In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-Émery Ricci tensor bounded be...In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-Émery Ricci tensor bounded below: One is $${u_t} = {\Delta _f}u + au\log u + bu$$ with a, b two real constants, and another is $${u_t} = {\Delta _f}u + \lambda {u^\alpha }$$ with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.展开更多
Abstract In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations...Abstract In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blow-up rate determined by the interaction between the diffusion and the boundary flux is obtained. Compared with previous results, the gradient term, whose exponent does not exceed two, does not affect the blow-up rate of the solutions.展开更多
In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are ...In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.展开更多
In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,...In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,t）＋h（x,t）u^p（x,t）=0（p 〉 1） on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang （[1], Bull. London Math. Soc. 38（2006）, 1045-1053） and the author （[2], Nonlinear Anal. 74 （2011）, 5141-5146）.展开更多
In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and co...In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and conservative gradient discretization method for spatial discretization.The method is based on a new cellcentered meshes,and it is locally conservative.It has smaller truncation error than the classical finite volume method on uniform meshes.We use the framework of the gradient discretization method to analyze the stability and convergence.The numerical experiments show that the new method has second-order convergence.Moreover,it is more accurate than the classical finite volume method in flux error,L2 error and L¥error.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
Let M be a noncompact complete Riemannian manifold.In this paper,we consider the following nonlinear parabolic equation on M ut(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 p...Let M be a noncompact complete Riemannian manifold.In this paper,we consider the following nonlinear parabolic equation on M ut(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).展开更多
We consider the Cauchy-Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain. The coefficients supposed to be only measurable in one of the space variables and small BMO wi...We consider the Cauchy-Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain. The coefficients supposed to be only measurable in one of the space variables and small BMO with respect to the others. We obtain Calderon-Zygmund type estimate for the gradient of the solution in generalized weighted Morrey spaces with Muckenhoupt weight.展开更多
In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (△-■t)u=pu+qu^(a+1) under the Yamabe flow. Here p,q∈C...In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (△-■t)u=pu+qu^(a+1) under the Yamabe flow. Here p,q∈C^(2,1)(M^(n)×[0,T]) and a is a positive constant.展开更多
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
文摘In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-Émery Ricci tensor bounded below: One is $${u_t} = {\Delta _f}u + au\log u + bu$$ with a, b two real constants, and another is $${u_t} = {\Delta _f}u + \lambda {u^\alpha }$$ with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.
基金Project supported by the Youth Foundation of the National Natural Science Foundation of China(No. 10701061)
文摘Abstract In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blow-up rate determined by the interaction between the diffusion and the boundary flux is obtained. Compared with previous results, the gradient term, whose exponent does not exceed two, does not affect the blow-up rate of the solutions.
基金supported by the National Natural Science Foundation of China(No.12271039)the Natural Science Foundation of universities of Anhui Province of China(Grant Nos.KJ2021A0927,2023AH040161).
文摘In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.
基金supported by the National Science Foundation of China(41275063 and 11401575)
文摘In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,t）＋h（x,t）u^p（x,t）=0（p 〉 1） on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang （[1], Bull. London Math. Soc. 38（2006）, 1045-1053） and the author （[2], Nonlinear Anal. 74 （2011）, 5141-5146）.
基金supported by the National Natural Science Foundation of China(No.11971069),NSAF(No.U1630249)and Science Challenge Project(No.TZ2016002).
文摘In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and conservative gradient discretization method for spatial discretization.The method is based on a new cellcentered meshes,and it is locally conservative.It has smaller truncation error than the classical finite volume method on uniform meshes.We use the framework of the gradient discretization method to analyze the stability and convergence.The numerical experiments show that the new method has second-order convergence.Moreover,it is more accurate than the classical finite volume method in flux error,L2 error and L¥error.
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））.
基金Supported partially by NSF of China(No.11171253).
文摘Let M be a noncompact complete Riemannian manifold.In this paper,we consider the following nonlinear parabolic equation on M ut(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).
基金partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan,Grant EIF-2013-9(15)-46/10/1the grant of Presidium Azerbaijan National Academy of Science 2015the research of L.Softova is partially supported by the grant INDAM-GNAMPA Project 2015
文摘We consider the Cauchy-Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain. The coefficients supposed to be only measurable in one of the space variables and small BMO with respect to the others. We obtain Calderon-Zygmund type estimate for the gradient of the solution in generalized weighted Morrey spaces with Muckenhoupt weight.
文摘In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (△-■t)u=pu+qu^(a+1) under the Yamabe flow. Here p,q∈C^(2,1)(M^(n)×[0,T]) and a is a positive constant.
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.
