Decay Estimates Of Solutions For Initial-Value Problem Of Dissipative Plate Equation

In this paper we focus on the initial value problem of linear plate equations in multidimensional space.The plate equation decays in polynomial order in low frequency region,and decays in exponential decay in high frequency region.On the basis of understanding the correlation property of the plate equation,by using Fourier transform,linear fundamental solution and corresponding solutions of point-wise estimates based on linear Fourier space solution of the problem,to obtain estimates and properties of operators.By using the energy method,finally we can get the optimal solution of the linear problem of the decay estimate.The transformation and analysis techniques used include the Fourier transform,the energy method and the superposition principle.In the first half of the paper,the initial value problem of linear plate equations is studied,which lays a foundation for the decay estimation of the initial value problem of semi-linear Equations in the second half of the paper.The prime objective of this paper are as follow:For the research of linear plate equation,the main tool is Fourier transform,and the solution formula of the fundamental solution,then the point-wise estimation of the solution of the linear problem is proved,and then the point-wise estimation of the fundamental solution is derived by using the point-wise estimation of the solution.Finally,the optimal decay estimation of the linear problem is obtained.For the research of semi-linear plate equation,on the basis of linear plate equation,a series of time-weighted Sobolev spaces are introduced.Under the assumption of small initial data and the fixed-point principle,the global existence and the optimal decay estimate of the solutions of the semi-linear problem are obtained.