Harnack inequality for parabolic Lichnerowicz equations on complete noncompact Riemannian manifolds
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Harnack inequality for parabolic Lichnerowicz equations on complete noncompact Riemannian manifolds Liang Zhao* *
Correspondence: [email protected] Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China
Abstract In this paper, we study the gradient estimates for positive solutions to the following parabolic Lichnerowicz equations
∂u = u + hu(x, t) + Aup (x, t) + Bu–q (x, t) ∂t on complete noncompact Riemannian manifolds, where h, p, q, A, B are real constants and p > 1, q > 0. MSC: Primary 58J05; secondary 58J35 Keywords: Lichnerowicz equation; positive solutions; Harnack inequality
1 Introduction Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we study the following nonlinear parabolic equation ut (x, t) = u(x, t) + hu(x, t) + Aup (x, t) + Bu–q (x, t),
(.)
where h, p, q, A, B are real constants and p > , q > . Gradient estimates play an important role in the study of PDE, especially the Laplace equation and heat equation. Li [] derived the gradient estimates and Harnack inequalities for positive solutions of nonlinear equations ( – ∂t∂ )u(x, t) + h(x, t)uα (x, t) = and Au + b∇u + huα = on Riemannian manifolds. The author in [] also obtained a theorem of Liouville-type for positive solutions of the nonlinear elliptic equation. Later, Yang [] gave the gradient estimates for the solution to the elliptic equation with singular nonlinearity u + cu–α = ,
(.)
where α > , c are two real constants. More precisely, the author [] obtained the following result. Theorem . (Yang []) Let M be a noncompact complete Riemannian manifold of dimension n without boundary. Let Bp (R) be a geodesic ball of radius R around p ∈ M. We denote –K(R), with K(R) ≥ , to be a lower bound of the Ricci curvature on Bp (R), © 2013 Zhao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zhao Boundary Value Problems 2013, 2013:190 http://www.boundaryvalueproblems.com/content/2013/1/190
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i.e., Ric(ξ , ξ ) ≥ –K(R)|ξ | for all tangent field ξ on Bp (R). Suppose that u(x) is a positive smooth solution of the equation (.) with α > , c being two real constants. Then we have: (i) If c > , then u(x) satisfies the estimate n(n + ) n(n – ) nν |∇u| + cu–(α+) ≤ + K(R) + + nK(R) u R R R on Bp (R), where > and ν > are some universal constants independent of geometry of M. (ii) If c < , then u(x) satisfies the estimate –α– √ |∇u| –(α+) + cu ≤ n(α + )(α + ) + n(α + ) |c| inf u Bp (R) u √ nν n K(R) + + n + R α+ √ n n + (n – ) K(R)R + (n + ) + R (α + ) on Bp (R), where > and ν > are some universal constants independent of geometry of M. For some interesting gradient estimates in this directio
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