In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differ...In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F>0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.展开更多
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 equationon tu=△u＋aulogu＋qu on M ...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 equationon tu=△u＋aulogu＋qu on M × （0, ∞）, where a is a constant and q is a C2 function. This result can be compared with the ones of Ma （JFA, 241, 374-382 （2006）） and Yang （PAMS, 136, 4095-4102 （2008））. Also, we obtain Hamilton＇s gradient estimate for the Schodinger equation. This can be compared with the result of Ruan （JGP, 58, 962-966 （2008））.展开更多
基金Supported by Universities Natural Science Foundation of Anhui Province(Grant No.KJ2016A310)
文摘In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F>0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.
文摘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 equationon tu=△u＋aulogu＋qu on M × （0, ∞）, where a is a constant and q is a C2 function. This result can be compared with the ones of Ma （JFA, 241, 374-382 （2006）） and Yang （PAMS, 136, 4095-4102 （2008））. Also, we obtain Hamilton＇s gradient estimate for the Schodinger equation. This can be compared with the result of Ruan （JGP, 58, 962-966 （2008））.
