| In this paper, we study three classes of fourth-order hyperbolic equations at high initial energy level. Currently, there are limited results about the high energy problems. We are more meticulous to demonstrate the structures and the properties of the solutions of fourth-order wave equations with nonlinear strain term in H01(Ω) space. The results have the theoretical instructions on the problems in physical and engineering models.First, we study the initial boundary value problem for class of a wave equations with the nonlinear strain term and dissipative term at high initial energy level. This fourth order wave equation arises in the elasto-plastic-microstructure models for tudinal motion of an elasto-plastic bar. Later it is frequently used in the vibration and simulation experiments of the elastic rod. This equation contains a weak damping term ut and it is required to prove that the function H’{t) is monotone increasing, thus we need to prove that H"(t)>0. However in the process of proving H"(t)=-2I(υ)-2γ(υt,υ)+2||υt||2>0, we only know that-2I(υ)>0and||υ||2>0, but it can not determine the sign of the term-2γ(υt,υ). Therefore we can not derive H"(t)>0, which is the first difficulty. Moreover, we can see that the equation con-tains the nonlinear strain term(?), hence it is required to test the equations with υ in the process of considering the global nonexistence of solutions. However the inner prod-uct(?) can not be simplified, which is the second difficulty. Furthermore in the low initial energy case E(0)<d it is needed to prove the finite time blow up of solu-tions by contradiction.In other words we should deduce that J(υ(t0))≥d is contradictory to J(υ(t0))≤E(0)<d. Clearly under the condition E(0)>0this contradiction is not estab-lished, which is the third difficulty. This paper successfully solves the above difficulties.By applying Fourier transformation, the energy conservation is obtained.And for both positive energy and non-positive energy we give some properties for potential wells and the invariance of some manifolds.Based on the above properties and the concavity method, this thesis proves a finite time blow up result of solutions at high initial energy level.Second, we study the high energy problem of a class of fourth order equations with non-linear source terms.This equation containing the generalized nonlinear source terms promotes the results of the first equation in this paper. Dealing with the nonlinear strain term and proving the invariance of some manifolds are the difficulties in considering this equation. Applying the potential well and the concavity method, for certain initial data in the unstable set, this paper proves the invariance of the unstable set and obtains the finite time blow up results of solutions with arbitrarily positive initial energy.In the end, we consider the initial boundary value problem for a class of fourth order strongly damped nonlinear wave equations. This equation contains the strong damping term Aut. To our knowledge there is little literature on the nonlinear wave equations with strong damping term. Using the convexity method to prove global non-existence of solutions, it is needed to introduce the auxiliary function M(t) and prove M(t)>0. However in the process of proving the term M(t)=2||υt||2-2I(υ)-2α(υ,△υt)>0, it can only be proved that-2I(υt)>0and||υ||2>0. In other words the sign of the term-2α(υ,△υt) can not be determined and the term M(t)>0can not be proved, which are the first difficulties. Moreover in the critical initial energy case E(0)=d it is needed to prove the invariance of the unstable set by contradiction.In other words it is needed to use the term (υ(t0),υt(t0))>0to derive the contradiction||υ(t0)|≠0, but the sign of2(υ(to),υt(to))-2α(υ(t0),△υt(t0))) can not be determined in the presence of the strong damping term. Hence the contradiction||υ(t0)||≠0can not be deduced, which is the second difficulty. Furthermore in the low initial energy case E(0)<d it is needed to prove the finite time blow up of solutions by contradiction. In other words we should deduce that J(υ(t0))≥d which is contradictory to J(υ(t0))5≤E(0)<d. Clearly under the condition E(0)>0this contradiction is not established, which is the third difficulty. The above difficulties are resolved in this paper. By introducing a family of potential wells this paper not only proves the finite time blow up of solutions at low initial energy level but also derives the global existence, asymptotic behaviour and finite time blow up of solutions in the critical case E(0)=d. Moreover, by using the convexity method the nonexistence of global solutions with arbitrary positive initial energy is obtained under the condition α=0. |
