In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation
$ \frac{\partial}{\partial t}f = \Delta f+af(\ln f)^{p} $
with Neumann boundary conditions on a compact Riemannian manifold satisfying the integral Ricci curvature assumption, where $ p\geq 0 $ is a real constant. This contrasts Olivé's gradient estimate, which works mainly for the heat equation rather than nonlinear parabolic equations and the result can be regarded as a generalization of the Li-Yau [P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153–201] and Olivé [X. R. Olivé, Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, Proc. Amer. Math. Soc., 147 (2019), 411–426] gradient estimates.
Citation: Hao-Yue Liu, Wei Zhang. Neumann gradient estimate for nonlinear heat equation under integral Ricci curvature bounds[J]. AIMS Mathematics, 2024, 9(2): 3881-3894. doi: 10.3934/math.2024191
In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation
$ \frac{\partial}{\partial t}f = \Delta f+af(\ln f)^{p} $
with Neumann boundary conditions on a compact Riemannian manifold satisfying the integral Ricci curvature assumption, where $ p\geq 0 $ is a real constant. This contrasts Olivé's gradient estimate, which works mainly for the heat equation rather than nonlinear parabolic equations and the result can be regarded as a generalization of the Li-Yau [P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator,
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Hao-Yue Liu, Wei Zhang. Neumann gradient estimate for nonlinear heat equation under integral Ricci curvature bounds[J]. AIMS Mathematics, 2024, 9(2): 3881-3894. doi: 10.3934/math.2024191
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