Reaction-Diffusion Equations with Hysteresis. Higher Spatial Dimensions

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1 in Higher Spatial Dimensions Mark Curran Under the supervision of PD. Dr. Pavel Gurevich Free University Berlin 2 SFB910 (Sonderforschungsbereich 910) 3 Berlin Mathematical School Patterns of Dynamics, Berlin, July 2016

2 Hysteresis in Biology Bacteria (Ja ger, Hoppensteadt 80, 83): Non-diffusing: Bacterium Diffusing: Nutrient, ph Thresholds: α < β Figure: Chiu, Hoppensteadt, Ja ger, Analysis and Computer Simulation of Accretion Patterns in Bacterial Cultures J. Math. Biol. 32, No.8 pp (1994)

3 Hysteresis in Biology Hydra (Marciniak-Czochra 06): Non-diffusing: Cells Diffusing: Ligands

4 Hysteresis in Biology Hydra (Marciniak-Czochra 06): Non-diffusing: Cells Diffusing: Ligands Figure: Reaction-diffusion equations and biological pattern formation, Anna Marciniak-Czochra, lecture notes, University of Wroclaw, 2011

5 Hysteresis in Biology Hydra (Marciniak-Czochra 06): Non-diffusing: Cells Diffusing: Ligands Thresholds: α < β Figure: Reaction-diffusion equations and biological pattern formation, Anna Marciniak-Czochra, lecture notes, University of Wroclaw, 2011

6 Model Problem x domain Rn, n 2, t 0, u(x, t), v (x, t) R, Neumann B.C., ξ0 {red, blue}, ut v u t=0 = u + f (u, v ), = H(ξ0, u), = ϕ. Thresholds: α < β

7 Model Problem x domain Rn, n 2, t 0, u(x, t), v (x, t) R, Neumann B.C., ξ0 {red, blue}, ut v u t=0 Non-Ideal Relay: = u + f (u, v ), = H(ξ0, u), = ϕ. Thresholds: α < β

8 Model Problem x domain Rn, n 2, t 0, u(x, t), v (x, t) R, Neumann B.C., ξ0 {red, blue}, ut v u t=0 Non-Ideal Relay: = u + f (u, v ), = H(ξ0, u), = ϕ. Thresholds: α < β

9 Connection to Slow-Fast Systems Slow-Fast System u t = u + f (u, v), εv t = v v 3 3 u. Stable normally hyperbolic, slow manifolds: red, blue Fold Points: α < β

10 Connection to Slow-Fast Systems Slow-Fast System u t = u + f (u, v), εv t = v v 3 3 u. Stable normally hyperbolic, slow manifolds: red, blue NOTE: Unlike, e.g., travelling waves in Fitzhugh-Nagumo, the fast variable is not diffusing. Fold Points: α < β

11 Connection to Slow-Fast Systems Slow-Fast System u t = u + f (u, v), εv t = v v 3 3 u. Stable normally hyperbolic, slow manifolds: red, blue NOTE: Unlike, e.g., travelling waves in Fitzhugh-Nagumo, the fast variable is not diffusing. Fold Points: α < β GOAL: Develop a theoretical framework for systems of independent non-ideal relays coupled via diffusion.

12 Context u t = u + f (u, v), v = H(ξ 0, u), u t=0 = ϕ, Context: + Neumann B.C, domain R n, n 2. Numerics + modelling: Jäger et. al 80, 83, 94; Marciniak-Czochra 06; Lopes et al. 08. Existence of solns for multi-valued hysteresis: Alt 85; Visintin 86; Aiki, Kopfova 08.

13 Context u t = u + f (u, v), v = H(ξ 0, u), u t=0 = ϕ, Context: + Neumann B.C, domain R n, n 2. Numerics + modelling: Jäger et. al 80, 83, 94; Marciniak-Czochra 06; Lopes et al. 08. Existence of solns for multi-valued hysteresis: Alt 85; Visintin 86; Aiki, Kopfova 08. n = 1: Well-posedness for transverse ϕ (Gurevich, Tikhomirov, Shamin 12-14)

14 Difficulties u t = u + f (u, v), L q v = H(ξ 0, u), L u t=0 = ϕ, + Neumann B.C, domain R n, n 2. Difficulties: What is a sufficient condition for uniqueness? Definition of solution? u t, u L q, q large enough.

15 Difficulties u t = u + f (u, v), L q v = H(ξ 0, u), L u t=0 = ϕ, + Neumann B.C, domain R n, n 2. Difficulties: What is a sufficient condition for uniqueness? Definition of solution? u t, u L q, q large enough. What is the mechanism for pattern formation? How does the free boundary evolve explicitly?

16 Result (P) u t = u + f (u, v), v = H(ξ 0, u), u t=0 = ϕ. + Neumann B.C, domain R n, n 2 Theorem (Local existence of solutions) If ϕ is transverse then there is a T > 0 such that: 1 There is at least one transverse solution to (P) on (0, T ) 2 Any solution to (P) is transverse on (0, T )

17 Result (P) u t = u + f (u, v), v = H(ξ 0, u), u t=0 = ϕ. + Neumann B.C, domain R n, n 2 Theorem (Local existence of solutions) If ϕ is transverse then there is a T > 0 such that: 1 There is at least one transverse solution to (P) on (0, T ) 2 Any solution to (P) is transverse on (0, T ) Theorem (Global uniqueness of transverse solutions) Given any T > 0 such that u 1, u 2 are two transverse solutions to (P) on the time interval t (0, T ), then u 1 = u 2 on (0, T ).

18 Transverse Initial Data

19 Transverse Initial Data

20 Transverse Initial Data Assumption: Dx ϕ 6= 0

21 Free boundary evolution: Example, Regularity t = t1

22 Free boundary evolution: Example, Regularity t = t1 t = t2 > t1,

23 Free boundary evolution: Example, Regularity t = t1 t = t2 > t1, t = t3 > t2,

24 Conclusions u t = u + f (u, v), v = H(ξ 0, u), u t=0 = ϕ. Theorem: If ϕ is transverse then there is a time interval such that the solution exists and is unique on this interval. Preliminary Applications: Hydra: Stability of stationary solutions Bacteria: Stability of numerics

25 Outlook u t = u + f (u, v), v = H(ξ 0, u), u t=0 = ϕ Outlook: Continuous dependence on ξ 0 {red, blue}. How does hysteresis approximate a slow-fast system in the PDE setting. Role of transversality in pattern formation (Sergey Tikhomirov, Pavel Gurevich).

26 Outlook u t = u + f (u, v), v = H(ξ 0, u), u t=0 = ϕ Outlook: Continuous dependence on ξ 0 {red, blue}. How does hysteresis approximate a slow-fast system in the PDE setting. Role of transversality in pattern formation (Sergey Tikhomirov, Pavel Gurevich). Thank you for your attention.

27 References I [1] Toyohiko Aiki and Jana Kopfová. A mathematical model for bacterial growth described by a hysteresis operator. In Recent Advances in Nonlinear Analysis, pages 1 10, [2] Hans Wilhelm Alt. On the thermostat problem. Control Cybern., 14(1-3): , [3] Pavel Gurevich and Sergey Tikhomirov. Uniqueness of transverse solutions for reaction-diffusion equations with spatially distributed hysteresis. Nonlinear Anal., 75(18): , December 2012.

28 References II [4] Pavel Gurevich, Sergey Tikhomirov, and Roman Shamin. Reaction diffusion equations with spatially distributed hysteresis. Siam J. of Math. Anal., 45(3): , [5] F.C. Hoppensteadt and W. Jäger. Pattern Formation by Bacteria. In Willi Jäger, Hermann Rost, and Petre Tautu, editors, Biological Growth and Spread, volume 38 of Lecture Notes in Biomathematics, pages Springer Berlin Heidelberg, 1980.

29 References III [6] F.C. Hoppensteadt, W. Jäger, and C. Pöppe. A hysteresis model for bacterial growth patterns. In Willi Jäger and James D. Murray, editors, Modelling of Patterns in Space and Time, volume 55 of Lecture Notes in Biomathematics, pages Springer Berlin Heidelberg, [7] Alexandra Köthe. Hysteresis-Driven Pattern Formation in Reaction-Diffusion-ODE Models. PhD thesis, University of Heidelberg, [8] Francisco JP Lopes, Fernando Vieira, David M Holloway, Paulo M Bisch, and Alexander V Spirov. Spatial bistability generates hunchback expression sharpness in the drosophila embryo. PLoS Computational Biology, 4(9), 2008.

30 References IV [9] Anna Marciniak-Czochra. Receptor-based models with hysteresis for pattern formation in hydra. Mathematical Biosciences, 199(1):97 119, [10] Augusto Visintin. Differential Models of Hysteresis. Applied Mathematical Sciences. Springer-Verglag, Berlin Heidelberg, 1994.