Decay Property For Solutions To Plate Type Equations With Variable Coefficients

Nonlinear partial differential equations play a huge role in the study of practical problems,it is very important in the research of mathematics and other related fields.The plate type equations are important partial differential equations,both in pure mathematic and in applied mathematics.In this thesis,we study the global existence and decay estimates of solutions to an initial value problem of a class of plate type equation with variable coefficients.For this problem,it is difficult to obtain explicitly the fundamental solution operators or their Fourier transform to the corresponding linear equations,due to the presence of the memory term and variable coefficients.However,in terms of the spectral resolution,we can obtain the pointwise estimate in the spectral space of the fundamental solution operators to the corresponding linear equation,from which we get the decay estimates of the solutions to the linear problem.And then by construct a suitable weighted Sobolev space and by the contracting mapping principle,the global existence and the decay estimates of solutions to the semilinear problem can be obtained.In particular,for the linear problem,we transfer it to an equivalent nonhomogeneous equation.Then by an energy estimates argument in the spectral space,we get a spectral functions by which the decay and regularity loss structure of the equation is determined.