In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also...In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also discussed.展开更多
This paper deals with the blow-up properties of positive solutions to a localized degenerate and singular parabolic equation with weighted nonlocal boundary condi- tions. Under appropriate hypotheses, the global exist...This paper deals with the blow-up properties of positive solutions to a localized degenerate and singular parabolic equation with weighted nonlocal boundary condi- tions. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. Furthermore, the global blow-up behavior and the uniform blow-up profile of blow-up solutions are also described. We find that the blow-up set is the whole domain {0, a}, including the boundaries, and this differs from parabolic equations with local sources case or with homogeneous Dirichlet boundary conditions case.展开更多
This paper deals with the blow-up properties of positive solutions to a degenerate and singular nonlocal parabolic equation with weighted nonlocal boundary conditions. Under appropriate hypotheses, the global existenc...This paper deals with the blow-up properties of positive solutions to a degenerate and singular nonlocal parabolic equation with weighted nonlocal boundary conditions. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. Furthermore, by using the properties of Green＇s function, we find that the blow-up set of the blow-up solution is the whole domain （0, a）, and this differs from parabolic equations with local sources case.展开更多
This paper deals with the blow up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u t-(x αu x) x=∫ a 0f(u) d x in (0,a)×(0,T) under homogeneous Dirichl...This paper deals with the blow up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u t-(x αu x) x=∫ a 0f(u) d x in (0,a)×(0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow up in finite time of positive solutions are obtained. It is also proved that the blow up set is almost the whole domain. This differs from the local case. Furthermore, the blow up rate is precisely determined for the special case: f(u)=u p,p>1.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
In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular different...In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applica- tions to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spec- tral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.展开更多
基金Supported by National Natural Science Foundation of China(10171113 and 10471156)Natural Science Foundation of GuangDong 2004(4009793).
文摘In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also discussed.
基金supported by the research scheme of the natural science of the universities of Jiangsu province(08KJD110017 and 13KJB110028)
文摘This paper deals with the blow-up properties of positive solutions to a localized degenerate and singular parabolic equation with weighted nonlocal boundary condi- tions. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. Furthermore, the global blow-up behavior and the uniform blow-up profile of blow-up solutions are also described. We find that the blow-up set is the whole domain {0, a}, including the boundaries, and this differs from parabolic equations with local sources case or with homogeneous Dirichlet boundary conditions case.
基金supported partially by the foundation of professors and doctors of Yancheng Normal University(14YSYJB0106)by the research scheme of the natural science of the universities of Jiangsu Province(08KJD110017 and 13KJB110028)
文摘This paper deals with the blow-up properties of positive solutions to a degenerate and singular nonlocal parabolic equation with weighted nonlocal boundary conditions. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. Furthermore, by using the properties of Green＇s function, we find that the blow-up set of the blow-up solution is the whole domain （0, a）, and this differs from parabolic equations with local sources case.
文摘This paper deals with the blow up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u t-(x αu x) x=∫ a 0f(u) d x in (0,a)×(0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow up in finite time of positive solutions are obtained. It is also proved that the blow up set is almost the whole domain. This differs from the local case. Furthermore, the blow up rate is precisely determined for the special case: f(u)=u p,p>1.
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 National Natural Science Foundation of China(Grant No.11171227)Fund for Doctoral Authority of China(Grant No.20123127110001)+1 种基金Fund for E-institute of Shanghai Universities(Grant No.E03004)Leading Academic Discipline Project of Shanghai Municipal Education Commission(Grant No.J50101)
文摘In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applica- tions to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spec- tral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.
