Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we will give the elliptic gradient estimate for positive smooth solutions to the non-homogeneous heat equation(?_t-△)u(x, t) = A(x, t)...Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we will give the elliptic gradient estimate for positive smooth solutions to the non-homogeneous heat equation(?_t-△)u(x, t) = A(x, t)when the metric evolves under the Ricci flow. As applications, we get Harnack inequalities to compare solutions at the same time.展开更多
Let （M,g, e^-fdv） be a smooth metric measure space. In this paper, we con- sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the followi...Let （M,g, e^-fdv） be a smooth metric measure space. In this paper, we con- sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equationand f is a smooth function on M under the assumptionthat the m-dimensional nonnegative Bakry-Emery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Emery Ricci curva- ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].展开更多
This paper considers a compact Finsler manifold(Mn,F(t),m)evolving under a Finsler-geometric flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation(a)tu(x,t)=△m...This paper considers a compact Finsler manifold(Mn,F(t),m)evolving under a Finsler-geometric flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation(a)tu(x,t)=△mu(x,t),(x,t)∈M×[0,T],where△m is the Finsler-Laplacian.By integrating the gradient estimates,we derive the corresponding Harnack inequalities.Our results generalize and correct the work of S.Lakzian,who established similar results for the Finsler-Ricci flow.Our results are also natural extension of similar results on Riemannian-geometric flow,previously studied by J.Sun.Finally,we give an application to the Finsler-Yamabe flow.展开更多
Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） e...Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） evolving by the Ricci flow gij/ t=-2Rij.In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u~ = /△u - aulogu - bu on M x （0,T], where a 〉 0 and b ∈ R.展开更多
The authors first derive the gradient estimates and Harnack inequalties for positive solutions of the equation on(x, t)An(x, t) + b(x, t)' ac(￣, t) + h(x, t)u(x, t) ￣ m ￣ 0on complete Riemannian manifolds, and...The authors first derive the gradient estimates and Harnack inequalties for positive solutions of the equation on(x, t)An(x, t) + b(x, t)' ac(￣, t) + h(x, t)u(x, t) ￣ m ￣ 0on complete Riemannian manifolds, and then derive the upper bounds of any positive L2fundamental solution of the equation when h(x, t) and b(x, t) are independent of t.展开更多
文摘Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we will give the elliptic gradient estimate for positive smooth solutions to the non-homogeneous heat equation(?_t-△)u(x, t) = A(x, t)when the metric evolves under the Ricci flow. As applications, we get Harnack inequalities to compare solutions at the same time.
基金supported by the Fundamental Research Fund for the Central Universities
文摘Let （M,g, e^-fdv） be a smooth metric measure space. In this paper, we con- sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equationand f is a smooth function on M under the assumptionthat the m-dimensional nonnegative Bakry-Emery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Emery Ricci curva- ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].
基金supported by NSFC 11971415Nanhu Scholars Program for Young Scholars of XYNU.
文摘This paper considers a compact Finsler manifold(Mn,F(t),m)evolving under a Finsler-geometric flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation(a)tu(x,t)=△mu(x,t),(x,t)∈M×[0,T],where△m is the Finsler-Laplacian.By integrating the gradient estimates,we derive the corresponding Harnack inequalities.Our results generalize and correct the work of S.Lakzian,who established similar results for the Finsler-Ricci flow.Our results are also natural extension of similar results on Riemannian-geometric flow,previously studied by J.Sun.Finally,we give an application to the Finsler-Yamabe flow.
基金Supported by National Natural Science Foundation of China （Grant N0s. 10926109 and 11001268） and Chinese Universities Scientific Fund （2009JS32 and 2009-2-05）
文摘Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） evolving by the Ricci flow gij/ t=-2Rij.In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u~ = /△u - aulogu - bu on M x （0,T], where a 〉 0 and b ∈ R.
文摘The authors first derive the gradient estimates and Harnack inequalties for positive solutions of the equation on(x, t)An(x, t) + b(x, t)' ac(￣, t) + h(x, t)u(x, t) ￣ m ￣ 0on complete Riemannian manifolds, and then derive the upper bounds of any positive L2fundamental solution of the equation when h(x, t) and b(x, t) are independent of t.
