Reaction-Di usion Equations with Hysteresis in Higher...

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

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

Diffusing: Ligands

Thresholds: α < β

Diffusing: Ligands

Thresholds: α < β

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

Model Problem

ut = ∆u + f (u, v),v = H(ξ0, u),

u|t=0 = ϕ.

Non-Ideal Relay:

Thresholds: α < β

Model Problem

ut = ∆u + f (u, v),v = H(ξ0, u),

u|t=0 = ϕ.

Non-Ideal Relay:

Thresholds: α < β

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.

ut = ∆u + f (u, v),

εvt = v − v3

3 − u.

Slow-Fast System

ut = ∆u + f (u, v),

εvt = v − v3

3 − u.

Slow-Fast System

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)

Context

ut = ∆u + f (u, v),v = H(ξ0, u),

u|t=0 = ϕ,

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)

Difficulties

ut = ∆u + f (u, v), ∈ Lqv = H(ξ0, u), ∈ L∞

u|t=0 = ϕ,

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?

Difficulties

ut = ∆u + f (u, v), ∈ Lqv = H(ξ0, u), ∈ L∞

u|t=0 = ϕ,

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?

Result

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 ).

Result

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 ).

Transverse Initial Data

Assumption: Dxϕ 6= 0

Free boundary evolution: Example, Regularity

t = t1

t = t2 > t1,

t = t3 > t2,

t = t1

t = t2 > t1,

t = t3 > t2,

t = t1

t = t2 > t1,

t = t3 > t2,

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

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.

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.

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.

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.

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.

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.

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