-formula>\n et nous \u00e9tudions des propri\u00e9t\u00e9s de d\u00e9croissance \u00e0 l\u2019infini et des situations d\u2019explosion de la solution. Une version non lin\u00e9aire particuli\u00e8re a \u00e9t\u00e9 propos\u00e9e dans [6].\n <\/jats:p>\n <\/jats:sec>","DOI":"10.1016\/s0294-1449(16)30223-2","type":"journal-article","created":{"date-parts":[[2017,2,16]],"date-time":"2017-02-16T21:36:08Z","timestamp":1487280968000},"page":"99-107","source":"Crossref","is-referenced-by-count":2,"title":["The Cauchy problem for a nonlinear Wheeler\u2013DeWitt equation"],"prefix":"10.4171","volume":"10","author":[{"given":"J.-P.","family":"Dias","sequence":"first","affiliation":[{"name":"CMAF, Av. Prof. Gama Pinto, 2 1699 Lisboa Codex, Portugal"}]},{"given":"M.","family":"Figueira","sequence":"additional","affiliation":[{"name":"CMAF, Av. Prof. Gama Pinto, 2 1699 Lisboa Codex, Portugal"}]}],"member":"2673","reference":[{"key":"10.1016\/S0294-1449(16)30223-2_bib1","first-page":"17","article-title":"The Simplified Wheeler-DeWitt Equation: the Cauchy Problem and Some Spectral Properties","volume":"Vol. 54","author":"Dias","year":"1991","journal-title":"Ann. Inst. H. Poincar\u00e9"},{"key":"10.1016\/S0294-1449(16)30223-2_bib2","doi-asserted-by":"crossref","first-page":"736","DOI":"10.1016\/0550-3213(89)90405-7","article-title":"What is a Typical Wave Function for the Universe?","volume":"Vol. 313","author":"Gibbons","year":"1989","journal-title":"Nucl. Phy. B"},{"key":"10.1016\/S0294-1449(16)30223-2_bib3","doi-asserted-by":"crossref","first-page":"2960","DOI":"10.1103\/PhysRevD.28.2960","article-title":"Wave Function of the Universe","volume":"Vol. 28","author":"Hartle","year":"1983","journal-title":"Phys. Rev. D"},{"year":"1983","series-title":"Semigroups of Linear Operators and Applications to Partial Differential Equations","author":"Pazy","key":"10.1016\/S0294-1449(16)30223-2_bib4"},{"year":"1975","series-title":"Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness","author":"Reed","key":"10.1016\/S0294-1449(16)30223-2_bib5"},{"key":"10.1016\/S0294-1449(16)30223-2_bib6","unstructured":"L. Susskind, Lectures at the Trieste School on String Theories and Quantum Gravity, April 1990."}],"container-title":["Annales de l'Institut Henri Poincar\u00e9 C, Analyse non lin\u00e9aire"],"original-title":[],"link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0294144916302232?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0294144916302232?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,7,31]],"date-time":"2025-07-31T18:49:16Z","timestamp":1753987756000},"score":1,"resource":{"primary":{"URL":"https:\/\/ems.press\/doi\/10.1016\/s0294-1449(16)30223-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,2]]},"references-count":6,"journal-issue":{"issue":"1"},"URL":"https:\/\/doi.org\/10.1016\/s0294-1449(16)30223-2","relation":{},"ISSN":["0294-1449","1873-1430"],"issn-type":[{"type":"print","value":"0294-1449"},{"type":"electronic","value":"1873-1430"}],"subject":[],"published":{"date-parts":[[1993,2]]}}}