Spikes for the Gierer-Meinhardt System with Discontinuous Diffusion Coefficients
نویسندگان
چکیده
Abstract. We rigorously prove results on spiky patterns for the Gierer-Meinhardt system [8] with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini and Sherratt [1], [3], [11]. Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable. Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper. To derive these new effects in a mathematically rigorous way, we use analytical tools like LiapunovSchmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work [24]. Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.
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ورودعنوان ژورنال:
- J. Nonlinear Science
دوره 19 شماره
صفحات -
تاریخ انتشار 2009