The biwave maps are a class of fourth order hyperbolic equations.In this paper,we are interested in the solution formulas of the linear homogeneous biwave equations.Based on the solution formulas and the weighted ener...The biwave maps are a class of fourth order hyperbolic equations.In this paper,we are interested in the solution formulas of the linear homogeneous biwave equations.Based on the solution formulas and the weighted energy estimate,we can obtain the L∞(R^(n))−WN,1(R^(n))and L∞(R^(n))−WN,2(R^(n))estimates,respectively.By our results,we find that the biwave maps enjoy some different properties compared with the standard wave equations.展开更多
In one space-and in one time -dimension a beam-like equation is solved, where the second time derivative is replaced by the α- fractional time derivative, 1 〈 α ≤ 2. The solution is given in closed form in terms o...In one space-and in one time -dimension a beam-like equation is solved, where the second time derivative is replaced by the α- fractional time derivative, 1 〈 α ≤ 2. The solution is given in closed form in terms of the Mttag-Leffler functions in two parameters.展开更多
Global existence of small amplitude solution and nonlinear scattering result for the Canchy problem of the generalized IMBq equation were considered in the paper titled ＂Small amplitude solutions of the generalized I...Global existence of small amplitude solution and nonlinear scattering result for the Canchy problem of the generalized IMBq equation were considered in the paper titled ＂Small amplitude solutions of the generalized IMBq equation＂ [1]. It is a pity that the authors overlooked the bad behavior of low frequency part of S（t）ψ which causes troubles in L^∞ and H^＊ estimates. In this note, we will present a new proof of global existence under same conditions as in [1] but for space dimension n ≥ 3.展开更多
基金the Zhejiang Provincial Outstanding Youth Science Foundation(Grant No.LR22A010004)the Natural Science Foundation of Zhejiang Province(Grant No.LY20A010026)the National Natural Science Foundation of China(Grant Nos.12071435 and 11871212).
文摘The biwave maps are a class of fourth order hyperbolic equations.In this paper,we are interested in the solution formulas of the linear homogeneous biwave equations.Based on the solution formulas and the weighted energy estimate,we can obtain the L∞(R^(n))−WN,1(R^(n))and L∞(R^(n))−WN,2(R^(n))estimates,respectively.By our results,we find that the biwave maps enjoy some different properties compared with the standard wave equations.
基金Supported by the Natural Science Foundation of Fujian Province(2001J009, Z0511015)the fund of Fuzhou University.
文摘In one space-and in one time -dimension a beam-like equation is solved, where the second time derivative is replaced by the α- fractional time derivative, 1 〈 α ≤ 2. The solution is given in closed form in terms of the Mttag-Leffler functions in two parameters.
文摘Global existence of small amplitude solution and nonlinear scattering result for the Canchy problem of the generalized IMBq equation were considered in the paper titled ＂Small amplitude solutions of the generalized IMBq equation＂ [1]. It is a pity that the authors overlooked the bad behavior of low frequency part of S（t）ψ which causes troubles in L^∞ and H^＊ estimates. In this note, we will present a new proof of global existence under same conditions as in [1] but for space dimension n ≥ 3.
