Study On The Decay Estimates Of Solutions For Semi-linear Plate Equation

With the development of science and technology,partial differential equations play an increasingly important role and become an important part of contemporary mathematics.As a kind of partial differential equations,the plate equation has gradually gained attention,and domestic and foreign countries have begun to pay attention to its research.The main research object of this paper is the initial value problem of semi-linear plate equations in multidimensional space.The transformation and analysis techniques used therein include the Fourier transform,the energy method,and the superposition principle of the solution.The prime objective of this paper are as follow:Firstly we study the linear plate equation.The main tool is Fourier transform and the solution formula of the fundamental solution.Then,the point-wise estimation of the fundamental solution is derived by using the point-wise estimation of the solutions.Finally,the optimal decay estimation of the linear problem is obtained.Then we study the semi-linear plate equation on the basis of linear plate equation.In this paper,a series of time-weighted Sobolev spaces are introduced.Under the assumption of small initial values and the fixed point principle,the global existence and the optimal decay estimates of the solutions of the semi-linear problem are obtained.