Gradient Estimates and Harnack Inequalities of a Nonlinear Heat Equation for the Finsler-Laplacian
DOI:
https://doi.org/10.15407/mag17.04.521Ключові слова:
градiєнтнi оцiнки Лi–Яу, нерiвнiсть Гарнака, нелiнiйне рiвняння теплопровiдностi, потiк Фiнслера–РiччiАнотація
Нехай $(M^{n}, F, m)$ є $n$-вимiрним компактним фiнслеровим многовидом. У цiй роботi ми вивчаємо нелiнiйне рiвняння теплопровiдностi
$$ \partial_{t}u=\Delta_{m} u\quad\text{on}\ M^n\times[0, T], $$
де $\Delta_{m}$ є фiнслеровим лапласiаном. Одержано градiєнтнi оцiнки типу Лi–Яу для позитивних глобальних розв’язкiв цього рiвняння на статичних фiнслерових многовидах, а також пiд дiєю потоку Фiнслера–Рiччi. Як наслiдок, в обох випадках також одержано вiдповiднi нерiвностi Гарнака.
Mathematics Subject Classification: 35K55, 53C21
Посилання
S. Azami and A. Razavi, Existence and uniqueness for solutions of Ricci flow on Finsler manifolds, Int. J. Geom. Methods Mod. Phys. 10(2013), 21 pages. https://doi.org/10.1142/S0219887812500910
B. Bidabad and M. Yarahmadi, On quasi-Einstein Finsler spaces, Bull. Iranian Math. Soc. 40 (2014), 921–930.
D. Bao, S.S. Chern, and Z.M. Shen, An Introduction to Riemannian–Finsler Geometry, Grad. Texts in Math., 200, Springer-Verlag, 2000. https://doi.org/10.1007/978-1-4612-1268-3
B. Bao and C. Robles, Ricci and Flag Curvatures in Finsler Geometry. A sampler of Riemann–Finsler geometry. Math. Sci. Res. Inst. Publ., 50, Cambridge Univ. Press, Cambridge, 2004.
B. Bao, On two curvature-driven problems in Riemann–Finsler geometry, in Finsler Geometry: In memory of Makoto Matsumoto, Advanced Studies in Pure Mathematics, 48, Math. Soc., Japan, Tokyo, 2007, 19–71.
S.-Y. Cheng and S.-T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math. 28 (1975), 333–354. https://doi.org/10.1002/cpa.3160280303
M. Bailesteanu, X. D. Cao and A. Pulemotov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal. 258(2010), 3517–3542. https://doi.org/10.1016/j.jfa.2009.12.003
E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math, 92, Camb. Univ. Press, 1989. https://doi.org/10.1017/CBO9780511566158
R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), 113–125. https://doi.org/10.4310/CAG.1993.v1.n1.a6
G.Y. Huang, Z.J. Huang, and H.Z. Li, Gradient estimates for the porous medium equations on Riemannian manifolds, J. Geom. Anal. 23 (2013), 1851–1875. https://doi.org/10.1007/s12220-011-9284-y
G.Y. Huang, Z.J. Huang, and H.Z. Li, Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds, Ann. Global Anal. Geom. 43 (2013), 209–232. https://doi.org/10.1007/s10455-012-9342-0
P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153–201. https://doi.org/10.1007/BF02399203
S. P. Liu, Gradient estimates for solutions of the heat equation under Ricci flow, Pacific J. Math. 243 (2009), 165–180. https://doi.org/10.2140/pjm.2009.243.165
J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal. 100 (1991), 233– 256. https://doi.org/10.1016/0022-1236(91)90110-Q
J. F. Li and X. J. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math. 226 (2011), 4456–4491. https://doi.org/10.1016/j.aim.2010.12.023
L. Ma, Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds, J. Funct. Anal. 241 (2006), 374–382. https://doi.org/10.1016/j.jfa.2006.06.006
B.Q. Ma and F.Q. Zeng, Hamilton–Souplet–Zhang’s gradient estimates and Liouville theorems for a nonlinear parabolic equation, C. R. Math. Acad. Sci. Paris 356 (2018), 550–557. https://doi.org/10.1016/j.crma.2018.04.003
S. Ohta, Vanishing s-curvature of randers spaces, Differential Geom. Appl. 29 (2011), 174–178. https://doi.org/10.1016/j.difgeo.2010.12.007
S. Ohta and K.-T. Sturm, Bochner-Weitzenbock formula and Li–Yau estimates on Finsler manifolds, Adv. Math. 252 (2014), 429–448. https://doi.org/10.1016/j.aim.2013.10.018
S. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math. 62 (2009), 1386–1433. https://doi.org/10.1002/cpa.20273
S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009), 211–249. https://doi.org/10.1007/s00526-009-0227-4
S. Lakzian, Differential Harnack estimates for positive solutions to heat equation under Finsler–Ricci flow, Pacific J. Math. 278 (2015), 447–462. https://doi.org/10.2140/pjm.2015.278.447
B. Qian, Remarks on differential Harnack inequalities, J. Math. Anal. Appl. 409 (2014), 556–566. https://doi.org/10.1016/j.jmaa.2013.07.043
J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, Pacific J. Math. 253 (2011), 489–510. https://doi.org/10.2140/pjm.2011.253.489
G.F. Wang and C. Xia, A sharp lower bound for the first eigenvalue on Finsler manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013) 983–996. https://doi.org/10.1016/j.anihpc.2012.12.008
Y.Y. Yang, Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds, Proc. Amer. Math. Soc. 136 (2008), 4095–4102. https://doi.org/10.1090/S0002-9939-08-09336-2
C.J. Yu and F.F. Zhao, A note on Li-Yau-type gradient estimate, Acta Math. Sci. 39 (2019), 273–282. https://doi.org/10.1007/s10473-019-0420-2
S.T. Yin, Q. He, and Y.B. Shen, On the first eigenvalue of Finsler-Laplacian in a Finsler manifold with nonnegative weighted Ricci curvature, Sci. China Math. 57 (2014), 1057–1070. https://doi.org/10.1007/s11425-014-4804-4
S.T. Yin, Q. He, and D.X. Zheng, Some new lower bounds of the first eigenvalue on Finsler manifolds, Kodai Math. J. 39 (2016), 318–339. https://doi.org/10.2996/kmj/1467830140
F.Q. Zeng, Gradient estimates of a nonlinear elliptic equation for the V -Laplacian, Bull. Korean Math. Soc. 56 (2019), 853–865.
F.Q. Zeng and Q. He, Gradient estimates for a nonlinear heat equation under the Finsler–Ricci flow, Math. Slovaca 69 (2019), 409–424. https://doi.org/10.1515/ms-2017-0233
