Summary
The reaction-diffusion system considered involves only one nonlinear term and is a gradient system. In a bifurcation analysis for the equilibrium states, the global existence of infinitely many solution branches can be shown by the method of Ljusternik-Schnirelmann. Their stability is studied. Using a Ljapunov functional it can be shown that the solutions of the time-dependent system converge to the equilibrium states.
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Rothe, F. Some analytical results about a simple reaction-diffusion system for morphogenesis. J. Math. Biol. 7, 375–384 (1979). https://doi.org/10.1007/BF00275155
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DOI: https://doi.org/10.1007/BF00275155