Abstract
Stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains. It is shown that a stationary solution with minimal “energy” is stable in a weak sense if the inhibitor reacts sufficiently fast, while it is unstable whenever the reaction of the inhibitor is slow. Moreover, the loss of stability results in a Hopf bifurcation.
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References
M.G. Crandall and P.H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions. Arch. Rational Mech. Anal.,67 (1977), 53–72.
E.N. Dancer, On stability and Hopf bifurcations for chemotaxis systems. Preprint, 2000.
A. Doelman, R.A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J.,49 (2000).
B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ℝn. Mathematical Analysis and Applications, Part A; Adv. Math. Suppl. Stud.,7A (1981), 369–402.
A. Gierer and H. Meinhardt, A theory of biological pattern formation. Kybernetik (Berlin),12 (1972), 30–39.
M. Grossi, Uniqueness of the least-energy solutions for a semilinear Neumann problem. Proc. Amer. Math. Soc,128 (2000), 1665–1672.
D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math.840, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1981.
D. Iron and M.J. Ward, A metastable spike solutions for a nonlocal reaction-diffusion model. SIAM J. Appl. Math.,60 (2000), 778–802.
M.K. Kwong, Uniqueness of positive solutions of Δu −u +u p = 0 in ℝn. Arch. Rational Mech. Anal.,105 (1989), 243–266.
C.-S. Lin and W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem. Calculus of Variations and Partial Differential Equations (eds. S. Hildebrandt, D. Kinderlehrer and M. Miranda), Lecture Notes in Math.1340, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1988, 160–174.
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations,72 (1988), 1–27.
C.-S. Lin and I. Takagi, Method of rotating planes applied to a singularly perturbed Neumann problem. Calc. Var. Partial Differential Equations, to appear.
W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math.,44 (1991), 819–851.
W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions. Duke Math. J.,70 (1993), 247–281.
W.-M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains. Japan J. Indust. Appl. Math.,12 (1995), 327–365.
W.-M. Ni, I. Takagi, J. Wei and E. Yanagida, in preparation.
W.-M. Ni, I. Takagi, and E. Yanagida, Stability analysis of point-condensation solutions to a reaction-diffusion system proposed by Gierer and Meinhardt. Preprint.
A.M. Turing, The chemical basis of morphogenesis. Philos. Trans. Roy. Soc. London, Ser. B,237 (1952), 37–72.
J. Wei, On a nonlocal eigenvalue problem and its applications to point-condensations in reaction-diffusion systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg.,10 (2000), 1485–1496.
J. Wei, Uniqueness and critical spectrum of boundary spike solutions. Proc. Roy. Soc. Edinburgh, Sect. A (Mathematics), to appear.
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Ni, WM., Takagi, I. & Yanagida, E. Stability of least energy patterns of the shadow system for an activator-inhibitor model. Japan J. Indust. Appl. Math. 18, 259–272 (2001). https://doi.org/10.1007/BF03168574
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DOI: https://doi.org/10.1007/BF03168574