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Updated: Feb 20, 2026
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Published on: September 26, 2016
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Weighted Aronson-Bénilan estimates and Harnack inequalities for slow diffusion equations with a nonlinear forcing
Ali Taheri1, Vahideh Vahidifar1
1School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, United Kingdom.
Nonlinear Differential Equations and Applications : Nodea
|February 19, 2026
Summary
This study introduces novel gradient estimates for nonlinear slow diffusion equations, enhancing understanding of their behavior in complex spaces. These findings improve existing theories and offer new insights into parabolic inequalities.
Area of Science:
- Nonlinear Partial Differential Equations
- Geometric Analysis
- Differential Geometry
Background:
- Nonlinear slow diffusion equations model various physical phenomena.
- Existing gradient estimates have limitations in complex geometric settings.
- Understanding solution dynamics is crucial for applications.
Purpose of the Study:
- To develop new Aronson-Bénilan and Li-Yau type gradient estimates.
- To extend these estimates to smooth metric measure spaces (weighted manifolds).
- To unify and improve existing results on slow diffusion equations.
Main Methods:
- Formulation and proof of novel gradient estimates.
- Utilizing Harnack quantities with time-variable coefficients.
- Exploiting the interplay between geometry, nonlinearity, and equation dynamics.
Main Results:
- New Aronson-Bénilan and Li-Yau type gradient estimates for positive solutions.
- Demonstrated applicability within smooth metric measure spaces.
- Extension, unification, and improvement of prior estimates.
Conclusions:
- The developed estimates offer a significant advancement in the analysis of nonlinear diffusion.
- Implications for parabolic Harnack inequalities and global bounds are established.
- Provides a unified framework for studying these equations in geometric settings.
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