Existence And Decay Estimates Of Solutions For The Nonlinear Fourth Order Wave Equation With Damped Term

We study the existence, uniqueness and asymptotic properties of global solutions to the Cauchy problem of the nonlinear fourth order wave equation with damped term in Rn. The first chapter introduces the development history of wave equation. We bring out the research object utt-△u-△utt+△2u+ut=△f(u),x∈Rn, t>0,(0.3) and the main achievements of this paper. In chapter2, we give some preliminaries, mainly including some definitions and lemmas. In the third chapter, we research the existence and uniqueness of weak solutions to the Cauchy problem of the linear equation utt-△u-△utt+△2u+ut=0, x∈Rn, t>0.(0.4)We get asymptotic properties and decay estimates to the Cauchy problem of the equation (0.4) by using the energy and multiplier method in the fourth chapter. In section5, we explore the local existence to the equation (0.3) by using the semigroup theory. In the last chapter, we discuss the existence, uniqueness of global solution and asymptotic behavior for the nonlinear problem. Firstly, according to Duhamel principle, the solution of the nonlinear problem can be written as an equivalence integral equation. Then using the integral estimates we get the decay estimates of the nonlinear problem will be the same as the linear problem.