<i>L</i><sup>2</sup> Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity
This article studies the Schrödinger equation with an inhomogeneous combined term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><msub><mo>∂</mo><mi>t</mi&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/7/1060 |
| _version_ | 1797212242430656512 |
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| author | Baoli Xie Congming Peng Caochuan Ma |
| author_facet | Baoli Xie Congming Peng Caochuan Ma |
| author_sort | Baoli Xie |
| collection | DOAJ |
| description | This article studies the Schrödinger equation with an inhomogeneous combined term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><msub><mo>∂</mo><mi>t</mi></msub><mi>u</mi><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mo>Δ</mo><mo stretchy="false">)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><msub><mi>λ</mi><mn>1</mn></msub><msup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><mrow><mo>−</mo><mi>b</mi></mrow></msup><msup><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow><mi>p</mi></msup><mi>u</mi><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><msup><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow><mi>q</mi></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>λ</mi><mn>2</mn></msub><mo>=</mo><mo>±</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo><</mo><mi>b</mi><mo><</mo><mrow><mo>{</mo><mn>2</mn><mi>s</mi><mo>,</mo><mi>N</mi><mo stretchy="false">}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><msub><mi>λ</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>λ</mi><mn>2</mn></msub></semantics></math></inline-formula> have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters. |
| first_indexed | 2024-04-24T10:39:16Z |
| format | Article |
| id | doaj.art-90ab95d2205345b3a00e9b7e89531ca1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T10:39:16Z |
| publishDate | 2024-04-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-90ab95d2205345b3a00e9b7e89531ca12024-04-12T13:22:45ZengMDPI AGMathematics2227-73902024-04-01127106010.3390/math12071060<i>L</i><sup>2</sup> Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-LinearityBaoli Xie0Congming Peng1Caochuan Ma2School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, ChinaSchool of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, ChinaSchool of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, ChinaThis article studies the Schrödinger equation with an inhomogeneous combined term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><msub><mo>∂</mo><mi>t</mi></msub><mi>u</mi><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mo>Δ</mo><mo stretchy="false">)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><msub><mi>λ</mi><mn>1</mn></msub><msup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><mrow><mo>−</mo><mi>b</mi></mrow></msup><msup><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow><mi>p</mi></msup><mi>u</mi><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><msup><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow><mi>q</mi></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>λ</mi><mn>2</mn></msub><mo>=</mo><mo>±</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo><</mo><mi>b</mi><mo><</mo><mrow><mo>{</mo><mn>2</mn><mi>s</mi><mo>,</mo><mi>N</mi><mo stretchy="false">}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><msub><mi>λ</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>λ</mi><mn>2</mn></msub></semantics></math></inline-formula> have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.https://www.mdpi.com/2227-7390/12/7/1060inhomogeneous Schrödinger equationL2 concentrationlimit behaviour |
| spellingShingle | Baoli Xie Congming Peng Caochuan Ma <i>L</i><sup>2</sup> Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity Mathematics inhomogeneous Schrödinger equation L2 concentration limit behaviour |
| title | <i>L</i><sup>2</sup> Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity |
| title_full | <i>L</i><sup>2</sup> Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity |
| title_fullStr | <i>L</i><sup>2</sup> Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity |
| title_full_unstemmed | <i>L</i><sup>2</sup> Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity |
| title_short | <i>L</i><sup>2</sup> Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity |
| title_sort | i l i sup 2 sup concentration of blow up solutions for the nonlinear schrodinger equation with an inhomogeneous combined non linearity |
| topic | inhomogeneous Schrödinger equation L2 concentration limit behaviour |
| url | https://www.mdpi.com/2227-7390/12/7/1060 |
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