Matematyka stosowana matematyka teoretyka Grzegorz Karch wyk lad - PowerPoint PPT Presentation

Matematyka stosowana matematyka teoretyka Grzegorz Karch wyk� lad o wynikach uzyskanych w Heidelbergu wsp´ olnie z Ann¸ a Marciniak-Czochr¸ a i Kanako Suzuki April 20, 2017

Basic model av � � u t = u + v − d c u , v t = − d b v + u 2 w − dv , (RD) w t = 1 γ w xx − d g w − u 2 w + dv + κ 0 for x ∈ (0 , 1), t > 0 with the homogeneous Neumann boundary conditions for the function w = w ( x , t ) w x (0 , t ) = w x (1 , t ) = 0 for all t > 0 , and with positive initial conditions u ( x , 0) = u 0 ( x ) , v ( x , 0) = v 0 ( x ) , w ( x , 0) = w 0 ( x ) .

Biological system ◮ Cell proliferation ( e.g. in lungs) is influenced by growth factor ◮ Growth factor is externally supplied or produced by the cells ◮ Growth factor diffuses along the structure formed by the cells and binds to cell membrane receptors ◮ Hypothesis: The diffusion of this growth factor may significantly influence the dynamics of the whole cell population

Spatial profiles of the solutions

Kinetic system. Boundedness of solutions ◮ Solutions are nonnegative and uniformly bounded (change of variables ( u , v u , uw )). � � ◮ The trivial steady state ( u 0 , v 0 , w 0 ) ≡ 0 , 0 , κ 0 is locally asymptotically d g stable. 0 ≥ Θ, where Θ = 4 d g d b d 2 c ( d b + d ) ◮ Assume a > d c and κ 2 ( a − d c ) 2 . Then, the kinetic system has two positive constant stationary solutions ( u ± , v ± , w ± ) , where � κ 2 v ± = d 2 w ± = κ 0 ± 0 − Θ c ( d b + d ) 1 u ± = a − d c , w ± , v ± . 2 d g ( a − d c ) 2 d c ◮ ( u − , v − , w − ) is stable, and ( u + , v + , w + ) is unstable.

Model with diffusion. Boundedness of mass Theorem Let κ 0 ≥ 0 . The solution ( u , v , w ) of (RD) satisfies � 1 u ( x , t ) dx ≤ κ 0 lim sup µ d c , t →∞ 0 � 1 v ( x , t ) dx ≤ κ 0 lim sup µ , t →∞ 0 � � Cd + 1 lim sup t →∞ � w ( t ) � ∞ ≤ κ 0 . µ d 1 / 2 d g g Here µ = min { d g , d b } > 0 .

Stationary problem � � aV U + V − d c U = 0 , (1) − d b V + U 2 W − dV = 0 , (2) 1 γ W xx − d g W − U 2 W + dV + κ 0 = 0 (3) and the boundary condition W x (0) = W x (1) = 0. ◮ We are interested only in U ( x ) > 0 and V ( x ) > 0, ◮ Let a > d c , ◮ We obtain V ( x ) = d 2 U ( x ) = a − d c c ( d b + d ) 1 V ( x ) and (4) W ( x ) . d c ( a − d c ) 2

Two-point boundary value problem The boundary value problem for W ( x ) γ W ′′ − d g W − d b d 2 1 c ( d b + d ) 1 W + κ 0 = 0 , ( a − d c ) 2 W x (0) = W x (1) = 0 . We find explicit γ 0 such that ◮ for all γ ∈ (0 , γ 0 ], the above problem has only constant solutions, ◮ for all γ > γ 0 , we describe all positive solutions of the problem.

Construction of patterns Definition Let k ∈ N and k ≥ 2 . We call a function W ∈ C ([0 , 1]) a periodic function on � 0 , 1 � [0 , 1] with k modes if W = W ( x ) is monotone on and if k � 2 j  � x − 2 j � k , 2 j +1 � W for x ∈  k k W ( x ) = � 2 j +2 � 2 j +1 � k , 2 j +2 � W − x for x ∈  k k for every j ∈ { 0 , 1 , 2 , 3 , ... } such that 2 j + 2 ≤ k.

Instability of patterns Let W ( x ) be one of the functions from the previous theorem, and ( U ( x ) , V ( x ) , W ( x )) be a stationary solution of our system, where V ( x ) = d 2 U ( x ) = a − d c c ( d b + d ) 1 V ( x ) and W ( x ) . d c ( a − d c ) 2 This stationary solution appears to be unstable solution of the reaction-diffusion equations (RD).

Instability of patterns Let W ( x ) be one of the functions from the previous theorem, and ( U ( x ) , V ( x ) , W ( x )) be a stationary solution of our system, where V ( x ) = d 2 U ( x ) = a − d c c ( d b + d ) 1 V ( x ) and W ( x ) . d c ( a − d c ) 2 This stationary solution appears to be unstable solution of the reaction-diffusion equations (RD). Let us be more precise.

Instability of patterns Linearized operator The linearization of system (RD) at the steady state ( U , V , W )  0 0 0   + A ( x ) . 0 0 0 L =  γ ∂ 2 1 0 0 x We consider L as an operator in the Hilbert space H = L 2 (0 , 1) ⊕ L 2 (0 , 1) ⊕ L 2 (0 , 1) with the domain D ( L ) = L 2 (0 , 1) ⊕ L 2 (0 , 1) ⊕ W 2 , 2 (0 , 1) . L has infinitely many positive eigenvalues.

Instability of patterns Spectrum of L Together with the matrix   � d c � ( a − d c ) 2 d c a − 1 0 a   K 2 A ( x ) = ( a ij ) i , j =1 , 2 , 3 = 2 K − d b − d  ,  W 2 ( x ) K 2 − 2 K d − d g − W 2 ( x ) we consider its sub-matrix � a 11 � a 12 A 12 ≡ . a 21 a 22 Lemma Let λ be an eigenvalue of the matrix A 12 . Then λ belongs to the continuous spectrum of the operator L .

Instability of patterns Spectrum of L Together with the matrix   � d c � ( a − d c ) 2 d c a − 1 0 a   K 2 A ( x ) = ( a ij ) i , j =1 , 2 , 3 = 2 K − d b − d  ,  W 2 ( x ) K 2 − 2 K d − d g − W 2 ( x ) we consider its sub-matrix � a 11 � a 12 A 12 ≡ . a 21 a 22 Lemma Let λ be an eigenvalue of the matrix A 12 . Then λ belongs to the continuous spectrum of the operator L . The matrix A 12 has a positive eigenvalue λ 0 .

Instability of patterns Spectrum of L - the crucial lemma Lemma A complex number λ is an eigenvalue of the operator L if and only if the following two conditions are satisfied ◮ λ is not an eigenvalue of the matrix A 12 , ◮ the boundary value problem has a nontrivial solution: 1 det( A − λ I ) γ η ′′ + det( A 12 − λ I ) η = 0 , x ∈ (0 , 1) η ′ (0) = η ′ (1) = 0 . Proof. Study the system ( a 11 − λ ) ϕ + a 12 ψ = 0 a 21 ϕ + ( a 22 − λ ) ψ + a 23 η = 0 1 γ ∂ 2 x η + a 31 ϕ + a 32 ψ + ( a 33 − λ ) η = 0 , supplemented with the boundary condition η x (0) = η x (1) = 0 �

Instability of patterns Spectrum of L - main result Theorem Denote by λ 0 the positive eigenvalue of the matrix A 12 . There exists a sequence { λ n } n ∈ N of positive eigenvalues of the operator L that satisfy λ n → λ 0 as n → ∞ . Recall that λ 0 belongs to the continuous spectrum of the operator L . Idea of the proof. Analysis of solutions of the generalized Sturm-Liouville problem 1 γ η ′′ + q ( x , λ ) η = 0 , x ∈ (0 , 1) η ′ (0) = η ′ (1) = 0 , where q ( x , λ ) = det( A ( x ) − λ I ) det( A 12 − λ I ) . �

Existence of discontinuous patterns � aV � U + V − d c U = 0 , (5) − d b V + U 2 W − dV = 0 , (6) 1 γ W xx − d g W − U 2 W + dV + κ 0 = 0 (7) Theorem Assume that a > d c and κ 2 0 > Θ . There exists a continuum of weak solutions of the stationary system with some γ > 0 . Each such solution ( U , V , W ) ∈ L ∞ (0 , 1) × L ∞ (0 , 1) × C 1 ([0 , 1]) has the following property: there exists a sequence 0 = x 0 < x 1 < x 2 < ... < x N = 1 such that for each k ∈ { 0 , N − 1 } either ◮ for all x ∈ ( x k , x k +1 ) , U ( x ) = V ( x ) = 0 and W ( x ) satisfies γ W ′′ − d g W + κ 0 = 0 , 1 or ◮ for all x ∈ ( x k , x k +1 ) , U ( x ) > 0 , V ( x ) > 0 and W are solutions of the stationary equation.

Instability of discontinuous stationary solutions Theorem Every discontinuous weak stationary solution ( U I , V I , W I ) with a null set I ⊂ [0 , 1] , is an unstable solution of the nonlinear system considered in the Hilbert space H I . ◮ For a null set I , we define the associate L 2 -space L 2 I (0 , 1) = { v ∈ L 2 (0 , 1) : v ( x ) = 0 on I} , supplemented with the usual L 2 -scalar product, which is a Hilbert space as the closed subspace of L 2 (0 , 1). ◮ If u 0 ( x ) = v 0 ( x ) = 0 for some x ∈ [0 , 1] then u ( x , t ) = v ( x , t ) = 0 for all t ≥ 0. Hence, the space H I = L 2 I (0 , 1) × L 2 I (0 , 1) × L 2 (0 , 1) is invariant for the flow generated by the system.

Main result: instability of ALL stationary solutions A.M-C, G.K., K.S., J.Math.Pures et Appl., 2013

Reaction-diffusion-ODE system (A. Marciniak-Czochra, G.K., K. Suzuki)

. The point of departure: a general system of reaction-diffusion (reaction-diffusion-ODE) equations: u t = f ( u , v ) , for x ∈ Ω , t > 0 x ∈ Ω , t > 0 v t = D ∆ v + g ( u , v ) for in a bounded domain Ω ⊂ R n . The Neumann boundary condition: x ∈ ∂ Ω , t > 0 ∂ n v = 0 for Initial data: u ( x , 0) = u 0 ( x ) , v ( x , 0) = v 0 ( x ) . ◮ D > 0 – a constant diffusion coefficient. (We can set D = 1.) ◮ arbitrary C 1 -nonlinearities f = f ( u , v ) and g = g ( u , v ).

Constant stationary solutions – Turing instability u t = f ( u , v ) , v t = ∆ v + g ( u , v ) ∂ n v = 0 x ∈ ∂ Ω , t > 0 u ( x , 0) = u 0 ( x ) , v ( x , 0) = v 0 ( x ) . Theorem Assume that the constant vector (¯ u , ¯ v ) is a (stationary) solution of the initial-boundary value problem for this ordinary-PDE system. If f u (¯ u , ¯ v ) > 0 , then (¯ u , ¯ v ) is an unstable solution of this problem. Remark. Autocatalysis leads to the instability of stationary solutions.

Regular stationary solutions – standing assumption We consider only regular stationary solutions , namely, we assume, that we can solve the equation f ( U ( x ) , V ( x )) = 0 to have U ( x ) = k ( V ( x )) for a C 1 -function k = k ( V ). Under this assumption, regular stationary solutions of f ( u , v ) = 0 , ∆ v + g ( u , v ) = 0 ∂ n v = 0 x ∈ ∂ Ω satify the boundary value problem ∆ V + h ( V ) = 0 , where h ( V ) = g ( k ( V ) , V ) , ∂ n V = 0 on ∂ Ω .