| Substantial progress had been made in the last two decades in the theory of nonlinear systems of partial differential equations. Much of the developments are motivated by applications to the natural sciences of biology, physics and chemistry. In this dissertation, we main discuss the stationary patterns, periodic, blowup and decay properties of some kinds of evolution equations which are came from biology, physics and chemistry. We divide the dissertation into three parts:In the first part (Chapter 2-Chapter 4), we are concerned with the stationary patterns of some reaction-diffusion equations or fraction-diffusion equations. For the problem of Neumann boundary condition, we mainly study the effects of diffusion on the pattern formation (namely, positive non-constant steady-state solutions). When the boundary condition is of Dirichlet type, we mainly investigate the existence of coexistence solutions and stability of positive steady-state solutions (PSS), and determine the asymptotic behavior of PSS. The specific arrangements are as follows:In Chapter 2, we consider the pattern formation of two-cell Brusselator model. Pattern formation now becomes an important research aspect of modern science and technology. It can be used to describe the structure changes of interacting species or reactants of ecology, chemical reaction and gene formation in nature. For the cited model, we establish the fine upper and lower bounds of PSS and then study the existence and non-existence of non-constant PSS. As a consequence, our results show that, under some cases, diffusion can create pattern formation. (The main results of this chapter are published in J. Math. Anal. Appl., 2010 (366): 679–693)In Chapter 3, we consider the stationary patterns of a prey-predator model with Dirichlet boundary condition, and are mainly concerned about the existence of coexistence solutions. As it is known, such problems are very interesting in both mathematics and application, although they are usually quite difficult and full of challenges. By meticulously analyzing the asymptotic behaviors of solutions, we find the necessary and sufficient conditions to the existence of coexistence solutions by the classical Leray-Schauder degree theory and bifurcation theory. (The main results of this chapter are published in J. Math. Anal. Appl., 2010 (369): 555–563)In Chapter 4, we discuss the stationary patterns of a Lotka-Volterra model with nonlinear diffusion of fraction type. First, we give some priori estimates for the steady state solutions. Second, we give some conditions for the non-existence of coexistence solutions by using the priori estimates. At last, we give some sufficient conditions for the existence of coexistence solutions by using the bifurcation theory. (The main results of this chapter are submitted to Math. Model Anal.)In the second part (Chapter 5-Chapter 7), we discuss the periodic properties and blowup properties of nonlinear parabolic equations. We first discuss the periodic solutions, and then we study the blowup solutions. The specific arrangements are as follows:In Chapter 5, we discuss the periodic solutions to a porous medium equation of logistic type. We establish the existence of nontrivial periodic solution by Leray- Schauder fixed point theory. We also show that the supports of these solutions are independent of time by providing a priori estimates for their upper bounds by using Moser iteration. Furthermore, we establish the attractivity of the maximal periodic solution by using the monotonicity method. (The main results of this chapter are published in Math. Meth. Appl. Sci., 2010 (33): 1942-1954)In Chapter 6, we deal with the weakly coupled degenerate and singular parabolic equations with localized source. The existence of a unique classical non-negative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained. Furthermore, under certain conditions, we study the blowup set. At last, we also obtain the blowup rate under appropriate assumptions. (The main results of this chapter are published in Z. Ange. Math. Phy., 2011 (62): 47-66)In Chapter 7, we deal with the global existence and blowup properties of the non-Newton polytropic filtration system with nonlocal source. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depending on the initial data and the parameters; we also give a criterion for the solution exists globally or blows up in finite time in the critical case. (The main results of this chapter are published in ANZIAM J., 2008 (50): 13-29)In the third part (Chapter 8-Chapter 9), we discuss the blowup, life-span and decay properties of hyperbolic equations. Since hyperbolic equations have strong physical background, many mathematic works focus on them for a long time. They study the relationship between the life-span and nonlinear terms or spatial dimensions since John introduced the concept of life-span (the existence time of local solution). In this part, we will discuss the the blowup, life-span and decay properties of two kinds of hyperbolic equations. The specific arrangements are as follows:In Chapter 8, we study the initial-boundary value problem for a system of nonlinear hyperbolic equations, involving nonlinear damping terms, in a bounded domain ?. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results of Agre and Rammaha (Diff. Inte. Equation, 2006), specially, the blowup of weak solutions in the case of non-negative energy. (The main results of this chapter are submitted to Nonlinear Anal)In Chapter 9, we consider a class of nonlinear higher-order wave equation with nonlinear damping in ?. We show that the solution is global in time under some conditions. We also show that the local solution blows up in finite time with some assumptions on the initial energy. The decay estimate of the energy function for the global solution and the lifespan for the blowup solution are given. These extend the recent results of Ye (J. Inequa. Appl., 2010). (The main results of this chapter are submitted to Nonlinear Anal)... |
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