Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds
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Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds Jiaxian Wu · Yi-Hu Yang
Received: 26 November 2013 / Revised: 30 January 2014 / Accepted: 10 February 2014 / Published online: 21 March 2014 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2014
Abstract Let M be a noncompact complete Riemannian manifold. In this paper, we consider the following nonlinear parabolic equation on M u t (x, t) = u(x, t) + au(x, t) ln u(x, t) + bu α (x, t). We prove a Li–Yau type gradient estimate for positive solutions to the above equation; as an application, we also derive the corresponding Harnack inequality. These results generalize the corresponding ones proved by Li (J Funct Anal 100:233–256, 1991). Keywords Gradient estimate · Ricci curvature · Harnack inequality · Nonlinear parabolic equation Mathematics Subject Classification (2010)
58J35 · 35K10 · 35K55
1 Introduction Let M be an n-dimensional Riemannian manifold with nonnegative Ricci curvature, in particular, the n-dimensional Euclidean space. In [3], Gidas and Spruck studied the following elliptic equation on M u + u α = 0
(1.1)
J. Wu Department of Mathematics, Tongji University, Shanghai 200092, China e-mail: [email protected] Y.-H. Yang (B) Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China e-mail: [email protected]
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J. Wu, Y.-H. Yang
where 1 ≤ α ≤ n+2 n−2 . They showed that any nonnegative solution of (1.1) have to be zero. A natural question is what the situation is for the corresponding parabolic equation of (1.1). In [6], Li, by generalized the results of [5], considered this question. More precisely, he studied the following parabolic equation u t (x, t) = u(x, t) + h(x, t)u α (x, t),
(1.2)
on M × [0, ∞), where α is a positive constant and h(x, t) is a function defined on M × [0, ∞), which is C 2 in the x-variable and C 1 in the t-variable. His main result can be stated as follows: Theorem A Let (M, g = gi j ) be an n-dimensional complete Riemannian manifold with Ricci tensor Ri j ≥ −kgi j (k ≥ 0). Let h(x, t) be a nonnegative function defined on M × [0, ∞), which is C 2 in the x-variable and C 1 in the t-variable. Assume that 0 < α < n/(n − 1) and ( + ∂/∂t)h(x, t) ≥ 0. If u(x, t) is a positive solution of (1.2), then 1 ut 1 2n hu α−1 2n K |∇u|2 − ≤ + + u2 α α u t α2 (2 − α)α
(1.3)
and 2n (t2 − t1 )2n K t2 α ρ2 u(x1 , t1 ) ≤ u(x2 , t2 ) + , exp t1 4α(t2 − t1 ) (2 − α)
(1.4)
where x1 , x2 ∈ M, 0 < t1 < t2 < ∞, and ρ(x1 , x2 ) is the geodesic distance between x1 and x2 . On the other hand, to understand the gradient Ricci soliton [4], Ma [7] studied the following elliptic equation u(x) + au(x) ln u(x) + bu(x) = 0,
(1.5)
where a and b are constants and a < 0. The corresponding parabolic equation of (1.5) was studied by Yang [8] and he also got a Li–Yau type gradient estimate. In this paper, we want to study the following parabolic equation u t (x, t) = u(x, t) + au(x, t) ln u(x, t
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