Differential Harnack inequalities for nonlinear heat equations with potentials under the Ricci flow (Q442055)

\textit{P. Li} and \textit{S.-T. Yau} [Acta Math. 156, 153--201 (1986; Zbl 0611.58045)] proved a gradient estimate for solutions of heat equations on a Riemannian manifold using the maximum principle. They also derived a Harnack inequality from this by integrating the gradient estimate along space-time paths. These so-called differential Harnack inequalities have been generalized to nonlinear heat type equations by Yau and by X. Xu and J. Li, and to various curvature flow problems by Hamilton, Chow, Cao and Andrews, among others.NEWLINENEWLINEAnother direction of research has been to derive differential Harnack inequalities for backward heat equations on a manifold with metric evolving under the Ricci flow NEWLINE\[NEWLINE \frac{\partial}{\partial t} g_{ij} = -2R_{ij}, \leqno(1) NEWLINE\]NEWLINE for example, by Hamilton, Chow and Knopf, Cheng, and others. In particular, \textit{G. Perelman} [``The entropy formula for the Ricci flow and its geometric applications'', arXiv e-print service, Cornell University Library, Paper No. 0211159, 39 p. (2002; Zbl 1130.53001)] proved a Harnack inequality for the fundamental solution of the backward heat equation coupled with the Ricci flow without curvature assumptions. This result has many important applications.NEWLINENEWLINEThe forward heat equation coupled with the Ricci flow has also been studied by various authors. More recently, nonlinear forward heat equations of the form NEWLINE\[NEWLINE \frac{\partial }{\partial t}f = \Delta f - af\ln f - bf \leqno(2) NEWLINE\]NEWLINE with real constants \(a,b\), coupled with the Ricci flow, were studied by \textit{L. Ma} [J. Funct. Anal. 241, No. 1, 374--382 (2006; Zbl 1112.58023)]. Many related results have been proved. We mention only that of \textit{X. Cao} and \textit{Z. Zhang} [Springer Proc. Math. 8, 87--98 (2011; Zbl 1228.53078)].NEWLINENEWLINEHere the author proves several differential Harnack inequalities for various forward and backward heat type equations. The first result is for positive solutions of the forward heat equation NEWLINE\[NEWLINE \frac{\partial }{\partial t}f = \Delta f - f\ln f + \varepsilon Rf \leqno(3) NEWLINE\]NEWLINE on a closed surface with metric evolving by the \(\varepsilon\)-Ricci flow NEWLINE\[NEWLINE \frac{\partial}{\partial t} g_{ij} = -\varepsilon Rg _{ij}, \leqno(4) NEWLINE\]NEWLINE where \(R\) is the scalar curvature, and \(\varepsilon\geq 0\). Assuming \(R>0\) and that \(f\) is a positive solution of (4), the author shows that if \(u=-\ln f\), \(H_\varepsilon=\Delta u -\varepsilon R\), then \(H_\varepsilon \leq \frac{1}{t}\). This gives interesting corollaries in the cases \(\varepsilon=0\) and \(\varepsilon=1\). In particular, it improves (in two dimensions only) a result of Cao and Zhang.NEWLINENEWLINEThe remainder of the paper deals with differential Harnack inequalities for the nonlinear backward heat equation NEWLINE\[NEWLINE \frac{\partial }{\partial t}f = - \Delta f + f\ln f + m Rf \leqno(5) NEWLINE\]NEWLINE with \(m=1\) or \(m=2\), coupled with the Ricci flow (1). These results are related to shrinking gradient Ricci solitons in the case \(m=2\). The results are an extension to nonlinear backward heat equations of results of \textit{X. Cao} [J. Funct. Anal. 255, No. 4, 1024--1038 (2008; Zbl 1146.58014)] and \textit{S. Kuang} and \textit{Q. S. Zhang} [J. Funct. Anal. 255, No. 4, 1008--1023 (2008; Zbl 1146.58017)] for the linear case.