Harnack Inequalities for Simple Heat Equations on Riemannian Manifolds
Tóm tắt
In this paper, we consider Harnack inequalities (the gradient estimates) of positive solutions for two different heat equations via the use of the maximum principle. In the first part, we obtain the gradient estimate for positive solutions to the following nonlinear heat equation problem
$\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$
on the compact Riemannian manifold (M, g) of dimension n and with Ric(M) ≥−K. Here a > 0 and K are some constants and V is a given smooth positive function on M. Similar results are showed to be true in case when the manifold (M, g) has compact convex boundary or (M, g) is a complete non-compact Riemannian manifold. In the second part, we study Harnack inequality (gradient estimate) for positive solution to the following linear heat equation on a compact Riemannian manifold with non-negative Ricci curvature:
$ \partial _{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $
where Wi and V only depend on the space variable x ∈ M. The novelties of our paper are the refined global gradient estimates for the corresponding evolution equations, which are not previously considered by other authors such as Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010).
Tài liệu tham khảo
Aubin, T.: Nonlinear Analysis on Manifolds. Springer, New York (1982)
Bailesteanu, M., Cao, X., Pulemotov, A.: Gradient estimates for the heat equation under the Ricci flow. J. Funct. Anal. 258, 3517–3542 (2010)
Chow, B., Hamilton, R.S.: Constrained and linear Harnack inequalities for parabolic equations. Invent. Math. 129(2), 213–238 (1997)
Cao, X., Hamilton, R.S.: Differential Harnack estimates for time dependent heat equations with potentials. Geom. Funct. Anal. 19(4), 989–1000 (2009)
Cao, X., Fayyazuddin Ljungbergb, B., Liu, B.: Differential Harnack estimates for a nonlinear heat equation. J. Funct. Anal. 265, 2312–2330 (2013)
Cao, X., Zhang, Q.S.: The conjugate heat equation and ancient solutions of the Ricci flow. Adv. Math. 228(5), 2891–2919 (2011)
Cao, H.-D., Ni, L.: Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds. Math. Ann. 331(4), 795–807 (2005)
Chen, L., Chen, W.: Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds. Ann. Global Anal. Geom. 35, 397–404 (2009)
Hamilton, R.S.: A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1(1), 113–126 (1993)
Hamilton, R.S.: The Information of Singularities in the Ricci Flow. Survey in Differential Geometry, vol. 2, pp. 7–136. International Press, Boston (1995)
Huang, G., Ma, B.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Arch. Math. (Basel) 94, 265–275 (2010)
Huang, G., Huang, Z., Li, H.: Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds. Ann. Global Anal. Geom. 43(3), 209–232 (2013)
Kuang, S., Zhang, Q.S.: A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow. J. Funct. Anal. 255(4), 1008–1023 (2008)
Ni, L.: The entropy formula for linear heat equation. J. Geom. Anal. 14, 369–374 (2004)
Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986)
Li, J.: Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J. Funct. Anal. 100, 233–256 (1991)
Ma, L.: Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds. J. Funct. Anal. 241(1), 374–382 (2006)
Ma, L.: Gradient estimates for a simple nonlinear heat equation on manifolds. Appl. Anal. 96, 225–230 (2017)
Ma, L.: Harnack’s inequality and Green’s functions on locally finite graphs. Nonlinear Anal. 170, 226–237 (2018)
Ma, L., Zhao, L., Song, X.: Gradient estimate for the degenerate parabolic equation u t = ΔF(u) + H(u) on manifolds. J. Differ. Equ. 244, 1157–1177 (2008)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1 (2002)
Qian, B.: A uniform bound for the solutions to a simple nonlinear equation on Riemannian manifolds. Nonlinear Anal. 73, 1538–1542 (2010)
Souplet, P., Zhang, Q.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. Lond. Math. Soc. 38(6), 1045–1053 (2006)
Wu, J.: Elliptic gradient estimates for a ninlinear harat equation and application. Nonlinear Anal. 151, 1–17 (2017)
Xu, X.: Gradient estimates for on manifolds and some Liouville-type theorems. J. Differ. Equ. 252(2), 1403–1420 (2012)
Yau, S.-T.: On the Harnack inequalities of partial differential equations. Comm. Anal. Geom. 2(3), 431–450 (1994)
Yau, S.-T.: Harnack inequality for non-self-adjoint evolution equations. Math. Res. Lett. 2(4), 387–399 (1995)
Yau, S.-T.: Harmonic function on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)
Yang, Y.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Proc. Am. Math. Soc. 136, 4095–4102 (2008)
Zhu, X., Li, Y.: Li-Yau estimates for a nonlinear parabolic equation on manifolds. Math. Phys. Anal. Geom. 17, 273–288 (2014)