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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/10/1683 |
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| Summary: | 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 |