In the present study, we establish novel gradient bounds reminiscent of Li–Yau inequalities for strictly positive solutions of a specific category of weighted nonlinear diffusion equations incorporating a potential term, formulated as
$ \partial_{t}u(z, t) = \Delta_{\phi} u^{p}(z, t) + C u^{q}(z, t), \quad (z, t) \in M \times [0, T], $
where the underlying space is a weighted Riemannian manifold $ (M^n, g(t), e^{-\phi} \mathrm{d}v) $ evolving under a geometric flow governed by $ \frac{\partial g}{\partial t} = 2h(t) $. As a direct consequence, we further establish associated Harnack-type inequalities for such evolving settings using partial differential equations.
Citation: Majid Ali Choudhary, Foued Aloui, Maged Z. Youssef, Mohammad Nazrul Islam Khan. Li–Yau gradient estimates of weighted nonlinear diffusion equations under geometric flow[J]. AIMS Mathematics, 2026, 11(6): 16008-16026. doi: 10.3934/math.2026659
In the present study, we establish novel gradient bounds reminiscent of Li–Yau inequalities for strictly positive solutions of a specific category of weighted nonlinear diffusion equations incorporating a potential term, formulated as
$ \partial_{t}u(z, t) = \Delta_{\phi} u^{p}(z, t) + C u^{q}(z, t), \quad (z, t) \in M \times [0, T], $
where the underlying space is a weighted Riemannian manifold $ (M^n, g(t), e^{-\phi} \mathrm{d}v) $ evolving under a geometric flow governed by $ \frac{\partial g}{\partial t} = 2h(t) $. As a direct consequence, we further establish associated Harnack-type inequalities for such evolving settings using partial differential equations.
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Majid Ali Choudhary, Foued Aloui, Maged Z. Youssef, Mohammad Nazrul Islam Khan. Li–Yau gradient estimates of weighted nonlinear diffusion equations under geometric flow[J]. AIMS Mathematics, 2026, 11(6): 16008-16026. doi: 10.3934/math.2026659
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