Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds

In this paper, we study the local gradient estimate for the positive solution to the following equation:

     

Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds

In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation

Complement of gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds

In this paper, along the idea of Souplet and Zhang, we deduce a local elliptic‐type gradient estimates for positive solutions of the nonlinear parabolic equation: on for α ≥ 1 and α ≤ 0. As applications, related Liouville‐type theorem is exported. Our results are complement of known results. Copyright © 2016 John Wiley & Sons, Ltd.     

Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds

In the first part of this paper, we prove the sharp global Li‐Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman‐type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu–Ni–Vázquez–Villani (JMPA 2009) and Kotschwar–Ni (Annales Scientifiques de l'École Normale Supérieure 2009). Copyright © 2013 John Wiley & Sons, Ltd.     

Gradient estimates and Harnack inequalities for diffusion equation on Riemannian manifolds

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.     

Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds

Let (M,g) be a complete noncompact Riemannian manifold with the m-dimensional Bakry-Émery Ricci curvature bounded below. In this paper, we give a local Li-Yau type gradient estimate for the positive solutions to a general nonlinear parabolic equation

ut=Δu−∇?⋅∇u−aulogu−qu     

Riemann流形上改进的p-Laplace方程的梯度估计

本文研究Riemann流形上的改进的p-Laplace方程,运用截断函数的估计、Hessian比较定理和Laplace比较定理,得到该方程正解的梯度估计.并应用该结论,得到在Riemann流形上关于改进的p-Laplace方程正解的Harnack不等式和Liouville型定理.     

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation $ \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu$ on ${M \times [0, + \infty)}Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation

\frac?u?t = Dfu +au log u + bu \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu      10. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds      Yunyan Yang 《Proceedings of the American Mathematical Society》2008,136(11):4095-4102 Let be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation on , where , are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).       11. Global Hessian estimates for an Allen-Cahn equation on Riemannian manifolds      Liangdi Zhang 《Journal of Mathematical Analysis and Applications》2021,493(2):124573       12. Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds      Bin Qian 《数学学报(英文版)》2011,27(6):1071-1078 Let (M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation $ \partial _t u = \Delta u + au\log u + qu $ \partial _t u = \Delta u + au\log u + qu       13. Strichartz estimates for the wave equation on manifolds with boundary      Matthew D. Blair  Hart F. Smith  Christopher D. Sogge   《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009,26(5):1817-1829 We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.      14. 黎曼流形上一类非线性抛物方程的Hamilton 梯度估计      蒋新荣  廖蔡生 《数学研究及应用》2015,35(4):435-447 本文我们得到了黎曼流形上一类非线性抛物方程的局部Hamilton梯度估计. 利用这个局部估计,我们得到了一个Harnack型不等式和一个Liouville型定理.      15. Some gradient estimates and Harnack inequalities for nonlinear parabolic equations on Riemannian manifolds      《Mathematische Nachrichten》2017,290(11-12):1905-1917 In this paper, by the method of J. F. Li and X. J. Xu (Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456–4491 ), we shall consider the nonlinear parabolic equation on Riemannian manifolds with , . First of all, we shall derive the corresponding Li–Xu type gradient estimates of the positive solutions for . As applications, we deduce Liouville type theorem and Harnack inequality for some special cases. Besides, when , our results are different from Li and Yau's results. We also extend the results of J. F. Li and X. J. Xu, and the results of Y. Yang.      16. 流形上的微分Harnack 估计          下载免费PDF全文 牛艳艳 《中国科学:数学》2013,43(5):423-429 In the thesis, we study the differential Harnack estimate for the heat equation of the Hodge Laplacian deformation of (p, p)-forms on both fixed and evolving (by Kähler-Ricci flow) Kähler manifolds, which generalize the known differential Harnack estimates for (1, 1)-forms. On a Kähler manifold, we define a new curvature cone Cp and prove that the cone is invariant under Kähler-Ricci flow and that the cone ensures the preservation of the nonnegativity of the solutions to Hodge Laplacian heat equation. After identifying the curvature conditions, we prove the sharp differential Harnack estimates for the positive solution to the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow after obtaining some interpolating matrix differential Harnack type estimates for curvature operators between Hamilton’s and Cao’s matrix Harnack estimates. Similarly, we define another new curvature cone, which is invariant under Ricci flow, and prove another interpolating matrix differential Harnack estimates for curvature operators on Riemannian manifolds.      17. Existence of travelling wave solutions for the heat equation in infinite cylinders with a nonlinear boundary condition      Mads Kyed 《Mathematische Nachrichten》2008,281(2):253-271 The existence of travelling wave solutions for the heat equation ∂t u –Δu = 0 in an infinite cylinder subject to the nonlinear Neumann boundary condition (∂u /∂n) = f (u) is investigated. We show existence of nontrivial solutions for a large class of nonlinearities f. Additionally, the asymptotic behavior at ∞ is studied and regularity properties are established. We use a variational approach in exponentially weighted Sobolev spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)      18. Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds      Guangyue Huang  Zhijie Huang  Haizhong Li 《Annals of Global Analysis and Geometry》2013,43(3):209-232 Let (M n , g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: $$u_t=\Delta u+au\log u+bu$$ on M n  × [0,T], where a, b are two real constants. We derive local gradient estimates of the Li-Yau type for positive solutions of the above equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results extend the ones of Davies in Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol 92, Cambridge University Press, Cambridge,1989, and Li and Xu in Adv Math 226:4456–4491 (2011).      19. Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds      WU Jiayong 《偏微分方程(英文版)》2010,(1):68-79 Let （M,g） be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Emery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation ut = △u - △↓ φ· △ ↓u - aulogu- bu,where φ is a C^2 function, and a ≠ 0 and b are two real constants. This work generalizes the results of Souplet and Zhang （Bull. London Math. Soc., 38 （2006）, pp. 1045-1053） and Wu （Preprint, 2008）.      20. Gradient estimates on the weighted p-Laplace heat equation      Lin Feng Wang 《Journal of Differential Equations》2018,264(1):506-524 In this paper, by a regularization process we derive new gradient estimates for positive solutions to the weighted p-Laplace heat equation when the m-Bakry–Émery curvature is bounded from below by ?K for some constant $K\ge 0$. When the potential function is constant, which reduce to the gradient estimate established by Ni and Kotschwar for positive solutions to the p-Laplace heat equation on closed manifolds with nonnegative Ricci curvature if $K\searrow 0$, and reduce to the Davies, Hamilton and Li–Xu's gradient estimates for positive solutions to the heat equation on closed manifolds with Ricci curvature bounded from below if $p=2$.

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Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation

on , where , are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).

     

Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let (M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation

Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.     

黎曼流形上一类非线性抛物方程的Hamilton 梯度估计

本文我们得到了黎曼流形上一类非线性抛物方程的局部Hamilton梯度估计. 利用这个局部估计,我们得到了一个Harnack型不等式和一个Liouville型定理.     

Some gradient estimates and Harnack inequalities for nonlinear parabolic equations on Riemannian manifolds

In this paper, by the method of J. F. Li and X. J. Xu (Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456–4491 ), we shall consider the nonlinear parabolic equation on Riemannian manifolds with , . First of all, we shall derive the corresponding Li–Xu type gradient estimates of the positive solutions for . As applications, we deduce Liouville type theorem and Harnack inequality for some special cases. Besides, when , our results are different from Li and Yau's results. We also extend the results of J. F. Li and X. J. Xu, and the results of Y. Yang.     

流形上的微分Harnack 估计

In the thesis, we study the differential Harnack estimate for the heat equation of the Hodge Laplacian deformation of (p, p)-forms on both fixed and evolving (by Kähler-Ricci flow) Kähler manifolds, which generalize the known differential Harnack estimates for (1, 1)-forms. On a Kähler manifold, we define a new curvature cone C_p and prove that the cone is invariant under Kähler-Ricci flow and that the cone ensures the preservation of the nonnegativity of the solutions to Hodge Laplacian heat equation. After identifying the curvature conditions, we prove the sharp differential Harnack estimates for the positive solution to the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow after obtaining some interpolating matrix differential Harnack type estimates for curvature operators between Hamilton’s and Cao’s matrix Harnack estimates. Similarly, we define another new curvature cone, which is invariant under Ricci flow, and prove another interpolating matrix differential Harnack estimates for curvature operators on Riemannian manifolds.     

Existence of travelling wave solutions for the heat equation in infinite cylinders with a nonlinear boundary condition

The existence of travelling wave solutions for the heat equation ∂_t u –Δu = 0 in an infinite cylinder subject to the nonlinear Neumann boundary condition (∂u /∂n) = f (u) is investigated. We show existence of nontrivial solutions for a large class of nonlinearities f. Additionally, the asymptotic behavior at ∞ is studied and regularity properties are established. We use a variational approach in exponentially weighted Sobolev spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds

Let (M ⁿ , g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: $$u_t=\Delta u+au\log u+bu$$ on M ⁿ  × [0,T], where a, b are two real constants. We derive local gradient estimates of the Li-Yau type for positive solutions of the above equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results extend the ones of Davies in Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol 92, Cambridge University Press, Cambridge,1989, and Li and Xu in Adv Math 226:4456–4491 (2011).     

Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds

Let （M,g） be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Emery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation ut = △u - △↓ φ· △ ↓u - aulogu- bu,where φ is a C^2 function, and a ≠ 0 and b are two real constants. This work generalizes the results of Souplet and Zhang （Bull. London Math. Soc., 38 （2006）, pp. 1045-1053） and Wu （Preprint, 2008）.     

Gradient estimates on the weighted p-Laplace heat equation

In this paper, by a regularization process we derive new gradient estimates for positive solutions to the weighted p-Laplace heat equation when the m-Bakry–Émery curvature is bounded from below by ?K for some constant $K\ge 0$. When the potential function is constant, which reduce to the gradient estimate established by Ni and Kotschwar for positive solutions to the p-Laplace heat equation on closed manifolds with nonnegative Ricci curvature if $K\searrow 0$, and reduce to the Davies, Hamilton and Li–Xu's gradient estimates for positive solutions to the heat equation on closed manifolds with Ricci curvature bounded from below if $p=2$.