Existence And Blow Up Of Weak Solutions To Wave Equations With Damping Terms

In this paper,a fourth order wave equation with linear damping terms and a general nonlinearity is studied,and the main concerns are the local existence and uniqueness of weak solutions to the problem and the finite time blow-up property.The whole paper is divided into three chapters.In Chapter 1,we briefly recall the background of the problem considered in this paper and the progress achieved by mathematicians both in China and aboard.Some preliminaries,including some definitions,notations and necessary lemmas when proving the main results,are presented in Chapter 2.In Chapter 3,the main results of this paper are stated and proved.We first show the existence and uniqueness of weak solutions to the corresponding linear problem,by combining the classical Galerkin's method with a priori estimate.The local existence and uniqueness of weak solutions to the original problem follows by an application of Banach Contraction Mapping Principle.Then we show that the unstable set is invariant when the initial data satisfy some specific conditions.On the basis of this invariance and Levine's concavity method,we show that the weak solution blows up in finite time,and an upper bound for the blow-up time is also derived at the same time.Finally,a lower bound for the blow-up time is also obtained by using the first order differential inequality and some embedding inequalities.It is worthy pointing out that the blow-up results obtained in this paper imply,at least for some specific nonlinearity,that the problem admits finite time blowup solutions with arbitrarily high initial energy.