Exponential Scattering for a Damped Hartree Equation
This note studies the linearly damped generalized Hartree equation <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mover accent="true"><mi>u</mi><mo>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/1/51 |
| _version_ | 1827625594221756416 |
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| author | Talal Alharbi Salah Boulaaras Tarek Saanouni |
| author_facet | Talal Alharbi Salah Boulaaras Tarek Saanouni |
| author_sort | Talal Alharbi |
| collection | DOAJ |
| description | This note studies the linearly damped generalized Hartree equation <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi mathvariant="sans-serif">Δ</mi><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><mi>i</mi><mi>a</mi><mi>u</mi><mo>=</mo><mo>±</mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mrow><mo stretchy="false">(</mo></mrow><msub><mi>J</mi><mi>γ</mi></msub><msup><mrow><mo>∗</mo><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>p</mi></msup><mrow><mo>)</mo><mi>u</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mn>0</mn><mo><</mo><mi>s</mi><mo><</mo><mn>1</mn><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>p</mi><mo>≥</mo><mn>2</mn><mo>.</mo></mrow></mrow></semantics></math></inline-formula> Indeed, one proves an exponential scattering of the energy global solutions, with spherically symmetric datum. This means that, for large time, the solution goes exponentially to the solution of the associated free problem <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi mathvariant="sans-serif">Δ</mi><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><mi>i</mi><mi>a</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, in <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>H</mi><mi>s</mi></msup></semantics></math></inline-formula> norm. The radial assumption avoids a loss of regularity in Strichartz estimates. The exponential scattering, which means that <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>:</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>a</mi><mi>t</mi></mrow></msup><mi>u</mi></mrow></semantics></math></inline-formula> scatters in <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>H</mi><mi>s</mi></msup></semantics></math></inline-formula>, is proved in the energy sub-critical defocusing regime and in the mass-sub-critical focusing regime. This result is presented because of the gap due to the lack of scattering in the mass sub-critical regime, which seems not to be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and the last one is a damping term <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mi>a</mi><mi>u</mi></mrow></semantics></math></inline-formula>. In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kind of damping terms. |
| first_indexed | 2024-03-09T12:39:51Z |
| format | Article |
| id | doaj.art-682c0b4c40ba4a7eb54ab48f90766175 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T12:39:51Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-682c0b4c40ba4a7eb54ab48f907661752023-11-30T22:19:33ZengMDPI AGFractal and Fractional2504-31102023-01-01715110.3390/fractalfract7010051Exponential Scattering for a Damped Hartree EquationTalal Alharbi0Salah Boulaaras1Tarek Saanouni2Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah 52571, Saudi ArabiaDepartment of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass 58822, Saudi ArabiaDepartment of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah 52571, Saudi ArabiaThis note studies the linearly damped generalized Hartree equation <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi mathvariant="sans-serif">Δ</mi><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><mi>i</mi><mi>a</mi><mi>u</mi><mo>=</mo><mo>±</mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mrow><mo stretchy="false">(</mo></mrow><msub><mi>J</mi><mi>γ</mi></msub><msup><mrow><mo>∗</mo><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>p</mi></msup><mrow><mo>)</mo><mi>u</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mn>0</mn><mo><</mo><mi>s</mi><mo><</mo><mn>1</mn><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>p</mi><mo>≥</mo><mn>2</mn><mo>.</mo></mrow></mrow></semantics></math></inline-formula> Indeed, one proves an exponential scattering of the energy global solutions, with spherically symmetric datum. This means that, for large time, the solution goes exponentially to the solution of the associated free problem <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi mathvariant="sans-serif">Δ</mi><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><mi>i</mi><mi>a</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, in <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>H</mi><mi>s</mi></msup></semantics></math></inline-formula> norm. The radial assumption avoids a loss of regularity in Strichartz estimates. The exponential scattering, which means that <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>:</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>a</mi><mi>t</mi></mrow></msup><mi>u</mi></mrow></semantics></math></inline-formula> scatters in <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>H</mi><mi>s</mi></msup></semantics></math></inline-formula>, is proved in the energy sub-critical defocusing regime and in the mass-sub-critical focusing regime. This result is presented because of the gap due to the lack of scattering in the mass sub-critical regime, which seems not to be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and the last one is a damping term <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mi>a</mi><mi>u</mi></mrow></semantics></math></inline-formula>. In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kind of damping terms.https://www.mdpi.com/2504-3110/7/1/51Choquard equationdampingglobal existenceexponential scattering |
| spellingShingle | Talal Alharbi Salah Boulaaras Tarek Saanouni Exponential Scattering for a Damped Hartree Equation Fractal and Fractional Choquard equation damping global existence exponential scattering |
| title | Exponential Scattering for a Damped Hartree Equation |
| title_full | Exponential Scattering for a Damped Hartree Equation |
| title_fullStr | Exponential Scattering for a Damped Hartree Equation |
| title_full_unstemmed | Exponential Scattering for a Damped Hartree Equation |
| title_short | Exponential Scattering for a Damped Hartree Equation |
| title_sort | exponential scattering for a damped hartree equation |
| topic | Choquard equation damping global existence exponential scattering |
| url | https://www.mdpi.com/2504-3110/7/1/51 |
| work_keys_str_mv | AT talalalharbi exponentialscatteringforadampedhartreeequation AT salahboulaaras exponentialscatteringforadampedhartreeequation AT tareksaanouni exponentialscatteringforadampedhartreeequation |