Abstract
Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the L 2 norm of the solution. This method is appropriate for the equations with time dependent coefficients. It yields new results and is applicable when the usual linearization method is not applicable.
References
Burenkov, V.: Sobolev Spaces on Domains, pp. 192–193. Teubner, Leipzig (1998),
Daleckii, Yu., Krein, M.G.: Stability of Solutions to Differential Equations in Banach Space. Am. Math. Soc., Providence (1974)
Friedman, A.: Partial Differential Equations. Krieger, New York (1976)
Hoang, N.S., Ramm, A.G.: Some nonlinear inequalities and applications. J. Abstr. Differ. Equ. Appl. 2(1), 84–101 (2011)
Klainerman, S.: Long-time behavior of solutions to nonlinear evolution equations. Arch. Ration. Mech. Anal. 78, 73–98 (1982)
Ladyzhenskaya, O.A., Solonnikov, V., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow (1973)
Meinhardt, H.: Models of biological pattern formation: from elementary steps to the organization of embryonic axes. In: Current Topics in Developmental Biology, vol. 81 (2008)
Murray, J.D.: Mathematical Biology: I. An Introduction, 3rd edn. Springer, Berlin (2002)
Murray, J.D.: Mathematical Biology: II. Spatial Models and Biomedical Applications, 3rd edn. Springer, Berlin (2003)
Othmer, H.G., Painter, K., Umulis, D., Xue, C.: The intersection of theory and application in elucidating pattern formation in developmental biology. Math. Model. Nat. Phenom. 4(4), 3–82 (2009)
Ramm, A.G.: Asymptotic stability of solutions to abstract differential equations. J. Abstr. Differ. Equ. Appl. 1(1), 27–34 (2010)
Ramm, A.G.: Stability of solutions to some evolution problems. Chaotic Model. Simul. (CMSIM) 1, 17–27 (2011)
Ramm, A.G.: Stability result for abstract evolution problems. Math. Methods Appl. Sci. (to appear)
Showalter, R.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Am. Math. Soc., Providence (1997)
Turing, A.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 237, 37–72 (1952)
Zheng, S.: Nonlinear Evolution Equations. CRC Press, Boca Raton (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ramm, A.G., Volpert, V. Convergence of Time-Dependent Turing Structures to a Stationary Solution. Acta Appl Math 123, 31–42 (2013). https://doi.org/10.1007/s10440-012-9711-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-012-9711-5