Research On Parabolic Equations With High Energy And Global Well-posedness For Nonlinear Hyperbolic Equations

In the first part of this thesis, we study the initial boundary value problem of a class of semilinear parabolic equation. Our main tools are the comparison principle and variational methods. The initial boundary value problem of this class of equation plays a very important role in the research on applied mechanics and astrophysics. However, because of the limitation of our research methods, recently researches are mostly limited to the strict control of the initial energy, i.e. J(u0)< d. Our work is to use the analysis of the Nehari manifold which is a combination of comparison principle and variational methods to remove the energy limit. Finally we find both finite time blow up and global solutions at high energy.In the second part of this thesis, we study the initial boundary value problem of semilinear hyperbolic equations with dissipative term. Currently the normal convexity method is always employed to study the global nonexistence of the solutions for the hyperbolic equations without dissipative term, which may not work while the dissipative term is introduced. So the main difficulty in this part is to improve the classical convexity method for obtaining a sharp condition of global nonexistence of solutions for the equations with dissipative term. In addition, due to the fact that almost all of the relative works focus on the case E(0)< d, where d is the depth of the potential well defined for the problem. It is also a difficult open problem to prove the global nonexistence of solutions for the equations with critical initial data E(0)= d. This is also solved in this thesis.