Some remarks on spatial uniformity of solutions of reaction-diffusion PDEs. (English) Zbl 1355.35103
Summary: In this paper, we present a condition which guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction-diffusion partial differential equation (PDE) with Neumann boundary conditions in one dimension, using the Jacobian matrix of the reaction term and the first Dirichlet eigenvalue of the Laplacian operator on the given spatial domain. The estimates are based on logarithmic norms in non-Hilbert spaces, which allow, in particular for a class of examples of interest in biology, tighter estimates than other previously proposed methods.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
spatial uniformity; synchronization; contraction; reaction-diffusion PDEs; logarithmic norm; logarithmic Lipschitz constantReferences:
| [1] | Aminzare, Z., On synchronous behavior in complex nonlinear dynamical systems (2015), Rutgers University, The State University of New Jersey, (Dissertation) |
| [2] | Aminzare, Z.; Shafi, Y.; Arcak, M.; Sontag, E. D., Guaranteeing spatial uniformity in reaction-diffusion systems using weighted \(L_2\)-norm contractions, (Kulkarni, V.; Stan, G.-B.; Raman, K., A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations (2014), Springer-Verlag), 73-101 |
| [3] | Aminzare, Z.; Sontag, E. D., Logarithmic Lipschitz norms and diffusion-induced instability, Nonlinear Anal. TMA, 83, 31-49 (2013) · Zbl 1263.35129 |
| [4] | Aminzare, Z.; Sontag, E. D., Synchronization of diffusively-connected nonlinear systems: Results based on contractions with respect to general norms, IEEE Trans. Netw. Sci. Eng., 1, 2, 91-106 (2014) |
| [5] | Arcak, M., Certifying spatially uniform behavior in reaction-diffusion PDE and compartmental ode systems, Automatica, 47, 6, 1219-1229 (2011) · Zbl 1235.93124 |
| [6] | Cantrell, R. S.; Cosner, C., (Spatial Ecology via Reaction-Diffusion Equations. Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology (2003)) · Zbl 1059.92051 |
| [7] | Conway, E.; Hoff, D.; Smoller, J., Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35, 1-16 (1978) · Zbl 0383.35035 |
| [8] | Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations (1965), D. C. Heath and Co.: D. C. Heath and Co. Boston, Mass. · Zbl 0154.09301 |
| [9] | Dahlquist, G., Stability and error bounds in the numerical integration of ordinary differential equations (1958), University of Stockholm, Almqvist & Wiksells Boktryckeri AB: University of Stockholm, Almqvist & Wiksells Boktryckeri AB Uppsala, (Inaugural dissertation) |
| [10] | Daners, D.; Koch Medina, P., (Abstract Evolution Equations, Periodic Problems and Applications. Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series, vol. 279 (1992), Longman Scientific & Technical, Harlow; Copublished in the United States with John Wiley & Sons, Inc.: Longman Scientific & Technical, Harlow; Copublished in the United States with John Wiley & Sons, Inc. New York) · Zbl 0789.35001 |
| [11] | Deimling, K., Nonlinear Functional Analysis (1985), Springer · Zbl 0559.47040 |
| [12] | Del Vecchio, D.; Ninfa, A. J.; Sontag, E. D., Modular cell biology: Retroactivity and insulation, Nat. Mol. Syst. Biol., 4, 161 (2008) |
| [13] | Desoer, C. A.; Vidyasagar, M., Feedback Synthesis: Input-Output Properties (2009), SIAM: SIAM Philadelphia · Zbl 0327.93009 |
| [14] | Friedman, A., Partial Differential Equations (1976), Robert E. Krieger Publishing Co.: Robert E. Krieger Publishing Co. Huntington, NY · Zbl 0347.34001 |
| [15] | Goodwin, B., Oscillatory behavior in enzymatic control processes, Adv. Enzyme Regul., 3, 425-439 (1965) |
| [16] | Hadeler, K. P., Diffusion Equations in Biology (2011), Springer Berlin, Heidelberg: Springer Berlin, Heidelberg Berlin, Heidelberg · Zbl 1225.92062 |
| [17] | Henrot, A., Extremum Problems for Eigenvalues of Elliptic Operators (2006), Birkhauser · Zbl 1109.35081 |
| [18] | Ingalls, B., Mathematical Modeling in Systems Biology: An Introduction (2013), MIT Press · Zbl 1312.92003 |
| [19] | Lozinskii, S. M., Error estimate for numerical integration of ordinary differential equations. I, Izv. Vtssh. Uchebn. Zaved. Mat., 4, 52-90 (1958), Erratum in 5 (1959) 222 · Zbl 0198.21202 |
| [20] | Martin, R. H., Bounds for solutions of a class of nonlinear differential equations, J. Differential Equations, 8, 3, 416-430 (1970) · Zbl 0213.36802 |
| [21] | Michel, A. N.; Liu, D.; Hou, L., Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems (2007), Springer-Verlag: Springer-Verlag New York |
| [22] | Morgan, J., Boundedness and decay results for reaction-diffusion systems, SIAM J. Math. Anal., 21, 5, 1172-1189 (1990) · Zbl 0723.35039 |
| [23] | Othmer, H. G., Current problems in pattern formation, (Levin, S. A., Some Mathematical Questions in Biology, VIII. Some Mathematical Questions in Biology, VIII, Lectures on Math. in the Life Sciences, vol. 9 (1977), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 57-85 · Zbl 0398.92003 |
| [24] | Russo, G.; di Bernardo, M.; Sontag, E. D., Global entrainment of transcriptional systems to periodic inputs, PLoS Comput. Biol., 6, e1000739 (2010) |
| [25] | Smith, H., Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (1995), American Mathematical Society · Zbl 0821.34003 |
| [26] | Smoller, J., Shock Waves and Reaction-Diffusion Equations (1994), Springer New York: Springer New York New York, NY · Zbl 0807.35002 |
| [27] | Soderlind, G., Bounds on nonlinear operators in finite-dimensional Banach spaces, Numer. Math., 50, 1, 27-44 (1986) · Zbl 0585.47047 |
| [28] | Soderlind, G., The logarithmic norm. History and modern theory, BIT, 46, 3, 631-652 (2006) · Zbl 1102.65088 |
| [29] | Strom, T., On logarithmic norms, SIAM J. Numer. Anal., 12, 741-753 (1975) · Zbl 0321.15012 |
| [30] | Thron, C., The secant condition for instability in biochemical feedback control - parts i and ii, Bull. Math. Biol., 53, 383-424 (1991) · Zbl 0751.92002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
