{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:45:28Z","timestamp":1760237128098,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,4]],"date-time":"2020-03-04T00:00:00Z","timestamp":1583280000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The third author is supported by Researchers Supporting Project number (RSP- 6 2019\/4), King Saud University, Riyadh, Saudi Arabia","award":["RSP- 6 2019\/4"],"award-info":[{"award-number":["RSP- 6 2019\/4"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study the large-time behavior of solutions to the nonlinear exterior problem     L u  ( t , x )  =   \u03ba | u  ( t , x )  |  p  ,   ( t , x )  \u2208  ( 0 , \u221e )  \u00d7  D c      under the nonhomegeneous Neumann boundary condition       \u2202 u   \u2202 \u03bd    ( t , x )  = \u03bb  ( x )  ,   ( t , x )  \u2208  ( 0 , \u221e )  \u00d7 \u2202 D ,     where     L : = i  \u2202 t  + \u0394     is the Schr\u00f6dinger operator,     D = B ( 0 , 1 )     is the open unit ball in     R N    ,     N \u2265 2    ,      D c  =  R N   \u2216 D     ,     p &gt; 1    ,     \u03ba \u2208 C    ,     \u03ba \u2260 0    ,     \u03bb \u2208  L 1   ( \u2202 D , C )      is a nontrivial complex valued function, and     \u2202 \u03bd     is the outward unit normal vector on     \u2202 D    , relative to     D c    . Namely, under a certain condition imposed on     ( \u03ba , \u03bb )    , we show that if     N \u2265 3     and     p &lt;  p c     , where      p c  =  N  N \u2212 2   ,     then the considered problem admits no global weak solutions. However, if     N = 2    , then for all     p &gt; 1    , the problem admits no global weak solutions. The proof is based on the test function method introduced by Mitidieri and Pohozaev, and an adequate choice of the test function.<\/jats:p>","DOI":"10.3390\/sym12030394","type":"journal-article","created":{"date-parts":[[2020,3,4]],"date-time":"2020-03-04T03:24:20Z","timestamp":1583292260000},"page":"394","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Nonexistence of Global Weak Solutions for a Nonlinear Schr\u00f6dinger Equation in an Exterior Domain"],"prefix":"10.3390","volume":"12","author":[{"given":"Awatif","family":"Alqahtani","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabi"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6095-5875","authenticated-orcid":false,"given":"Mohamed","family":"Jleli","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabi"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bessem","family":"Samet","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabi"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5836-6847","authenticated-orcid":false,"given":"Calogero","family":"Vetro","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1794","DOI":"10.1063\/1.523491","article-title":"On the blowing-up of solutions to the Cauchy problem for the nonlinear Schr\u00f6dinger equation","volume":"18","author":"Glassey","year":"1977","journal-title":"J. Math. 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