Systems of Nonlinear Partial Differential Equations

We begin with an introduction to background methods and techniques which will be widely used in this book. Among the most fundamental and important tools are the maximum principles. In the calculus of one variable we know that a function which is concave up in (a,b) and continuous in [a,b] must attain its maximum at x = a or b at the boundary. In more than one independent variables, similar situations occur. We will generalize and clarify such principles for twice continuously differentiable functions in this section.

We will use the techniques described in the last chapter to study reaction-diffusion systems related to ecology. We consider steady states and stabilities for prey-predator and competing-species systems. In this chapter, we are primarily concerned with the case when values for the species are prescribed on the boundary (i.e., Dirichlet boundary conditions). In the next chapter, more elaborate problems and other boundary conditions are treated, together with certain asymptotic approximations. The special case of zero-flux boundary condition (i.e. homogeneous Neumann condition) is studied in Chapter 7. Numerical approximations and calculations by finite difference is presented in Chapter 6.

In this Chapter, we consider various extensions of the theories in the last chapter in order to include more realistic and general problems. In section 3.2, we study the situation where the boundary condition is coupled, mixed and nonlinear. In prey-predator interaction for example, the predator may be under control at the boundary of the medium, while the prey cannot move across the boundary. Diffusion of predators at the boundary may be adjusted nonlinearly according to populations present, and there might also be some physical limitations to the process. In section 3.3, we analyze the problem when the diffusion rate is density dependent, and thus the Laplacian operator will be modified to become nonlinear and u-dependent. Moreover, the nonlinear nonhomogeneous terms become highly spatially dependent. In section 3.4, we consider the case when diffusion rate of some component is small. More thorough results concerning large-time behavior can be obtained by asympptotic methods. Estimates can be obtained by using an appropriate “reduced”problem.

In sections 4.2 to 4.4, we will consider the application of reactiondiffusion systems to the study of neutron fission reactors. We also discuss some ecological mutualist species interactions whose equations sometimes has similar structure. For the fission reactor theory, we investigate multigroup neutron-flux equations describing fission, scattering and absorption for n energy groups. The reactor core is represented by a bounded domain Ω in R^d, d ≥ 2. The functions u_i(x) or ũ_i(x,t), i=1,...,n,x=(x₁,...,x_d)ɛ Ω are the neutron flux of the i^th energy group (decreasing energy for increasing i). T(x) is the core temperature above average coolant temperature.

In the previous chapters, the major method for proving the existence of steady state solutions for elliptic systems is Theorem 1.4-2 of the type of intermediate value theorem. It essentially uses maximum principle and the homo-topic invariance of degree. Another important technique for analyzing solutions of elliptic systems is the method of monotone schemes. Besides existence, it can be adapted to study uniqueness and stability for corresponding parabolic systems. Moreover, an analogous theory can be developed for finite difference systems. The corresponding monotone schemes provide numerical method for studying elliptic systems. The finite difference theory will be described in Chapter 6.

In this chapter we adapt the monotone schemes method to find approximate solutions for semilinear elliptic systems. We combine finite difference method with the monotone procedures developed in the last chapter. Accelerated version of the schemes is also considered in Section 6.3. We will consider up to two dimensional domain in Section 6.4, the method can naturally extend to higher dimensions. We will be only concerned with positive solutions to systems with Volterra-Lotka type ecological interactions. The method can however carry over to other interactions with similar monotone properties (c.f. Section 5.3). Further, the acceleration method can be applied to nonlinear interactions, with the appropriate convexity property (c.f. Equation (6.3-4)).

In Sections 7.2 and 7.3, we will give a careful treatment of large parabolic systems of Volterra-Lotka prey-predator type under zero Neumann boundary condition. We follow the methods in [191], [192], [193] and [194], combining graph-theoretic technique with the use of Lyapunov functions. The earlier use of Lyapunov functions to study such reaction-diffusion systems began with [197], [229], [134], [190], and others. The recent results presented here give very general and elegant insight into the problem. Such systems had also been investigated by many others by invariant rectangles and comparison methods as indicated in the notes at the end of this chapter. Results using these other techniques had been summarized in other books, e.g. [211], [31], and are thus not included here.

A-priori bounds for spatial and time derivatives of possible solutions of parabolic equations and systems are crucial for the existence proof for solutions of nonlinear problems. The following theorems in this section concerning intial value problem with Dirichlet boundary condition have been used in Chapter 2, e.g. Theorem 2.1-1.