The Exponential Decay Properties Of Solutions To The Cauchy Problem Of Zakharov-Kuznetsov Equation

In this thesis,the exponential decay properties of solutions to the initial value problem associated with the generalized Zakharov-Kuznetsov equation and three-dimensional Zakharov-Kuznetsov equation are studied,respectively.Concretely,the exponential decay of order 3/2 is investigated in some time,which is related with the persistence properties in weighted Sobolev spaces and sharp unique continuation properties of solutions to this problem.The exponential decay properties imply that the strong solutions,initially decaying together with its spacial derivative at infinity,must retain this properties.The method of estimate with weight function is an effective one to study these properties,by which properties of many typical equations have been proved.On the basis of the wellposedness results,the purpose of this thesis is to establish the well-posedness properties and exponential decay properties of the solutions for these Cauchy problems above.The main contents of this thesis are as follows:Firstly,we briefly describe the purpose and significance of the decay properties of the solutions and the results which the researchers have obtain about the decay properties both at home and abroad,and give the theorems and lemmas which are needed in the proof of these properties.Secondly,we prove the exponential decay properties of solutions to the initial value problem associated with the generalized Zakharov-Kuznetsov equation.At last,we establish the decay properties of solutions to the initial value problem associated with the three-dimensional Zakharov-Kuznetsov equation.