In the present paper, we show that there exists a unique common fixed point for four self maps in a fuzzy metric space where two of the maps are reciprocally continuous and the other two maps are z-asymptotically comm...In the present paper, we show that there exists a unique common fixed point for four self maps in a fuzzy metric space where two of the maps are reciprocally continuous and the other two maps are z-asymptotically commuting.展开更多
The local well-posedness of the Cauchy problem for the fifth order shallow water equation δtu＋αδx^5u＋βδx^3u＋rδxu＋μuδru=0,x,t∈R, is established for low regularity data in Sobolev spaces H^s（s≥-3/8） by ...The local well-posedness of the Cauchy problem for the fifth order shallow water equation δtu＋αδx^5u＋βδx^3u＋rδxu＋μuδru=0,x,t∈R, is established for low regularity data in Sobolev spaces H^s（s≥-3/8） by the Fourier restriction norm method. Moreover, the global well-posedness for L^2 data follows from the local well-posedness and the conserved quantity. For data in H^s（s〉0）, the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L^2 norm of initial data.展开更多
In this paper,for the 3D quadratic nonlinear Klein-Gordon equation on the product space R^(2)×T,we focus on the lower bound of the lifespan of the smooth solution with slowly decaying initial data.When the size o...In this paper,for the 3D quadratic nonlinear Klein-Gordon equation on the product space R^(2)×T,we focus on the lower bound of the lifespan of the smooth solution with slowly decaying initial data.When the size of initial data is bounded by ε_(0)>0,it is shown that a smooth solution exists up to the time C_(0)/20 with0 being sufficiently small and e^(c0)/ε_(0)^(2)>0 being some suitable constant.Note that the solution of the corresponding 3D linear homogeneous Klein-Gordon equation on R^(2)×T only admits the optimal time-decay rate(1+t)−1,from which we generally derive the lifespan of the nonlinear Klein-Gordon equation up to e^(c0/ε0)rather than the more precise e^(c0/ε^(2)0) here.展开更多
文摘In the present paper, we show that there exists a unique common fixed point for four self maps in a fuzzy metric space where two of the maps are reciprocally continuous and the other two maps are z-asymptotically commuting.
文摘The local well-posedness of the Cauchy problem for the fifth order shallow water equation δtu＋αδx^5u＋βδx^3u＋rδxu＋μuδru=0,x,t∈R, is established for low regularity data in Sobolev spaces H^s（s≥-3/8） by the Fourier restriction norm method. Moreover, the global well-posedness for L^2 data follows from the local well-posedness and the conserved quantity. For data in H^s（s〉0）, the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L^2 norm of initial data.
基金supported by National Natural Science Foundation of China(Grant No.11871030)supported by National Natural Science Foundation of China(Grant No.11731007)。
文摘In this paper,for the 3D quadratic nonlinear Klein-Gordon equation on the product space R^(2)×T,we focus on the lower bound of the lifespan of the smooth solution with slowly decaying initial data.When the size of initial data is bounded by ε_(0)>0,it is shown that a smooth solution exists up to the time C_(0)/20 with0 being sufficiently small and e^(c0)/ε_(0)^(2)>0 being some suitable constant.Note that the solution of the corresponding 3D linear homogeneous Klein-Gordon equation on R^(2)×T only admits the optimal time-decay rate(1+t)−1,from which we generally derive the lifespan of the nonlinear Klein-Gordon equation up to e^(c0/ε0)rather than the more precise e^(c0/ε^(2)0) here.
