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
The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div（（｜u｜^δ ＋ d0）｜↓△｜^p（x,t）-2↓△u） ＋ f（x, t） （0 〈 δ 〈 2）. They a...The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div（（｜u｜^δ ＋ d0）｜↓△｜^p（x,t）-2↓△u） ＋ f（x, t） （0 〈 δ 〈 2）. They apply the method of parabolic regularization and Galerkin＇s method to prove the existence of solutions to the mentioned problem and then prove the uniqueness of the weak solution by arguing by contradiction. The authors prove that the solution approaches 0 in L^2 （Ω） norm as t →∞.展开更多
We consider nonlinear parabolic equations with nonlinear non-Lipschitz＇s term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space...We consider nonlinear parabolic equations with nonlinear non-Lipschitz＇s term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space yC^1,W^2,2（[0,T）,R^n）,n ≤ 3. Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space yC^1,L^2（[0,T）,R^n）,n≤ 3.展开更多
This paper gives the sufficient and necessary conditions of existence of global solutions and decay estimates of the solutions for the initial boundary value problem of some nonlinear parabolic equations with small in...This paper gives the sufficient and necessary conditions of existence of global solutions and decay estimates of the solutions for the initial boundary value problem of some nonlinear parabolic equations with small initial energy and the nonlinear power less than Sobolev critical value. The existence, nonexistence and the decay estimates of global solutions are considered. The conditions that initial energy is small and nonlinear power is less than Sobolev critical value is imposed.展开更多
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
According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in...According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in the instability of the steady states.展开更多
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 NSFC (10771085)Graduate Innovation Fund of Jilin University(20111034)the 985 program of Jilin University
文摘The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div（（｜u｜^δ ＋ d0）｜↓△｜^p（x,t）-2↓△u） ＋ f（x, t） （0 〈 δ 〈 2）. They apply the method of parabolic regularization and Galerkin＇s method to prove the existence of solutions to the mentioned problem and then prove the uniqueness of the weak solution by arguing by contradiction. The authors prove that the solution approaches 0 in L^2 （Ω） norm as t →∞.
基金Supported by Ministry of Science of Republic Serbia
文摘We consider nonlinear parabolic equations with nonlinear non-Lipschitz＇s term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space yC^1,W^2,2（[0,T）,R^n）,n ≤ 3. Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space yC^1,L^2（[0,T）,R^n）,n≤ 3.
文摘This paper gives the sufficient and necessary conditions of existence of global solutions and decay estimates of the solutions for the initial boundary value problem of some nonlinear parabolic equations with small initial energy and the nonlinear power less than Sobolev critical value. The existence, nonexistence and the decay estimates of global solutions are considered. The conditions that initial energy is small and nonlinear power is less than Sobolev critical value is imposed.
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.
基金supported by National Natural Science Foundation of China(11126336 and 11201324)New Teachers’Fund for Doctor Stations,Ministry of Education(20115134120001)+1 种基金Fok Ying Tuny Education Foundation(141114)Youth Fund of Sichuan Province(2013JQ0027)
文摘According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in the instability of the steady states.
