Finite Time Blow Up Problems For Several Nonlinear Wave Equations At High Energy Level

The research topics of the present thesis are from the research projects supported by theNational Natural Science Foundation of China （11101102）. This work is concerned with theinitial boundary value problem for a class of nonlinear wave equations with nonlinear gen-eralized source terms, some nonlinear wave equations with several nonlinear source terms ofdifferent signs, the nonlinear fourth-order dispersive-dissipative wave equation,1-D nonlinearwave equation of sixth orders respectively.In order to obtain the blow up result of solutions for these problems at high energy level,it is required to prove the invariance of the unstable set under the flow of these problems re-spectively. In the low initial energy case E（0）<d it is needed to obtain the invariant unstableset by contradiction. In other words we should deduce that J（u（t₀））≥d is contradictory toJ（u（t₀））≤E（0）<d. Clearly under the condition E（0）>0this contradiction is not estab-lished, which is the main difficulty.For the nonlinear wave equations with nonlinear generalized source terms and some non-linear wave equations with several nonlinear source terms of different signs, this paper analyzesthe influence of complex source terms on global nonexistence of solutions to the initial bound-ary value problem of these two nonlinear wave equations at high energy level. By employing thepotential well concavity method, this theme gives that for which initial data the global solutionsof these two problems do not exist at high energy level.For the nonlinear fourth-order dispersive-dissipative wave equation, this paper presentsthat the norm of solutions in some spaces with respect to time is strictly monotone increasing.By using the potential well concavity method as well as the technique of anti-dissipativity, thiswork proves that some solutions of this problem blow up in finite time at high energy level.For1-D nonlinear wave equation of sixth orders, this work uses the invariant set and con-cavity method and derives a sufficient condition on the initial data with arbitrarily positive initialenergy such that the corresponding local solution of this problem blows up in finite time.