Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up p...Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up phenomenons for ut = ？u ＋ f（u） are discussed in terms of two cases：（i） an initial condition is given;（ii） a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.展开更多
This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the disc...This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.展开更多
Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certa...Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certain assumptions.Then under the functional framework we use the three-solution theorem and the variational method to prove that the above equation has at least three solutions,of which one is trivial and the others are strictly positive.展开更多
By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main re...By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main result of F.Münch who considers the case of CD(K,∞)curvature.Hence,we answer a question raised by Münch.For that purpose,we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded LaplacianΔand perpetual cutoff semigroup P w t in our setting.展开更多
Continuing our previous work （arXiv：1509.07981vl）, we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In ...Continuing our previous work （arXiv：1509.07981vl）, we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li-Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359-409 （2015）]. In many topics, such as eigenvalue estimate and heat kernel estimate （not including the Liouville type theorems）, replacing the Li-Yau inequality by the global gradient estimate, we can get similar results.展开更多
基金supported by the National Science Foundation of China(11671401)supported by the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(17XNH106)
文摘Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up phenomenons for ut = ？u ＋ f（u） are discussed in terms of two cases：（i） an initial condition is given;（ii） a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.
文摘This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.
文摘Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certain assumptions.Then under the functional framework we use the three-solution theorem and the variational method to prove that the above equation has at least three solutions,of which one is trivial and the others are strictly positive.
文摘By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main result of F.Münch who considers the case of CD(K,∞)curvature.Hence,we answer a question raised by Münch.For that purpose,we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded LaplacianΔand perpetual cutoff semigroup P w t in our setting.
基金supported by National Natural Science Foundation of China(Grant No.11271011)supported by National Natural Science Foundation of China(Grant Nos.11171347 and 11471014)
文摘Continuing our previous work （arXiv：1509.07981vl）, we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li-Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359-409 （2015）]. In many topics, such as eigenvalue estimate and heat kernel estimate （not including the Liouville type theorems）, replacing the Li-Yau inequality by the global gradient estimate, we can get similar results.
