This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation. By the priori estimates and the method in [9], It proves that the Cauchy problem admits a unique global classical solution...This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation. By the priori estimates and the method in [9], It proves that the Cauchy problem admits a unique global classical solution. And by the concave method, we give sufficient conditions on the blowup of the global solution for the Cauchy problem.展开更多
The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solu...The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in Arch.Rational Mech.Anal.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.展开更多
The purpose of the present paper is to call for attention to the following question: Which of the initial data (nonsmall) admit global smooth solutions to the Cauchy problem for nonlinear wave equations. A few cases a...The purpose of the present paper is to call for attention to the following question: Which of the initial data (nonsmall) admit global smooth solutions to the Cauchy problem for nonlinear wave equations. A few cases and examples are sketched, showing that the general answer of this question may be quite complicated.展开更多
In this paper, we consider the following equation ut=（um）xx＋（un）x, with the initial condition as Dirac measure. Attention is focused on existence, nonexistence, uniqueness and the asymptotic behavior near （0,0）...In this paper, we consider the following equation ut=（um）xx＋（un）x, with the initial condition as Dirac measure. Attention is focused on existence, nonexistence, uniqueness and the asymptotic behavior near （0,0） of solution to the Cauchy＇s problem. The special feature of this equation lies in nonlinear convection effect, i.e., the equation possesses nonlinear hyperbolic character as well as degenerate parabolic one. The situation leads to a more sophisticated mathematical analysis. To our knowledge, the solvability of singular solution to the equation has not been concluded yet. Here based on the previous works by the authors, we show that there exists a critical number n0=m＋2 such that a unique source-type solution to this equation exists if 0≤n展开更多
基金the Natural Science Foundation of Henan Province(0611050500)
文摘This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation. By the priori estimates and the method in [9], It proves that the Cauchy problem admits a unique global classical solution. And by the concave method, we give sufficient conditions on the blowup of the global solution for the Cauchy problem.
基金Project supported by the National Natural Science Foundation of China (No. 10728101)the Basic Research Program of China (No. 2007CB814800)+1 种基金the Doctoral Program Foundation of the Ministry of Education of Chinathe "111" Project (No. B08018) and SGST (No. 09DZ2272900)
文摘The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in Arch.Rational Mech.Anal.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.
基金Project supported by the Chinese SpecialFunds for Major State Basic Research Project"NonlinearScience"
文摘The purpose of the present paper is to call for attention to the following question: Which of the initial data (nonsmall) admit global smooth solutions to the Cauchy problem for nonlinear wave equations. A few cases and examples are sketched, showing that the general answer of this question may be quite complicated.
基金National Natural Science Foundation of China (Grant Nos. 10671103 and 11001142)
文摘In this paper, we consider the following equation ut=（um）xx＋（un）x, with the initial condition as Dirac measure. Attention is focused on existence, nonexistence, uniqueness and the asymptotic behavior near （0,0） of solution to the Cauchy＇s problem. The special feature of this equation lies in nonlinear convection effect, i.e., the equation possesses nonlinear hyperbolic character as well as degenerate parabolic one. The situation leads to a more sophisticated mathematical analysis. To our knowledge, the solvability of singular solution to the equation has not been concluded yet. Here based on the previous works by the authors, we show that there exists a critical number n0=m＋2 such that a unique source-type solution to this equation exists if 0≤n
