Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary
We first consider the damped wave inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>u</mi...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-12-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/5/4/258 |
| _version_ | 1797504462095384576 |
|---|---|
| author | Areej Bin Sultan Mohamed Jleli Bessem Samet |
| author_facet | Areej Bin Sultan Mohamed Jleli Bessem Samet |
| author_sort | Areej Bin Sultan |
| collection | DOAJ |
| description | We first consider the damped wave inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>≥</mo><msup><mi>x</mi><mi>σ</mi></msup><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>p</mi></msup><mo>,</mo><mspace width="1.em"></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace><mi>x</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>L</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></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><mn>1</mn></mrow></semantics></math></inline-formula>, under the Dirichlet boundary conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>L</mi><mo>)</mo><mo>)</mo><mo>=</mo><mo>(</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>,</mo><mspace width="1.em"></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> We establish sufficient conditions depending on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>, <i>p</i>, the initial conditions, and the boundary conditions, under which the considered problem admits no global solution. Two cases of boundary conditions are investigated: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>t</mi><mi>γ</mi></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>></mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Next, we extend our study to the time-fractional analogue of the above problem, namely, the time-fractional damped wave inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo><mi>α</mi></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>t</mi><mi>α</mi></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mo>∂</mo><mi>β</mi></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>t</mi><mi>β</mi></msup></mrow></mfrac><mo>≥</mo><msup><mi>x</mi><mi>σ</mi></msup><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>p</mi></msup><mo>,</mo><mspace width="1.em"></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace><mi>x</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>L</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><msup><mo>∂</mo><mi>τ</mi></msup><mrow><mo>∂</mo><msup><mi>t</mi><mi>τ</mi></msup></mrow></mfrac></semantics></math></inline-formula> is the time-Caputo fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>∈</mo><mo>{</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>}</mo></mrow></semantics></math></inline-formula>. Our approach is based on the test function method. Namely, a judicious choice of test functions is made, taking in consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm. |
| first_indexed | 2024-03-10T04:04:54Z |
| format | Article |
| id | doaj.art-013192c159564e4d884ffc6061e68a38 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T04:04:54Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-013192c159564e4d884ffc6061e68a382023-11-23T08:24:27ZengMDPI AGFractal and Fractional2504-31102021-12-015425810.3390/fractalfract5040258Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the BoundaryAreej Bin Sultan0Mohamed Jleli1Bessem Samet2Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaWe first consider the damped wave inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>≥</mo><msup><mi>x</mi><mi>σ</mi></msup><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>p</mi></msup><mo>,</mo><mspace width="1.em"></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace><mi>x</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>L</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></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><mn>1</mn></mrow></semantics></math></inline-formula>, under the Dirichlet boundary conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>L</mi><mo>)</mo><mo>)</mo><mo>=</mo><mo>(</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>,</mo><mspace width="1.em"></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> We establish sufficient conditions depending on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>, <i>p</i>, the initial conditions, and the boundary conditions, under which the considered problem admits no global solution. Two cases of boundary conditions are investigated: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>t</mi><mi>γ</mi></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>></mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Next, we extend our study to the time-fractional analogue of the above problem, namely, the time-fractional damped wave inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo><mi>α</mi></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>t</mi><mi>α</mi></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mo>∂</mo><mi>β</mi></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mi>t</mi><mi>β</mi></msup></mrow></mfrac><mo>≥</mo><msup><mi>x</mi><mi>σ</mi></msup><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>p</mi></msup><mo>,</mo><mspace width="1.em"></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace><mi>x</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>L</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><msup><mo>∂</mo><mi>τ</mi></msup><mrow><mo>∂</mo><msup><mi>t</mi><mi>τ</mi></msup></mrow></mfrac></semantics></math></inline-formula> is the time-Caputo fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>∈</mo><mo>{</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>}</mo></mrow></semantics></math></inline-formula>. Our approach is based on the test function method. Namely, a judicious choice of test functions is made, taking in consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm.https://www.mdpi.com/2504-3110/5/4/258time-fractional damped wave inequalitiesbounded domainsingularitynonexistence |
| spellingShingle | Areej Bin Sultan Mohamed Jleli Bessem Samet Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary Fractal and Fractional time-fractional damped wave inequalities bounded domain singularity nonexistence |
| title | Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary |
| title_full | Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary |
| title_fullStr | Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary |
| title_full_unstemmed | Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary |
| title_short | Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary |
| title_sort | nonexistence of global solutions to time fractional damped wave inequalities in bounded domains with a singular potential on the boundary |
| topic | time-fractional damped wave inequalities bounded domain singularity nonexistence |
| url | https://www.mdpi.com/2504-3110/5/4/258 |
| work_keys_str_mv | AT areejbinsultan nonexistenceofglobalsolutionstotimefractionaldampedwaveinequalitiesinboundeddomainswithasingularpotentialontheboundary AT mohamedjleli nonexistenceofglobalsolutionstotimefractionaldampedwaveinequalitiesinboundeddomainswithasingularpotentialontheboundary AT bessemsamet nonexistenceofglobalsolutionstotimefractionaldampedwaveinequalitiesinboundeddomainswithasingularpotentialontheboundary |