Chapter III is devoted to systems of elliptic and parabolic types. In Section 31, we study the questions of a priori estimates and existence for weakly coupled elliptic systems which are natural extensions of the scalar equations considered in Chapter I. In Section 32, we study a simple parabolic system which is the analogue of the scalar model problem (15.1) studied in Chapter II. For this system, we treat the questions of well-posedness, global existence and blow-up. In Section 33, we discuss the different possible effects of adding linear diffusion (and some boundary conditions) to a system of ODE’s. It will turn out that quite opposite effects can be observed. This will lead us to consider some systems arising in biological or physical contexts, such as mass-preserving and Gierer-Meinhardt systems.
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