Applications of Coupled Systems to Model Problems

The theory developed for coupled systems of parabolic and elliptic equations can be applied to a number of reaction diffusion models in several fields of applied science and engineering. This includes models arising from biology, ecology, and biochemistry as well as the classical fields of physical and engineering sciences. In fact, much of the theory developed in the previous chapters is motivated by these model problems. Although many of these models involve coupled system of two equations there are also a number of problems that are coupled through three or more equations, including ordinary differential equations. The basic idea of the application is to develop techniques for the construction of coupled upper and lower solutions for each individual model problem. For parabolic systems this construction leads to the global existence or nonexistence of a time-dependent solution and various qualitative properties of the solution. This includes the asymptotic stability or instability of a steady-state solution, boundedness and invariance properties of the system, the asymptotic limit of the time-dependent solution, and blowing-up behavior of the solution in finite time. In each case explicit conditions in terms of the physical parameters are obtained, and in some cases these conditions yield a bifurcation criterion for the convergence or blowing-up behavior of the solution. For elliptic systems upper and lower solutions are constructed to ensure the existence of steady-state solutions. Also discussed are the uniqueness and multiplicity of positive steady-state solutions, the bifurcation of positive solutions, and estimates of stability and instability regions of a given steady-state solution.