Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations
This article deals with finite-time quenching for the system of coupled semi-linear heat equations <inline-formula><math display="inline"><semantics><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><msub><mi&g...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/12/2075 |
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| author | Ratinan Boonklurb Tawikan Treeyaprasert Aong-art Wanna |
| author_facet | Ratinan Boonklurb Tawikan Treeyaprasert Aong-art Wanna |
| author_sort | Ratinan Boonklurb |
| collection | DOAJ |
| description | This article deals with finite-time quenching for the system of coupled semi-linear heat equations <inline-formula><math display="inline"><semantics><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>v</mi><mi>t</mi></msub><mo>=</mo><msub><mi>v</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>f</i> and <i>g</i> are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given. |
| first_indexed | 2024-03-10T14:04:48Z |
| format | Article |
| id | doaj.art-56461f36a5e243c5bef5938daf72b764 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T14:04:48Z |
| publishDate | 2020-12-01 |
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| series | Symmetry |
| spelling | doaj.art-56461f36a5e243c5bef5938daf72b7642023-11-21T00:46:52ZengMDPI AGSymmetry2073-89942020-12-011212207510.3390/sym12122075Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat EquationsRatinan Boonklurb0Tawikan Treeyaprasert1Aong-art Wanna2Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, ThailandDepartment of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center Khlong Luang, Pathum Thani 12120, ThailandDepartment of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, ThailandThis article deals with finite-time quenching for the system of coupled semi-linear heat equations <inline-formula><math display="inline"><semantics><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>v</mi><mi>t</mi></msub><mo>=</mo><msub><mi>v</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>f</i> and <i>g</i> are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given.https://www.mdpi.com/2073-8994/12/12/2075simultaneous quenchingnonsimultaneous quenchingsemi-linear parabolic system |
| spellingShingle | Ratinan Boonklurb Tawikan Treeyaprasert Aong-art Wanna Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations Symmetry simultaneous quenching nonsimultaneous quenching semi-linear parabolic system |
| title | Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations |
| title_full | Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations |
| title_fullStr | Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations |
| title_full_unstemmed | Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations |
| title_short | Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations |
| title_sort | simultaneous and non simultaneous quenching for a system of multi dimensional semi linear heat equations |
| topic | simultaneous quenching nonsimultaneous quenching semi-linear parabolic system |
| url | https://www.mdpi.com/2073-8994/12/12/2075 |
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