Reaction-Diffusion Equations with Hysteresis inHigher Spatial Dimensions
Mark Curran 1 2 3
Under the supervision of PD. Dr. Pavel Gurevich 1 2
1Free University Berlin
2SFB910 (Sonderforschungsbereich 910)
3Berlin Mathematical School
Patterns of Dynamics, Berlin, July 2016
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Hysteresis in Biology
Bacteria(Jager, Hoppensteadt ’80, ’83):
Non-diffusing: Bacterium
Diffusing: Nutrient, pH
Figure: Chiu, Hoppensteadt, Jager, Analysis andComputer Simulation of Accretion Patterns in BacterialCultures J. Math. Biol. 32, No.8 pp. 841-855 (1994)
Thresholds: α < β
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Hysteresis in Biology
Hydra (Marciniak-Czochra ’06):
Non-diffusing: Cells
Diffusing: Ligands
Figure: ‘Reaction-diffusion equations and biologicalpattern formation’, Anna Marciniak-Czochra, lecturenotes, University of Wroc law, 2011
Thresholds: α < β
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Hysteresis in Biology
Hydra (Marciniak-Czochra ’06):
Non-diffusing: Cells
Diffusing: Ligands
Figure: ‘Reaction-diffusion equations and biologicalpattern formation’, Anna Marciniak-Czochra, lecturenotes, University of Wroc law, 2011
Thresholds: α < β
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Hysteresis in Biology
Hydra (Marciniak-Czochra ’06):
Non-diffusing: Cells
Diffusing: Ligands
Figure: ‘Reaction-diffusion equations and biologicalpattern formation’, Anna Marciniak-Czochra, lecturenotes, University of Wroc law, 2011
Thresholds: α < β
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Model Problem
x ∈ domain ⊂ Rn, n ≥ 2, t ≥ 0,u(x , t), v(x , t) ∈ R, NeumannB.C., ξ0 ∈ {red, blue},
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ.
Thresholds: α < β
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Model Problem
x ∈ domain ⊂ Rn, n ≥ 2, t ≥ 0,u(x , t), v(x , t) ∈ R, NeumannB.C., ξ0 ∈ {red, blue},
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ.
Non-Ideal Relay:
Thresholds: α < β
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Model Problem
x ∈ domain ⊂ Rn, n ≥ 2, t ≥ 0,u(x , t), v(x , t) ∈ R, NeumannB.C., ξ0 ∈ {red, blue},
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ.
Non-Ideal Relay:
Thresholds: α < β
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Connection to Slow-Fast Systems
ut = ∆u + f (u, v),
εvt = v − v3
3 − u.
Stable normally hyperbolic, slowmanifolds:red, blue
NOTE: Unlike, e.g., travellingwaves in Fitzhugh-Nagumo, the
fast variable is not diffusing.
Slow-Fast System
Fold Points: α < β
GOAL: Develop a theoretical framework for systems of independentnon-ideal relays coupled via diffusion.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Connection to Slow-Fast Systems
ut = ∆u + f (u, v),
εvt = v − v3
3 − u.
Stable normally hyperbolic, slowmanifolds:red, blue
NOTE: Unlike, e.g., travellingwaves in Fitzhugh-Nagumo, the
fast variable is not diffusing.
Slow-Fast System
Fold Points: α < β
GOAL: Develop a theoretical framework for systems of independentnon-ideal relays coupled via diffusion.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Connection to Slow-Fast Systems
ut = ∆u + f (u, v),
εvt = v − v3
3 − u.
Stable normally hyperbolic, slowmanifolds:red, blue
NOTE: Unlike, e.g., travellingwaves in Fitzhugh-Nagumo, the
fast variable is not diffusing.
Slow-Fast System
Fold Points: α < β
GOAL: Develop a theoretical framework for systems of independentnon-ideal relays coupled via diffusion.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Context
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ,
+ Neumann B.C, domain ⊂ Rn, n ≥ 2.
Context:
Numerics + modelling: Jager 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)
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Context
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ,
+ Neumann B.C, domain ⊂ Rn, n ≥ 2.
Context:
Numerics + modelling: Jager 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)
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Difficulties
ut = ∆u + f (u, v), ∈ Lqv = H(ξ0, u), ∈ L∞
u|t=0 = ϕ,
+ Neumann B.C, domain ⊂ Rn, n ≥ 2.
Difficulties:
What is a sufficient condition for uniqueness?
Definition of solution? ut ,∆u ∈ Lq, q large enough.
What is the mechanism for pattern formation?
How does the free boundary evolve explicitly?
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Difficulties
ut = ∆u + f (u, v), ∈ Lqv = H(ξ0, u), ∈ L∞
u|t=0 = ϕ,
+ Neumann B.C, domain ⊂ Rn, n ≥ 2.
Difficulties:
What is a sufficient condition for uniqueness?
Definition of solution? ut ,∆u ∈ Lq, q large enough.
What is the mechanism for pattern formation?
How does the free boundary evolve explicitly?
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Result
(P)
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ.
+ Neumann B.C, domain ⊂ Rn, 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 u1, u2 are two transverse solutions to(P) on the time interval t ∈ (0,T ), then u1 = u2 on (0,T ).
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Result
(P)
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ.
+ Neumann B.C, domain ⊂ Rn, 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 u1, u2 are two transverse solutions to(P) on the time interval t ∈ (0,T ), then u1 = u2 on (0,T ).
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Transverse Initial Data
Assumption: Dxϕ 6= 0
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Transverse Initial Data
Assumption: Dxϕ 6= 0
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Transverse Initial Data
Assumption: Dxϕ 6= 0
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Free boundary evolution: Example, Regularity
t = t1
t = t2 > t1,
t = t3 > t2,
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Free boundary evolution: Example, Regularity
t = t1
t = t2 > t1,
t = t3 > t2,
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Free boundary evolution: Example, Regularity
t = t1
t = t2 > t1,
t = t3 > t2,
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Conclusions
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ.
Theorem: If ϕ is transverse then there is a time interval such thatthe solution exists and is unique on this interval.
Preliminary Applications:
Hydra: Stability ofstationary solutions
Bacteria: Stability ofnumerics
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Outlook
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ
Outlook:
Continuous dependence on ξ0 ∈ {red , blue}.How does hysteresis approximate a slow-fast system in thePDE setting.
Role of transversality in pattern formation (SergeyTikhomirov, Pavel Gurevich).
Thank you for your attention.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
Outlook
ut = ∆u + f (u, v),v = H(ξ0, u),
u|t=0 = ϕ
Outlook:
Continuous dependence on ξ0 ∈ {red , blue}.How does hysteresis approximate a slow-fast system in thePDE setting.
Role of transversality in pattern formation (SergeyTikhomirov, Pavel Gurevich).
Thank you for your attention.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
References I
[1] Toyohiko Aiki and Jana Kopfova.A mathematical model for bacterial growth described by ahysteresis operator.In Recent Advances in Nonlinear Analysis, pages 1–10, 2008.
[2] Hans Wilhelm Alt.On the thermostat problem.Control Cybern., 14(1-3):171–193, 1985.
[3] Pavel Gurevich and Sergey Tikhomirov.Uniqueness of transverse solutions for reaction-diffusionequations with spatially distributed hysteresis.Nonlinear Anal., 75(18):6610–6619, December 2012.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
References II
[4] Pavel Gurevich, Sergey Tikhomirov, and Roman Shamin.Reaction diffusion equations with spatially distributedhysteresis.Siam J. of Math. Anal., 45(3):1328–1355, 2013.
[5] F.C. Hoppensteadt and W. Jager.Pattern Formation by Bacteria.In Willi Jager, Hermann Rost, and Petre Tautu, editors,Biological Growth and Spread, volume 38 of Lecture Notes inBiomathematics, pages 68–81. Springer Berlin Heidelberg,1980.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
References III
[6] F.C. Hoppensteadt, W. Jager, and C. Poppe.A hysteresis model for bacterial growth patterns.In Willi Jager and James D. Murray, editors, Modelling ofPatterns in Space and Time, volume 55 of Lecture Notes inBiomathematics, pages 123–134. Springer Berlin Heidelberg,1984.
[7] Alexandra Kothe.Hysteresis-Driven Pattern Formation inReaction-Diffusion-ODE Models.PhD thesis, University of Heidelberg, 2013.
[8] Francisco JP Lopes, Fernando Vieira, David M Holloway,Paulo M Bisch, and Alexander V Spirov.Spatial bistability generates hunchback expression sharpnessin the drosophila embryo.PLoS Computational Biology, 4(9), 2008.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis
References IV
[9] Anna Marciniak-Czochra.Receptor-based models with hysteresis for pattern formationin hydra.Mathematical Biosciences, 199(1):97 – 119, 2006.
[10] Augusto Visintin.Differential Models of Hysteresis.Applied Mathematical Sciences. Springer-Verglag, BerlinHeidelberg, 1994.
Mark Curran ([email protected]) Reaction-Diffusion Equations with Hysteresis