Convex compact sets in R N 1 give traveling fronts in R N in cooperation-diffusion systems. Masaharu Taniguchi

Transcription

1 BIRS Workshop 14w5017, May 29, 2014 Convex compact sets in R N 1 give traveling fronts in R N in cooperation-diffusion systems Masaharu Taniguchi (Dept of Mathematics, Okayama University) 1

2 Allen-Cahn equation Nagumo equation, scalar Ginzburg-Landau equation u t = u + f(u) in R N, t > 0, u(x, 0) = u 0 (x) in R N. Here N 2 and x = (x 1,..., x N ), u 0 BU(R N ), = N 2 i=1 x 2. i 2

3 Nonlinearity term f f(s) -1 -a 0 1 * s f C 1 (R) satisfies f(1) = 0, f( 1) = 0, f (1) < 0, f ( 1) < 0, 1 1 f > 0, f(s) < 0 if 1 < s < a or s > 1 f(s) > 0 for a constant a ( 1, 1). if a < s < 1 or s < 1 3

4 One-dimensional traveling front Φ(µ) 1 Φ(y) k y 0-1 There exists k > 0 and Φ(y) satisfying Φ (y) kφ (y) f(φ(y)) = 0 for all y R Φ (y) > 0 for all y R Φ( ) = +1, Φ( ) = 1 k : speed of one-dimensional traveling wave Φ 4

5 Explicit one-dimensional traveling front 1 Φ(y) k y 0-1 For f(u) = (u + 1)(u + a )(u 1), one has k = ( ) y 2a, Φ(y) = tanh 2. The Huxley solution (Nagumo, Yoshizawa, Arimoto 1965) 5

6 One-dimensional traveling fronts (existence and uniquenss for bistable f) Kanel (1961, 1962) Nagumo, Yoshizawa and Arimoto (1965) Aronson and Weinberger (1975, 1978) existence of traveling fronts Fife and McLeod (1977, 1981) stability of traveling fronts Xinfu Chen (1997) existence and stability of traveling fronts 6

7 A cooperation-diffusion system u = D u + f(u) x R N, t > 0, t u(x, 0) = u 0 (x) x R N. Here u(x, t) = D = ( u1 (x, t) u 2 (x, t) ( ) d1 0, d 0 d 1 > 0, d 2 > 0, 2 ) ( ) f1 (u), f(u) =, 0 = f 2 (u) ( 0 0 ), 1 = ( ) 1. 1 We write this solution as u(x, t; u 0 ). 7

8 Notations For u = (u 1, u 2 ) and v = (v 1, v 2 ), we write u v if and only if u 1 v 1 and u 2 v 2. We write u < v if and only if u 1 < v 1 and u 2 < v 2. 8

9 Assumptions (A1) f(0) = 0, f(1) = 1, f i min 0 i j. [0,1] 2 u j (A2) There exist k > 0, Φ = (Φ 1, Φ 2 ) with DΦ (y) + kφ (y) + f(φ(y)) = 0 y R, Φ( ) = 1, Φ( ) = 0. Φ (y) 0 y R. (A3) q R 2, q > 0, f (0)q < 0. det ( λ 2 D + kλi + f (0) ) 0 for all λ R. 9

10 Assumptions (continue) (A4) All roots of det ( λ 2 D + kλi + f (0) ) = 0 are nonzero real numbers. (A5) 0 < a < 1, f(a ) = 0, ( ) f α11 α (a ) = 12 α 21 α 22 α 12 > 0, α 21 > 0, α 12 α 21 α 11 α 22 > 0. 10

11 The comparision principle From the assumption (A1) the comparison principle gives if 0 u(x, t; u 0 ) 1 for all x R N, t > 0 0 u 0 (x) 1 for all x R N. The comparison principle also gives u(x, t; u 0 ) u(x, t; ũ 0 ) for all x R N, t > 0 if u 0 (x) ũ 0 (x) for all x R N. 11

12 Example 1. The Lotka-Volterra system u t = u + u(1 u c 0v) v t = d v + v(a 0 b 0 u v) x R N, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x) x R N. Positive constants a 0, b 0, c 0, d satisfy 1 c 0 < a 0 < b 0. u 0, v 0 BU(R N ) satisfy 0 u 0, 0 v 0 a 0. 12

13 Equilibrium points Two stable equilibrium points ( ) ( ) 0 1,. 0 a 0 One unstable equilibrium point a 0 c 0 1 b 0 c 0 1 b 0 a 0. b 0 c

14 The one-dimensional traveling front There exists s R and (U, V ) satisfying U (y) + su (y) + U(y)(1 U(y) c 0 V (y)) = 0 dv (y) + sv (y) + V (y)(a 0 b 0 U(y) V (y)) = 0 ( ) ( ) U (y) 0 > y R, ( U( ) V ( ) V (y) ) = ( 1 0 ), 0 ( ) U( ) V ( ) = ( 0 a 0 ). y R, See Kan-on (1995) and Kan-on and Fang (1996). Assumptions (A1) (A5) hold true if s 0. 14

15 Example 2: General Cases In addition to (A1), (A3), (A4) and (A5) assume that all eigenvalues of f (0) and those of f (1) lie in {λ C Reλ < 0}. Then there exist k R, Φ = (Φ 1, Φ 2 ) with DΦ (y) + kφ (y) + f(φ(y)) = 0 y R, Φ( ) = 1, Φ( ) = 0. Φ (y) 0 y R. See Volpert, Volpert and Volpert (1994). Assumptions (A1) (A5) hold true if k > 0. 15

16 The moving coordinate c 0 : the speed of a multi-dimensional traveling front µ = x N ct : the moving coordinate x = (x 1,..., x N 1 ) u(x, x N, t) = w(x, µ, t) w t N i=1 2 w x 2 i c w x N f(w) = 0 in R N, t > 0, w t=0 = u 0 in R N. We write the solution as w(x, t; u 0 ). 16

17 Profile equation for traveling front solutions A front profile v(x) satisfies N i=1 2 v x 2 i c v x N f(v) = 0 v xn = = 1, v x N = = 1 in R N Then v(x, x N ct) becomes a traveling front. c : the speed of a multi-dimensional traveling front 17

18 Planar traveling front solutions Planar traveling front solutions (c = k) J. Xin (CommPDE 1992), Kapitula (TransAMS 1997) Matano, Nara, Taniguchi (CommPDE 2009) Roquejoffre, Roussier-Michon (AnnaliMatematica2009) Stationary balls (c = 0) Nehari(1963), Berestycki-Lions-Peletier(1981), Berestycki- Lions (1983), Kwong(1989), Busca-Jendoubi-Polacik(2002) and so on. 18

19 Results for balanced f balanced cases: 1 1 f = 0 Cylindrically symmetric traveling fronts Chen,Guo,Hamel,Ninomiya,Roquejoffre (Ann. Poincaré 2007) cylindrically symmetric traveling fronts for balanced cases De Giorgi conjecture(statinary solutions) Savin (Annals of Math, 2009) monotone stationary solutions are hyperplanes if N 8. del Pino, Kowalczyk, Wei (Annals of Math, 2011) non-planar monotone stationary solutions in N 9. See also Gui(2003), Ghoussoub-Gui(2003) and other works. 19

20 The speed and the slope y ct kt θ x One has tan θ = m, where c 2 k 2 m =. k The slope determines the speed. 20

21 Munuzuri, Gomez-Gesteira, Munuzuri, Davydov and Perez-Villar(1995) 21

22 A 2-dimensional V-form wave speed c Fig.8: Contour lines of 2-dimensional V-form wave. (Ninomiya and Taniguchi, JDE2005, DCDS-A2006) 22

23 Multi-dimensional non-planar traveling waves Ninomiya, Taniguchi(JDE2005, DCDS-A2006) Two-dimensional V -form fronts Hamel, Monneau, Roquejoffre(DCDS-A2005, DCDS- A2006) Cylindrically symmetric traveling front Haragus, Scheel (Ann Poincaré 2006) almost planar V-form waves in R 2, bifurcation theory Z.-C. Wang(DCDS-A2012) Two-dimensional V -form fronts for a competition system 23

24 A 2-dimensional V-form wave speed c Fig.8: Contour lines of 2-dimensional V-form wave. (Ninomiya and Taniguchi, JDE2005, DCDS-A2006) 24

25 A 2-dimensional V-form wave Fig.9: A bird-eye view of 2-dimensional V-form wave. (Ninomiya and Taniguchi, JDE2005, DCDS-A2006) 25

26 The cylindrically symmetric traveling front Speed C z r Fig.11: The cylindrically symmetric traveling front U( x, x N ) (Hamel, Monneau, Roquejoffre(DCDS-A2005, DCDS-A2006)) 26

27 The cylindrically symmetric traveling front U(r, x N ) ( 2 r N 1 2 r r 2 x 2 N c ) U x N = f(u) r > 0, x N R, U(0, 0) = 0, U r (0, x N ) = 0 x N R. Define ϕ(r) by U(r, ϕ(r)) = 0. 27

28 A pyramidal traveling wave Speed C z y x Figure: The contour surface of a pyramidal traveling wave (Taniguchi, SIAM J.Math.Anal.2007, JDE2009) 28

29 Pyramidal traveling fronts Taniguchi (SIAM J. Math.Anal. 2007; JDE 2009) pyramidal traveling fronts in R 3 Kurokawa, Taniguchi (Proc.Roy.Soc.Edinburgh Sect.A, 2011) pyramidal traveling fronts in R N Wei-Ming Ni, Taniguchi (Networks and Heterogeneous Media, 2013) pyramidal traveling fronts for cooperation-diffusion systems in R N 29

30 Existence of a 3-dim square pyramidal traveling front h(x, y) = m 2 ( x + y ), S(x, y) = c 1 + φ(x, y) 2 k. φ(x, y) = (ρ h)(x, y), A subsolution v(x, y, z) = Φ ( k (z h(x, y))) c A supersolution ( ) 1 ζ φ(ξ, η) v(x, y, z) = Φ + εs(ξ, η) α 1 + φ(ξ, η) 2 Here ξ = αx, η = αy, ζ = αz, 0 < ε << 1 and 0 < α << 1. 30

31 A pyramidal traveling wave Speed C z y x Figure: The contour surface of a pyramidal traveling wave (Taniguchi, SIAM J.Math.Anal.2007, JDE2009) 31

32 Proposition 1 [Estimates on V (n) (x) uniform in n] Let V (n) be the pyramidal traveling front associated with a pyramid with n lateral faces. For any δ (0, 1) there exists γ > 0 such that V (n) (x) γ if 1 + δ V (n) (x) 1 δ. x N Here γ can be taken independent of n. 32

33 The limit of a pyramidal traveling front A pyramidal traveling front for a regular 2 n pyramid V (n) (x) converges to a cylindrically symmetric traveling front U(r, x N ) with lim n V (x, x N +z n ) = U( x, x N ) in Cloc 2 (RN ). Here lim n z n = +. 33

34 The cylindrically symmetric traveling front ϕ(r) defined by U(r, ϕ(r)) = 0 satisfies lim r ϕ (r) = m lim U(s + r, ϕ(s) + x N) = Φ s ( k c (x N m r) ) in C 2 loc (R2 ). Remark 1 ϕ(r) = m r k 0 log r + O(1) as r. Here k 0 > 0. (Hamel, Monneau, Roquejoffre (DCDS- A2006)) 34

35 Buckmaster and Ludford (1983) 35

36 Polyhedral flames in experiments Cross Section speed C Chrysler emblem five sided polyhedral flame (turned upside down) Smith and Pickering (1929), Buckmaster and Ludford (1983) 36

37 BUNSEN FLAMES OF UNUSUAL STRUCTURE* FRANCIS A. SMITH AND SAMUEL F. PICKERING 204 Chemistry Division, Bureau of Standards, Washington, D. C. An abstract of a description, illustrated by lantern slides and autochromes, of unusual flame structures, the causes for which have not yet been investigated. A mixture of air or oxygen with a combustible gas is forced through a burner tube, and the flame is observed as it burns in secondary air. Some examples of complex structure observed FIG. 1. Propane-air. Accurate and reproducible control of the composition and rate of flow of the gas mixtures, and very steady stream line flow in the burner tubes have been attained. Photographs of the above flames are presented herewith. Using propane and air, when secondary air is excluded the primary zone or combustion surface presents the appearance shown in Fig. 1 (5.31 per cent C3H8). It is perfectly stable and reproducible. The figure can he made to rotate rapidly, slowly, or to remain stationary, depending upon the composition of the gas mixture, to which it is very sensitive in acetylene-air flames are: four distinct zones of combustion, the intersection of two zones, and a hollow dark core extending upward from the tip of the primary zone. When secondary air is excluded, the primary combustion surface of some hydrocarbon-air flames becomes polyhedral. Flames having three, four, five, six, and seven sides have been observed, which will rotate or remain stationary. The number of sides is a function of the size of the burner tube and of the composition of the gas mixture. Burning in secondary air, the primary zone of some propane-oxygen flames becomes polyhedral, and luminous streamers rise from the tip and corners of the figure, which can be made to rotate slowly, rapidly, or remain stationary. * Publication approved by the Director of the Bureau of Standards of the U.S. Department of Commerce FIGS. 2 to 11. Acetylene-air.

38 Smith and Pickering (1929), Buckmaster and Ludford (1983) 37

39 Smith and Pickering (1929), Buckmaster and Ludford (1983) Does the Allen-Cahn equation (Nagumo equation) have such traveling front solutions? 37-a

40 Our Goal For g C 2 (S N 2 ), g 0 assume that {rξ 0 r g(ξ)ξ ξ S N 2 } is a strictly convex compact set and all principal curvatures of {g(ξ)ξ ξ S N 2 } are positive for any ξ S N 2. Then g G gives a traveling front of the Allen-Cahn equation in R N. An example of the graph of g Ellipsoid N 1 i=1 x 2 i a 2 i = 1 a i > 0 (1 i N 1). 38

41 An equivalence relation C g = {g(ξ)ξ ξ S N 2 }, D g = {rξ 0 r g(ξ), ξ S N 2 }. Let G be defined by { g C 2 (S N 2 ) g(ξ) 0, all principal curvatures of Cg are positive for any ξ S N 2} For g G and a 0 define g 1 G by C g1 = {x C g (R N 1 \D g ) dist(x, C g ) = a.} We call g 1 = τ a g. For g, g 1 G, define an equivalence relation g g 1 by g 1 = τ a g or g = τ a g 1 for some a 0. 39

42 An equivalence relation g g 1 C g1 a a C g Figure. The graphs of C g and C g1. 40

43 Our assertion An element of a quotient set G/ gives a traveling front in the Allen-Cahn equation in R N. 41

44 A traveling front Ũ associated with g Speed C Figure. the graph of Ũ. 42

45 The cylindrically symmetric traveling front ϕ(r) defined by U(r, ϕ(r)) = 0 satisfies lim r ϕ (r) = m lim U(s + r, ϕ(s) + x N) = Φ s Moreover, for every η 0 one has ( k c (x N m r) ) in C 2 loc (R2 ). U(r + η, x N + m η) U(r, x N ) for all r 0, x N R, lim R sup (U(r, x N ) U(r + η, x N + m η)) = 0. (r,x N ) [R, ) R 43

46 Theorem 1 (A traveling front associated with g in R 3 ) Let R(θ) C (S 1 ) be given and assume that the curvature of {(R(θ) cos θ, R(θ) sin θ) 0 θ 2π} is positive for all 0 θ 2π. Then there exists a unique Ũ with lim s x s ( 2 x 2 2 y 2 2 sup Ũ(x) Here x = (x, y, z). min U( 0 θ 2π z 2 c z ) Ũ f(ũ) = 0 in R 3, (x R(θ) cos θ) 2 + (y R(θ) sin θ) 2, z) = 0. (T, DCDS-A 2012 (vol 32, No 3)) 44

47 Theorem 2 (A traveling front associated with g in R N ) Let g C 2 (S N 2 ) and assume that all principal curvatures of {g(ξ)ξ ξ S N 2 } are positive for any ξ S N 2. Then there exists a unique Ũ that satisfies ( N i=1 lim sup s x s 2 x 2 i c x N ) Ũ f(ũ) = 0 in R N, Ũ(x) min U ( ) x g(ξ)ξ, x N ξ S = 0. N 2 45

48 Theorem 2 (continued) For j = 1, 2, let Ũ j be traveling fronts associated with g j, respectively. Then one has Ũ 2 (x 1,..., x N 1, x N ) = Ũ 1 (x 1,..., x N 1, x N ζ) for some ζ R if and only if g 1 g 2. (T, submitted) 46

49 Remark. If g(ξ) = (const.), one has for λ R. Ũ(x, x N ) = U(x, x N λ) For every g, there exists µ > 0 such that U(x, x N + µ) Ũ(x, x N ) U(x, x N µ). Thus there exists a cylindrically non-symmetric traveling front between two cylindrically symmetric traveling fronts. (cf. Hamel and Berestycki 2007) 47

50 A traveling front Ũ associated with g Speed C Figure. the graph of Ũ. 48

51 a a C g Figure. {g(ξ)ξ ξ R N 1 } and a surface lying with a constant distance. If x 0 N > 0 is large enough, {x R N 1 U(x, x 0 N ) = 0} is approximated by the red surface for large a > 0. 49

52 The proof of Proposition 1 F 1 (u) = u a f(s) ds. x 0 RN 1 : arbitrarily chosen Fix u κ ( 1, a ) by 2F (u κ ) = F ( a ). Ω ( (D N V )( V, ν) + 12 V 2 ν N F 1 (V )ν N ) ds = c Ω (D N V ) 2 dx Ω = { (x, x N ) R N x x 0 < l, u κ < V (x, x N ) < a } Γ 1 = { (x, x N ) R N x x 0 l, V (x, x N ) = a } Γ κ = { (x, x N ) R N x x 0 l, V (x, x N ) = u κ } Γ f = { (x, x N ) R N x x 0 = l, u κ V (x, x N ) a } 50

53 The proof of Proposition 1 (continued) Γ f Γ κ Ω Γ 1 Figure. a domain Ω and its boundary for N = 2 51

54 The proof of Proposition 1 (continued) Γ κ Γ f Γ 1 ( (D N V )( V, ν) + 12 V 2 ν N F 1 (V )ν N ) ( (D N V )( V, ν) + 12 V 2 ν N F 1 (V )ν N ) (( D N V )( V, ν) + 12 V 2 ν N F 1 (V )ν N ) Then one has = Γ 1 ds ( 1 ) 2 V D NV Γ ( D NV ) V ds kκ 2c V N 1l N 1. ds > 0. ds kκ c V N 1l N 1. ds 2m 0A N 2 l N 2. 52

55 A sketch of the proof of Theorem 2 a a C g A supersolution v(x, x N ) = min ξ S N 2 U( x g(ξ)ξ, x N ) for (x, x N ) R N A subsolution v(x, x N ) = max ξ S N 2 U( x g(ξ)ξ +η, x N +m η) for (x, x N ) R N 53

56 A traveling front Ũ associated with g Speed C Figure. the graph of Ũ. 54

57 Theorem 3 (A traveling front associated with g in R N ) Let g C 2 (S N 2 ) and assume that all principal curvatures of {g(ξ)ξ ξ S N 2 } are positive for any ξ S N 2. Then cooperation-diffusion system has a unique Ũ that satisfies ( ) N 2 x 2 c Ũ i=1 i x f(ũ) = 0 in RN, N lim sup s x s Ũ(x) min U ( ) x g(ξ)ξ, x N ξ S = 0. N 2 55

58 Theorem 3 (continued) For j = 1, 2, let Ũj be traveling fronts associated with g j, respectively. Then one has Ũ 2 (x 1,..., x N 1, x N ) = Ũ1(x 1,..., x N 1, x N ζ) for some ζ R if and only if g 1 g 2. (T, in preparation) 56

59 Thank you for listening! 57