Well-posedness On The Initial-Value Problem Of Plate Equation With Convolution

Plate equation is widely used in various fields of engineering and its research has been paid more and more attention in domestic and foreign countries.In this paper,we focus on the initial value problems of linear plate equations and semi-linear plate equations with convolution terms in multi-dimensional space.By using Fourier transform,Laplace transform,point-wise estimation,energy estimation,Duhamel principle and so on,the optimal decay estimation is finally obtained,which is helpful to solve practical problems.The prime objective of this paper are as follow:The research of linear plate equations with convolution terms.Firstly,by using Fourier transform and Laplace transform,we obtain the solution of linear plate equation.Then,wegain and proved the point-wise estimation of the solution to the linear equation,and then obtain the point-wise estimation of the fundamental solution,after that we obtain the decay estimation of the solution operator.Finally,we obtain the energy estimation of the solution,and then the optimal decay estimation of the linear solution is obtained.The research of semi-linear plate equations with convolution terms.On the basis of the linear plate equation problem,a series of time-weighted Sobolev Spaces are introduced.Under the assumption of small initial values,the global existence and optimal decay estimates of the solutions of the semi-linear plate equation are proved by using the contraction mapping principle.