Asymptotic Limit and Blowing-Up Behavior of Solutions

In a certain class of Neumann boundary-value problems where the reaction function f is given in the form (a1f, a2f) the steady-state problem possesses an infinite number of nonisolated constant solutions. The first part of this chapter shows the convergence of the time-dependent solution to one of these constant steady states and determines the form and exact value of the constant solution. This convergence property demonstrates that the asymptotic limit of the time-dependent solution depends solely on the ratio a1/a2 and the relative magnitude of the spatial average of the initial function and is independent of the diffusion mechanism and other physical parameters of the reaction function. The second part of this chapter is devoted to the blowing-up behavior of the solution for a more general coupled parabolic system, including a coupled parabolic-ordinary system. Sufficient conditions on the reaction function are given so that the solution blows-up in finite time for each of the three basic boundary conditions. These blowing-up results hold for quasimonotone functions as well as for nonquasimonotone functions. In certain system under consideration the solution blows-up for one class of initial functions while a global solution exists for another class of initial functions. Estimates for these two classes of initial functions and bounds of the blowing-up time are obtained. The blowing-up behavior and the global existence of a solution are extended to systems with coupled boundary conditions. Examples of the applicability of these results are given.