Li–Yau gradient estimates of weighted nonlinear diffusion equations under geometric flow

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

$${\mathrm{\partial }}_{t}u(z,t)={\mathrm{\Delta }}_{\varphi }{u}^p(z,t)+C{u}^q(z,t),(z,t)\in M\times [0,T],$$

where the underlying space is a weighted Riemannian manifold $(M^n,g(t),e^{-\varphi }\mathrm{d}v)$ evolving under a geometric flow governed by $\frac{\mathrm{\partial }g}{\mathrm{\partial }t}=2h(t)$. As a direct consequence, we further establish associated Harnack-type inequalities for such evolving settings using partial differential equations.