Dynamics of a class of
systems with an
order-preserving property
Hiroshi Matano
(Univ. of Tokyo)
“CMC Inaugural Conference at KAIST”
KAIST, December 5-6, 2013
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Outline
1. Introduction
Formulation of the problem
2. Basic concepts Order-preserving systems
3. Main theorems Order-preserving + mass conservation
4. Sketch of the proof
5. Applications
Molecular-motor model, reaction-diffusion systems, etc
Joint work with
Toshiko Ogiwara & Danielle Hilhorst
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1. Introduction
Some backgrounds
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GOAL
Evolution equations
comparison principle + mass conservation
Claim
These results follow immediately from a
general theory of order-preserving systems
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2. Basic concepts
Order-preserving systems
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Order structure
ordered metric space
Notation:
Examples
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Examples
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Order-preserving system
ordered metric space
Definition
A continuous map
implies
A continuous map
(SOP) if for any
such that
is called order-preserving (OP) if
is called strongly order-preserving
there exist neighborhoods U, V of
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Definition (for semiflow)
A semiflow
the map
A semiflow
for any
on X is called order-preserving if for any
is order-preserving.
on X is called strongly order-preserving (SOP) if
the map
is SOP.
(strong ) comparison principle
Examples
• quasilinear parabolic equations (on a bounded domain)
• cooperation RD systems
• competition RD systems (2 species)
• some delay-differential equations
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3. Main theorems
Order preserving systems with
mass conservation
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Basic setting
(T)
weaker than SOP
(M)
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Assume (T) (slightly strong comparison principle) and
(F) (mass conservation).
Corollary: Existence of a trivial solution implies
existence of a nontrivial solution.
If the equation (or the map T) is linear, then this implies the
existence of a positive solution.
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Assume (T) (slightly strong comparison principle) and
(F) (mass conservation).
Outline of proof:
• Any fixed point is stable but not asymptotically stable.
• Any bounded orbit converges; i.e., the omega-limit set is a
singleton.
(Analogue of a center manifold analysis without smoothness
assumption.)
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3. Sketch of the proof
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contradiction
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4. Applications
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(Nangaku, M. et al.,
Cell 79:1209- , 1994)
(1)
[Chipot-Hastings-Kinderhehrer (2004)]
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Molecular motor model
(Fokker Plank equation)
(2)
(3)
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Known results
Relative entropy method
Applying Krein-Rutman Theorem to an ajoint system
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Treatment by OP systems theory
fixed pt. of T = τ-periodic sol’n
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[Chipot-Hilhorst-Kinderlehrer-Olech (2009)]
(5)
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Application to cooperative reaction systems
[Chipot-Hilhorst-Kinderlehrer-Olech
(2009)]
(5)
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Application to a competition system
(6)
N.B.C
(6)
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(order-preserving, but not strongly order-preserving! )
Cannot apply the existing theory of “strongly
order-preserving dynamical systems”
( E : the set of all the T-periodic solution of (6) )
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Concluding remarks:
• Theory of order-preserving dynamical systems was developed
mainly in 1980’s and 90’s, and have found many applications.
• Recently, there have been new developments in this theory.
• In particular, if one combines the order-preserving property and a
mass conservation property, one can prove much stronger result
in a much simpler way.
• Among other things, one can show that existence of a trivial
solution automatically implies the existence of infinitely many nontrivial solutions.
Open questions:
• Can one apply the theory to a first-order conservation system,
which does not possess good compactness properties?
• How to deal with the p-Laplacian equation, which has strong
degeneracy?
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One would have to completely forgotten the history of
science so as not to remember that the desire to know
nature has had the most constant and the happiest
influence on the development of mathematics.
Jules Henri Poincaré (1854-1912)
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Thank you!
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