Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain
We study the large-time behavior of solutions to the nonlinear exterior problem <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">L</mi> <mi>u</mi> <mrow> <mo stretchy="false">...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-03-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/12/3/394 |
| _version_ | 1817993493240348672 |
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| author | Awatif Alqahtani Mohamed Jleli Bessem Samet Calogero Vetro |
| author_facet | Awatif Alqahtani Mohamed Jleli Bessem Samet Calogero Vetro |
| author_sort | Awatif Alqahtani |
| collection | DOAJ |
| description | We study the large-time behavior of solutions to the nonlinear exterior problem <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">L</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>κ</mi> <mo stretchy="false">|</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> <mo>,</mo> <mspace width="1.em"></mspace> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mi>D</mi> <mi>c</mi> </msup> </mrow> </semantics> </math> </inline-formula> under the nonhomegeneous Neumann boundary condition <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mi>ν</mi> </mrow> </mfrac> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1.em"></mspace> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <mo>∂</mo> <mi>D</mi> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">L</mi> <mo>:</mo> <mo>=</mo> <mi>i</mi> <msub> <mo>∂</mo> <mi>t</mi> </msub> <mo>+</mo> <mo>Δ</mo> </mrow> </semantics> </math> </inline-formula> is the Schrödinger operator, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>=</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is the open unit ball in <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>D</mi> <mi>c</mi> </msup> <mo>=</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> <mrow> <mo>∖</mo> <mi>D</mi> </mrow> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>κ</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>κ</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mo>∂</mo> <mi>D</mi> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is a nontrivial complex valued function, and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∂</mo> <mi>ν</mi> </mrow> </semantics> </math> </inline-formula> is the outward unit normal vector on <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∂</mo> <mi>D</mi> </mrow> </semantics> </math> </inline-formula>, relative to <inline-formula> <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>c</mi> </msup> </semantics> </math> </inline-formula>. Namely, under a certain condition imposed on <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula>, we show that if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo><</mo> <msub> <mi>p</mi> <mi>c</mi> </msub> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>c</mi> </msub> <mo>=</mo> <mfrac> <mi>N</mi> <mrow> <mi>N</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> then the considered problem admits no global weak solutions. However, if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, then for all <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, 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. |
| first_indexed | 2024-04-14T01:39:41Z |
| format | Article |
| id | doaj.art-bf0721dcc8d34287965571a279dc9aa2 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-14T01:39:41Z |
| publishDate | 2020-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-bf0721dcc8d34287965571a279dc9aa22022-12-22T02:19:48ZengMDPI AGSymmetry2073-89942020-03-0112339410.3390/sym12030394sym12030394Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior DomainAwatif Alqahtani0Mohamed Jleli1Bessem Samet2Calogero Vetro3Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiDepartment of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, ItalyWe study the large-time behavior of solutions to the nonlinear exterior problem <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">L</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>κ</mi> <mo stretchy="false">|</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> <mo>,</mo> <mspace width="1.em"></mspace> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mi>D</mi> <mi>c</mi> </msup> </mrow> </semantics> </math> </inline-formula> under the nonhomegeneous Neumann boundary condition <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>u</mi> </mrow> <mrow> <mo>∂</mo> <mi>ν</mi> </mrow> </mfrac> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1.em"></mspace> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <mo>∂</mo> <mi>D</mi> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">L</mi> <mo>:</mo> <mo>=</mo> <mi>i</mi> <msub> <mo>∂</mo> <mi>t</mi> </msub> <mo>+</mo> <mo>Δ</mo> </mrow> </semantics> </math> </inline-formula> is the Schrödinger operator, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>=</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is the open unit ball in <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>D</mi> <mi>c</mi> </msup> <mo>=</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> <mrow> <mo>∖</mo> <mi>D</mi> </mrow> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>κ</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>κ</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mo>∂</mo> <mi>D</mi> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is a nontrivial complex valued function, and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∂</mo> <mi>ν</mi> </mrow> </semantics> </math> </inline-formula> is the outward unit normal vector on <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∂</mo> <mi>D</mi> </mrow> </semantics> </math> </inline-formula>, relative to <inline-formula> <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>c</mi> </msup> </semantics> </math> </inline-formula>. Namely, under a certain condition imposed on <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula>, we show that if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo><</mo> <msub> <mi>p</mi> <mi>c</mi> </msub> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>c</mi> </msub> <mo>=</mo> <mfrac> <mi>N</mi> <mrow> <mi>N</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> then the considered problem admits no global weak solutions. However, if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, then for all <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, 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.https://www.mdpi.com/2073-8994/12/3/394nonlinear schrödinger equationexterior domainnonhomegeneous neumann boundary conditionglobal weak solution |
| spellingShingle | Awatif Alqahtani Mohamed Jleli Bessem Samet Calogero Vetro Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain Symmetry nonlinear schrödinger equation exterior domain nonhomegeneous neumann boundary condition global weak solution |
| title | Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain |
| title_full | Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain |
| title_fullStr | Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain |
| title_full_unstemmed | Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain |
| title_short | Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain |
| title_sort | nonexistence of global weak solutions for a nonlinear schrodinger equation in an exterior domain |
| topic | nonlinear schrödinger equation exterior domain nonhomegeneous neumann boundary condition global weak solution |
| url | https://www.mdpi.com/2073-8994/12/3/394 |
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