Second-Order Gradient Estimates for the Porous Medium Equation on Riemannian Manifolds
In this paper, we derive second-order gradient estimates for positive solutions of the porous medium equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaysty...
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2025-05-01
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| author | Jingjing Yang Guangwen Zhao |
| author_facet | Jingjing Yang Guangwen Zhao |
| author_sort | Jingjing Yang |
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| description | In this paper, we derive second-order gradient estimates for positive solutions of the porous medium equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac></mstyle><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mo>Δ</mo><mi>u</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mi>p</mi></msup><mo>,</mo><mspace width="1.em"></mspace><mi>p</mi><mo>∈</mo><mfenced separators="" open="(" close=")"><mn>1</mn><mo>,</mo><mn>1</mn><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><msqrt><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mfrac></mstyle></mfenced></mrow></semantics></math></inline-formula> on an <i>n</i>-dimensional Riemannian manifold under certain curvature conditions. |
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| issn | 2227-7390 |
| language | English |
| publishDate | 2025-05-01 |
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| spelling | doaj-art-3ea47eb455ac41aeae73dc390be426062025-08-20T03:14:32ZengMDPI AGMathematics2227-73902025-05-011310168310.3390/math13101683Second-Order Gradient Estimates for the Porous Medium Equation on Riemannian ManifoldsJingjing Yang0Guangwen Zhao1School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, ChinaSchool of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, ChinaIn this paper, we derive second-order gradient estimates for positive solutions of the porous medium equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac></mstyle><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mo>Δ</mo><mi>u</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mi>p</mi></msup><mo>,</mo><mspace width="1.em"></mspace><mi>p</mi><mo>∈</mo><mfenced separators="" open="(" close=")"><mn>1</mn><mo>,</mo><mn>1</mn><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><msqrt><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mfrac></mstyle></mfenced></mrow></semantics></math></inline-formula> on an <i>n</i>-dimensional Riemannian manifold under certain curvature conditions.https://www.mdpi.com/2227-7390/13/10/1683gradient estimateporous medium equationRiemannian manifold |
| spellingShingle | Jingjing Yang Guangwen Zhao Second-Order Gradient Estimates for the Porous Medium Equation on Riemannian Manifolds Mathematics gradient estimate porous medium equation Riemannian manifold |
| title | Second-Order Gradient Estimates for the Porous Medium Equation on Riemannian Manifolds |
| title_full | Second-Order Gradient Estimates for the Porous Medium Equation on Riemannian Manifolds |
| title_fullStr | Second-Order Gradient Estimates for the Porous Medium Equation on Riemannian Manifolds |
| title_full_unstemmed | Second-Order Gradient Estimates for the Porous Medium Equation on Riemannian Manifolds |
| title_short | Second-Order Gradient Estimates for the Porous Medium Equation on Riemannian Manifolds |
| title_sort | second order gradient estimates for the porous medium equation on riemannian manifolds |
| topic | gradient estimate porous medium equation Riemannian manifold |
| url | https://www.mdpi.com/2227-7390/13/10/1683 |
| work_keys_str_mv | AT jingjingyang secondordergradientestimatesfortheporousmediumequationonriemannianmanifolds AT guangwenzhao secondordergradientestimatesfortheporousmediumequationonriemannianmanifolds |