Decay Property For Solutions To Plate Type Equations With Memory-type Dissipation

The application of partial differential equations and physical disciplines closely relates in together in mathematical research,promote?promote each other,and the study of nonlinear partial differential equation has become one of the important subjects.For dissipation of partial differential equation(group)of the research has a long history,the result is more abundant.In recent years,the regularity-loss types of partial differential equation by the wide attention and research both at home and abroad.The vibration of the beam,plate and so on to meet the partial differential equation is regularity-loss type.At present,it has become a more active basic research subject.In this paper an friction term for a semilinear plate equation with memory-type dissipation(i.e.,the plate equation with time-delay)is considered,and the decay estimate of the plate equation as well as the regularity-loss property are studied.We first study the linear part of the equation.Through Fourier transform in frequency space can be converted to estimate.Ordinary differential equation and energy estimate method are used to get point estimates of the basic solution operators in frequency space.Due to the presence of the in the memory term,the usual method could not be used.To overcome this difficulty,the problem is transfered to a special inhomogeneous problem with the usual type of memory term.For nonhomogeneous estimate,we obtain the correlation of the decay and regularity-loss of the linear part of the equation.For the semilinear equation,we introduce a set of time-weighted Sobolev spaces.By using time-weighted Sobolev spaces and using the contraction mapping theorem,we obtain the global in-time existence and the optimal decay estimates of solutions to the semilinear problem.Innovation of this paper lies in: compared with the results of the traditional literatures,this paper both the decay and regularity are controlled by high frequency.Thus,a similar result holds without the 1(?9))assumption for the initial data.We study the decay of the plate equation with memory-type,we can get some ways to study the properties of the equations,is of great help to our future research.We hope the final results and method can be used to the study of related issues.We will combine multiple types of plate equation,according to the difference between them,looking for a different research method,strive to has a new breakthrough in the method and train of thought,obtain the nature of the optimal solution.