Abstract
We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation’s coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aronson, D.G., Bénilan, P.: Régularité des solutions de l’équation des milieux poreux dans \({\mathbb{R}}^n\). C. R. Acad. Sci. Paris. Sér. A B 288, A103–A105 (1979)
Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)
Bailesteanua, M.: A Harnack inequality for the parabolic Allen–Cahn equation. Ann. Global Anal. Geom. 51, 367–378 (2017)
Bailesteanua, M., Cao, X.-D., Pulemotov, A.: Gradient estimates for the heat equation under the Ricci flow. J. Funct. Anal. 258, 3517–3542 (2010)
Bakry, D., Emery, M.: Diffusion hypercontractivitives. In: Séminaire de Probabilités XIX, 1983/1984, Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)
Bismuth, S.: Prescribed scalar curvature on a complete Riemannian manifold in the negative case. J. Math. Pures Appl. 79, 941–951 (2000)
Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)
Calabi, E.: An extension of E. Hopf’s maximum principle with an application to Riemannian geometry. Duke Math. J. 25, 45–56 (1958)
Cantrell, R.S., Cosner, C.: Diffusive logistic equations with indefinite weights: population models in a disrupted environments. Proc. R. Soc. Edinb. 112A, 293–318 (1989)
Cao, H.-D.: Recent progress on Ricci solitons. In: Recent Advances in Geometric Analysis, Advanced Lectures in Mathematics (ALM), vol. 11, pp. 1–38. International Press, Somerville (2010)
Cao, X.-D., Cerenzia, M., Kazaras, D.: Harnack estimates for the endangered species equation. Proc. Am. Math. Soc. 143, 4537–4545 (2015)
Cao, X.-D., Fayyazuddin Ljungberg, B., Liu, B.: Differential Harnack estimates for a nonlinear heat equation. J. Funct. Anal. 265, 2312–2330 (2013)
Cao, X.-D., Liu, B., Pendleton, I., Ward, A.: Differential Harnack estimates for Fisher’s equation. Pac. J. Math. 290, 273–300 (2017)
Cao, X.-D., Zhang, Z.: Differential Harnack estimates for parabolic equations. In: Proceedings of Complex and Differential Geometry, pp. 87-98 (2011)
Case, J.: A Yamabe-type problem on smooth metric measure spaces. J. Differ. Geom. 101, 467–505 (2015)
Cheng, S.-Y., Yau, S.-T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28, 333–354 (1975)
Chow, B., Hamilton, R.: Constrained and linear Harnack inqualities for parabolic equations. Invent. Math. 129, 213–238 (1997)
Dung, N.-T., Khanh, N.-N., Ngo, Q.-A.: Gradient estimates for some \(f\)-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces. Manuscr. Math. 155, 471–501 (2018)
Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598 (1981)
Guo, Z.-M., Wei, J.-C.: Hausdoff dimension of ruptures for solutions of a semilinear equation with singular nonlinearity. Manuscr. Math. 120, 193–209 (2006)
Hamilton, R.: A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1, 113–126 (1993)
Hamilton, R.: The formation of singularities in the Ricci flow. Surv. Differ. Geom. 2, 7–136 (1995)
Jin, Z.-R.: A counterexample to the Yamabe problem for complete noncompact manifolds. In: Lecture Notes in Mathematics, vol. 1306, pp. 93–101 (1988)
Jin, Z.-R.: Prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature. Trans. Am. Math. Soc. 340, 785–810 (1993)
Lee, J., Parker, T.: The Yamabe problem. Bull. Am. Math. Soc. 17, 37–91 (1987)
Li, J.-F., Xu, X.-J.: Differential Harnack inequalities on Riemannian manifolds I: linear heat equation. Adv. Math. 226(5), 4456–4491 (2011)
Li, J.-Y.: Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J. Funct. Anal. 100, 233–256 (1991)
Li, P., Tam, L.-F., Yang, D.-G.: On the elliptic equation \(\Delta u+ku-Ku^p=0\) on complete Riemannian manifolds and their geometric applications. Trans. Am. Math. Soc. 350, 1045–1078 (1998)
Li, P., Yau, S.-T.: On the parabolic kernel of the Schrodinger operator. Acta Math. 156, 153–201 (1986)
Li, X.-D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pure. Appl. 84, 1295–1361 (2005)
Lott, J.: Some geometric properties of the Bakry–Émery–Ricci tensor. Comment. Math. Helv. 78, 865–883 (2003)
Lott, J.: Remark about scalar curvature and Riemannian submersions. Proc. Am. Math. Soc. 135, 3375–3381 (2007)
Mastrolia, P., Rigoli, M., Setti, A.G.: Yamabe-Type Equations on Complete, Noncompact Manifolds, Progress in Mathematics, vol. 302. Birkhäuser, Basel (2012)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002). arXiv:math.DG/0211159v1
Ratto, A., Rigoli, M., Veron, L.: Scalar curvature and conformal deformations of noncompact Riemannian manifolds. Math. Z. 225, 395–426 (1997)
Schoen, R.: Conformal deformations of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)
Schoen, R.: A report on some recent progress on nonlinear problems in geometry. In: Blaine Lawson, H., Shing-Tung, Yau., (eds.) Surveys in differential geometry, vol.1 (Cambridge, Massachusetts, 1990), pp. 201–241, Lehigh University, Pennsylvania (1991)
Souplet, P., Zhang, Q.S.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. Lond. Math. Soc. 38, 1045–1053 (2006)
Trudinger, N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Sc. Norm. Super Pisa 22, 265–274 (1968)
Wei, G.-F., Wylie, W.: Comparison geometry for the Bakry–Émery Ricci tensor. J. Differ. Geom. 83, 377–405 (2009)
Wu, J.-Y.: Li–Yau type estimates for a nonlinear parabolic equation on complete manifolds. J. Math. Anal. Appl. 369, 400–407 (2010)
Wu, J.-Y.: Elliptic gradient estimates for a weighted heat equation and applications. Math. Z. 280, 451–468 (2015)
Wu, J.-Y.: Elliptic gradient estimates for a nonlinear heat equation and applications. Nonlinear Anal. 151, 1–17 (2017)
Xu, X.-J.: Gradient estimates for the degenerate parabolic equation \(u_t=F(u)\) on manifolds and some Liouville-type theorems. J Differ. Equ. 252, 1403–1420 (2012)
Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)
Yang, Y.-Y.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Proc. Am. Math. Soc. 136(11), 4095–4102 (2008)
Yang, Y.-Y.: Gradient estimates for the equation \(\Delta u+cu^{-\alpha }=0\) on Riemannian manifolds. Acta Math. Sin. (Engl. Ser.) 26, 1177–1182 (2010)
Yau, S.-T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
Zhang, Q.S.: Positive solutions to \(\Delta u-Vu+Wu^p=0\) and its parabolic counterpart in noncompact manifolds. Pac. J. Math. 213, 163–200 (2004)
Zhu, X.-B.: Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds. Nonlinear Anal. 74, 5141–5146 (2011)
Zhu, X.-B.: Gradient estimates and Liouville theorems for linear and nonlinear parabolic equations on Riemannian manifolds. Acta Math. Sci. 36B(2), 514–526 (2016)
Acknowledgements
The author thanks Professor Jeffrey S. Case for helpful discussions. The author also thanks the referee for making valuable comments and suggestions and pointing out many errors which helped to improve the exposition of the paper. This work is supported by the NSFC (11671141) and the Natural Science Foundation of Shanghai (17ZR1412800).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, JY. Gradient estimates for a nonlinear parabolic equation and Liouville theorems. manuscripta math. 159, 511–547 (2019). https://doi.org/10.1007/s00229-018-1073-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-018-1073-5