Global And Blow-up Solutions Of Some Nonlinear Evolution Equations

The study for nonlinear phenomena in the field of physics, chemistry, biology and economy, etc., has been an important aspect in the field of nonlinear partial differential equations. This paper mainly concerns the qualitative properties for some nonlinear evolution equations. The main work include global existence and blow-up in finite time of solutions to the initial or initial-boundary value problem, the decay behavior of global solutions and the life span of blowup solutions.In Chapter 1, the background and history about the related work are introduced.In Chapter 2, we discuss the initial boundary problem of a porous equation in half space. By virtue of the supper and lower solution method, we obtain the global existence and finite time blow-up of solution to such a problem.Chapter 3 deals with the Cauchy problem of a higher-order parabolic system. Making use of the semigroup theory, we derive the global existence of solution with small initial data and the uniform decay estimates of global solution. Exploilting the test function method, we discuss the condition on the blow-up solution. Moreover, the life span of blow-up solution is estimated.Chapter 4 is for the Cauchy problem of hyperbolic system with weak damping. In the space of dimension N = 1,3, exploiting the semigroup method, the global existence and the decay behavior of the solution with small initial data is put forward. When N ≥ 1, the blow-up condition of solution is shown through the test function approach.Chapter 5 concerns with the initial-boundary problem of hyperbolic system with strong damping and a hyperbolic equation subject to boundary memory and boundary damping. For the first problem, with the successive approximation method, we first derive the existence and uniqueness of local solution. Secondly, using the potential well theory and via a difference inequality, we acquire the global existence and decay property of solution. Finally, utilizing the concave lemma, we discuss the blow-up behavior of solution. For the second problem, the global existence and decay estimates is derived with the help of the potential well theory and Galerkinprocedure.