In this note,we consider the following degenerate parabolic equation studied in[F.Chiarenza and R.Serapioni,Degenerate parabolic equations and Harnack inequality,Ann.Mat.Pura Appl.137(1984)]i.e.,■ where f=(f^(1),...,...In this note,we consider the following degenerate parabolic equation studied in[F.Chiarenza and R.Serapioni,Degenerate parabolic equations and Harnack inequality,Ann.Mat.Pura Appl.137(1984)]i.e.,■ where f=(f^(1),...,f^(n))andΩis a bounded domain in Rn with Lipschitz boundary,n≥2 and T>0.In this paper,we apply Moser iteration argument to build up the explicit relationship among the coefficients ai,j(x,t),f and the maximum norm of the solution.Meanwhile,we also find that the weighed Lebesgue space L^(2l/(l-1))to which f belongs is essentially sharp in order to establish local boundedness of the solution.Here the definition of l is found in Lemma 2.3.Our results cover the well-known results.展开更多
In this article, we investigate the existence of positive solutions of a singular quasilinear elliptic system for which the cooperative structure is not required. The approach is based on the Schauder fixed point theo...In this article, we investigate the existence of positive solutions of a singular quasilinear elliptic system for which the cooperative structure is not required. The approach is based on the Schauder fixed point theorem combined with perturbation arguments that involve the singular terms.展开更多
To obtain Lõe-estimates for the local solution of nonlinear parabolic equations local energyestimates and Moser iteration are used.Under some structural conditions a local sup-estimates of thelocal solution is es...To obtain Lõe-estimates for the local solution of nonlinear parabolic equations local energyestimates and Moser iteration are used.Under some structural conditions a local sup-estimates of thelocal solution is established in terms of some integral normal of the solution in a larger domain whichis independent of initial and boundary data.展开更多
This paper is concerned with the following fractional Schrodinger-Poisson system:{-(Δ)^(s)u+u+φu=λf(u)in R^(3)-(Δ)^(α)φu=u^(2)in R^(3)where s∈(3/4,1),α∈(0,1),λis a positive parameter,(-△)^(s),(-△)^(α)are ...This paper is concerned with the following fractional Schrodinger-Poisson system:{-(Δ)^(s)u+u+φu=λf(u)in R^(3)-(Δ)^(α)φu=u^(2)in R^(3)where s∈(3/4,1),α∈(0,1),λis a positive parameter,(-△)^(s),(-△)^(α)are fractional Laplacian operators.Under certain assumptions on f,we obtain the existence of at least one nontrivial solution of the system by using the methods of perturbation and Moser iterative method.展开更多
In this note,we generalize an extension theorem in [Le-Sesum] and [Xu-Ye-Zhao] of the mean curvature flow to the Hk mean curvature flow under some extra conditions.The main difficulty in proving the extension theorem ...In this note,we generalize an extension theorem in [Le-Sesum] and [Xu-Ye-Zhao] of the mean curvature flow to the Hk mean curvature flow under some extra conditions.The main difficulty in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the Hk mean curvature flow,and to do a suitable Moser iteration process.These two problems are overcome by imposing some extra conditions which may be weakened or removed in our forthcoming paper.On the other hand,we derive some estimates for the generalized mean curvature flow,which have their own interesting.展开更多
We consider a class of non-homogeneous ultraparabolic differential equations with singular drift terms arising from some physical models,and prove that weak solutions are H?lder continuous,which are sharp in some sens...We consider a class of non-homogeneous ultraparabolic differential equations with singular drift terms arising from some physical models,and prove that weak solutions are H?lder continuous,which are sharp in some sense and also generalize the well-known De Giorgi-Nash-Moser theory to degenerate parabolic equations satisfying the H?rmander hypoellipticity condition.The new ingredients are manifested in two aspects:on the one hand,for lower-order terms,we exploit a new Sobolev inequality suitable for the Moser iteration by improving the result of Pascucci and Polidoro(2004);on the other hand,we explore the G-function from an early idea of Kruzhkov(1964)and an approximate weak Poincaréinequality for non-negative weak sub-solutions to prove the H?lder regularity.展开更多
In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the s...In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").展开更多
By the use of Moser iteration and Campanato space estimate,the L~∞ and C~α regularity gradient of solutions of nonlinear parabolic system (u^2)/(t)-▽(g|▽u|▽u^2)=0 with nonstandard growth conditions are obtained u...By the use of Moser iteration and Campanato space estimate,the L~∞ and C~α regularity gradient of solutions of nonlinear parabolic system (u^2)/(t)-▽(g|▽u|▽u^2)=0 with nonstandard growth conditions are obtained under the natural structure constraints.展开更多
The authors consider a family of smooth immersions F（., t） ：M^n→R^n＋1 of closed hypersurfaces in R^n＋1 moving by the mean curvature flow F（p,t）/ t=-H（p,t）.v（p,t） for t E [0, T）. They show that if the norm...The authors consider a family of smooth immersions F（., t） ：M^n→R^n＋1 of closed hypersurfaces in R^n＋1 moving by the mean curvature flow F（p,t）/ t=-H（p,t）.v（p,t） for t E [0, T）. They show that if the norm of the second fundamental form is bounded above by some power of mean curvature and the certain subcritical quantities concerning the mean curvature integral are bounded, then the flow can extend past time T. The result is similar to that in [6-9].展开更多
基金supported by NSFC(Grant No.11301211)NSF of Jilin Province(Grant No.201500520056JH).
文摘In this note,we consider the following degenerate parabolic equation studied in[F.Chiarenza and R.Serapioni,Degenerate parabolic equations and Harnack inequality,Ann.Mat.Pura Appl.137(1984)]i.e.,■ where f=(f^(1),...,f^(n))andΩis a bounded domain in Rn with Lipschitz boundary,n≥2 and T>0.In this paper,we apply Moser iteration argument to build up the explicit relationship among the coefficients ai,j(x,t),f and the maximum norm of the solution.Meanwhile,we also find that the weighed Lebesgue space L^(2l/(l-1))to which f belongs is essentially sharp in order to establish local boundedness of the solution.Here the definition of l is found in Lemma 2.3.Our results cover the well-known results.
基金supported by the Europeanprogram Averroes-Erasmus Mundus(1872)
文摘In this article, we investigate the existence of positive solutions of a singular quasilinear elliptic system for which the cooperative structure is not required. The approach is based on the Schauder fixed point theorem combined with perturbation arguments that involve the singular terms.
文摘To obtain Lõe-estimates for the local solution of nonlinear parabolic equations local energyestimates and Moser iteration are used.Under some structural conditions a local sup-estimates of thelocal solution is established in terms of some integral normal of the solution in a larger domain whichis independent of initial and boundary data.
文摘This paper is concerned with the following fractional Schrodinger-Poisson system:{-(Δ)^(s)u+u+φu=λf(u)in R^(3)-(Δ)^(α)φu=u^(2)in R^(3)where s∈(3/4,1),α∈(0,1),λis a positive parameter,(-△)^(s),(-△)^(α)are fractional Laplacian operators.Under certain assumptions on f,we obtain the existence of at least one nontrivial solution of the system by using the methods of perturbation and Moser iterative method.
文摘In this note,we generalize an extension theorem in [Le-Sesum] and [Xu-Ye-Zhao] of the mean curvature flow to the Hk mean curvature flow under some extra conditions.The main difficulty in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the Hk mean curvature flow,and to do a suitable Moser iteration process.These two problems are overcome by imposing some extra conditions which may be weakened or removed in our forthcoming paper.On the other hand,we derive some estimates for the generalized mean curvature flow,which have their own interesting.
基金supported by National Natural Science Foundation of China(Grant No.12071054)National Support Program for Young Top-Notch Talents+1 种基金Dalian High-Level Talent Innovation Project(Grant No.2020RD09)supported by National Natural Science Foundation of China(Grant Nos.11471320,11631008 and 12031012)。
文摘We consider a class of non-homogeneous ultraparabolic differential equations with singular drift terms arising from some physical models,and prove that weak solutions are H?lder continuous,which are sharp in some sense and also generalize the well-known De Giorgi-Nash-Moser theory to degenerate parabolic equations satisfying the H?rmander hypoellipticity condition.The new ingredients are manifested in two aspects:on the one hand,for lower-order terms,we exploit a new Sobolev inequality suitable for the Moser iteration by improving the result of Pascucci and Polidoro(2004);on the other hand,we explore the G-function from an early idea of Kruzhkov(1964)and an approximate weak Poincaréinequality for non-negative weak sub-solutions to prove the H?lder regularity.
基金supported by the National Natural Science Foundation of China(Nos.11301399,11126189,11171259,11126190)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20120141120058)+1 种基金the China Postdoctoral Science Foundation(No.20110491212)the Fundamental Research Funds for the Central Universities(Nos.2042011111054,20420101101025)
文摘In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").
基金This work was supported by NNSF of China under Grant 39570223 and Grant 19675005
文摘By the use of Moser iteration and Campanato space estimate,the L~∞ and C~α regularity gradient of solutions of nonlinear parabolic system (u^2)/(t)-▽(g|▽u|▽u^2)=0 with nonstandard growth conditions are obtained under the natural structure constraints.
基金supported by the National Natural Science Foundation of China (Nos.10871069,10871070)the Shanghai Leading Academic Discipline Project (No.B407)
文摘The authors consider a family of smooth immersions F（., t） ：M^n→R^n＋1 of closed hypersurfaces in R^n＋1 moving by the mean curvature flow F（p,t）/ t=-H（p,t）.v（p,t） for t E [0, T）. They show that if the norm of the second fundamental form is bounded above by some power of mean curvature and the certain subcritical quantities concerning the mean curvature integral are bounded, then the flow can extend past time T. The result is similar to that in [6-9].
