The maximum principle is applied to prove the Harnack estimate of curvature flows of hypersurfaces in Rn＋1,where the normal velocity is given by a smooth function f depending only on the mean curvature.By use of the ...The maximum principle is applied to prove the Harnack estimate of curvature flows of hypersurfaces in Rn＋1,where the normal velocity is given by a smooth function f depending only on the mean curvature.By use of the estimate,some corollaries are obtained including the integral Harnack inequality.In particular,the conditions are given with which the solution to the flows is a translation soliton or an expanding soliton.展开更多
By an interpolation method,an intrinsic Harnack estimate and some supestimates are established for nonnegative solutions to a general singular parabolic equation.
An intrinsic Harnack estimate and some sup-estimates are established for nonnegative weak solutions of equations of non-Newtonian polytropic filtration ut -div(｜Dum ｜p-2Dum) =0, m(p- 1) ＜ 1, m＞0, p＞ 1.
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
We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow-up of the po...We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow-up of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.展开更多
In this paper, we prove an improved Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. As a consequence, we derive a lower estimate for Dirichlet eigenvalues usin...In this paper, we prove an improved Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. As a consequence, we derive a lower estimate for Dirichlet eigenvalues using the Harnack inequality, extending previous results of Chung and Yau for certain homogeneous graphs.展开更多
文摘The maximum principle is applied to prove the Harnack estimate of curvature flows of hypersurfaces in Rn＋1,where the normal velocity is given by a smooth function f depending only on the mean curvature.By use of the estimate,some corollaries are obtained including the integral Harnack inequality.In particular,the conditions are given with which the solution to the flows is a translation soliton or an expanding soliton.
基金Project supported by the Fujian Provincial Natural Science Foundation of China(No.2009J01009)the Natural Science Foundation of Jimei University
文摘By an interpolation method,an intrinsic Harnack estimate and some supestimates are established for nonnegative solutions to a general singular parabolic equation.
基金Project supported by the National Natural Science Foundation of China (No.19771069).
文摘An intrinsic Harnack estimate and some sup-estimates are established for nonnegative weak solutions of equations of non-Newtonian polytropic filtration ut -div(｜Dum ｜p-2Dum) =0, m(p- 1) ＜ 1, m＞0, p＞ 1.
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We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow-up of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.
基金Partially supported by National Natural Science Foundation of China(Grant No.11271011)
文摘In this paper, we prove an improved Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. As a consequence, we derive a lower estimate for Dirichlet eigenvalues using the Harnack inequality, extending previous results of Chung and Yau for certain homogeneous graphs.
