Gradient estimates and Harnack inequalities for diffusion equation on Riemannian manifolds

Abstract

We derive the gradient estimates and Harnack inequalities for positive solutions of the diffusion equation u_t = Δu^m on Riemannian manifolds. Then, we prove a Liouville type theorem.

Keywords

Gradient estimate / Harnack inequality / diffusion equation / Riemannian manifold

References

[1] Bonforte M, Grillo G. Asymptotics of the porous media equation via Sobolev inequalities. Journal of Functional Analysis, 2005, 225: 33-62

[2] Bonforte M, Vazquez J L. Global positivity estimates and Harnack inequalities for the fast diffusion equation. Journal of Functional Analysis, 2006, 240: 399-428

[3] Calabi E. An extension of E. Hopf’s maximum principle with an application to Riemannian geometry. Duke Math J, 1958, 25: 45-56

[4] Cheng S Y, Yau S T. Differential equations on Riemannian manifolds and their geometric applications. Comm Pure Appl Math, 1975, 28: 333-354

[5] Demange J. Porous media equation and Sobolev inequalities under negative curvature. Bull Sci Math, 2005, 129: 804-830

[6] Li J Y. Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. Journal of Functional Analysis, 1991, 100: 233-256

[7] Li P, Yau S T. On the parabolic kernel of the Schrödinger operator. Acta Math, 1986, 56(3-4): 153-201

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg