Global Existence, Uniform Decay And Blow-up Of Solutions For Nonlinear Petrovsky Equations

This work concerns with nonlinear Petrovsky equations and related results concerning local existence, global existence, asymptotic and blow-up of solutions have been established.In the first chapter we review some development of the related problems and summarize the main work of the present dissertation. Moreover, we introduce the spaces and notations.In Chapter 2, we consider the nonlinear Petrovsky equation with strong damping. We prove the global existence of solution under conditions without any relation between the exponents of nonlinear damping term and source term. Moreover, we establish an exponential decay rate. If the exponent of nonlinear source term is greater than the one of damping term, we show that the solution blows up in finite time and the initial energy less than the potential well depth.Chapter 3 are concerned with the Petrovsky viscoelastic equation with nonlinear boundary damping and source terms. Firstly, by using the the well-known contraction mapping theorem, we obtain the local existence. Secondly, we obtain the existence of global solution and prove that the solution decays exponentially or polynomially depending on the rate of the decay of the relaxation function of viscoelastic term by using perturbed energy technique. At last, if m