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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.
None:
We formulate and prove new Aronson-Bénilan and Li-Yau type gradient estimates for positive solutions to nonlinear slow diffusion equations. The framework is that of a smooth metric measure space (i.e., a weighted manifold) and the estimates make use of a range of Harnack quantities with suitable time-variable coefficients. The proofs exploit the intricate relation between geometry, nonlinearity and dynamics of the equation and the results extend, unify and improve various earlier estimates on slow diffusion equations. A number of important corollaries and implications, notably, to parabolic Harnack inequalities and global bounds are presented and discussed.
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