Bistable entire solutions in cylinder-like domains

Transcription

1 Bistable entire solutions in cylinder-like domains 3 ème journées de l ANR NonLocal Antoine Pauthier Institut de Mathématique de Toulouse Thèse dirigée par Henri Berestycki (EHESS) et Jean-Michel Roquejoffre (IMT) Supportée par le projet ERC ReaDi Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

2 Problem under study Bistable reaction-diffusion equation: { tu(t, x) u(t, x) = f (u), t R, x Ω, νu(t, x) = 0, t R, x Ω. The domain Ω is assumed to be a smooth infinite domain in the x 1 direction, i.e. { Ω = (x 1, x ), x 1 R, x ω(x 1 ) R N 1}. We also make a cylinder-like assumption: ω(x 1 ) x 1 ω The domain is, in one direction at infinity, the straight cylinder R ω. (1) x x 1 Figure : Example of asymptotically cylindrical domain Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

3 Assumptions: The reaction term f is of bistable kind, with 1 f (s)ds > 0. 0 f (u) There exists a C 2α diffeomorphism that sends Ω onto R ω = Ω. 0 θ 1 u Question Existence and uniqueness of an entire (i.e. eternal) solution in such a domain connecting 0 to 1? Influence on the dynamics in such domains? Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

4 Plan 1 Introduction The problem Motivations Homogeneous case Previous and current results 2 A one dimensional case study 3 Proof of the theorem Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

5 Influence of the geometry: biological motivations Population dynamics Population going through an isthmus or a straight (fishes) Cortical Spreading Depression CSD: transient and large depolarisation of the membrane of neurons. Propagation in the grey mater, absorption in the white matter. Aura in migraine with aura. Blocking of CSD in rodent: inefficient in human. These images are from the University of Wisconsin and Michigan State Comparative Mammalian Brain Collections, and from the National Museum of Health and Medicine Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

6 Homogeneous case In a straight cylinder R ω, uniqueness (up to translation) of the solution given by the bistable wave: u(t, x) = ϕ(x 1 ct). The couple (c, ϕ) is the unique solution of { cϕ ϕ = f (ϕ) ϕ( ) = 1, ϕ(+ ) = 0, ϕ(0) = θ. u(t = 0) = ϕ(x) u(t = 1) = ϕ(x c) c The travelling wave attracts the dynamics: if u 0 is front-like, there exists x 0 such that sup u(t, x) ϕ(x 1 ct + x 0 ) t + 0. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

7 Previous mathematical result (Berestycki-Bouhours-Chapuisat) Under a strong cylinder-like assumption: { } Ω x R N, x 1 < 0 = R ω, ω R N 1. This assumption asserts that the domain is equal to a cylinder in the left half space. ε Figure : Two examples of domains Ω There exists a unique solution of (1) defined for all t R such that u(t, x) ϕ(x 1 ct) 0, uniformly in Ω. t This soution is increasing in time and converges to a steady state u. Blocking phenomenon by a norrow passage. Partial invasion in sufficiently large domain. Complete invasion in star-shaped domains and domains with decreasing cross-section. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

8 Result Aim: get rid of the strong cylinder-like assumtion. We suppose that ω(x 1 ) converges to ω at some exponential rate for the C 2,α topology. Theorem (P.) There exists a unique function u(t, x) defined for t R and x Ω such that sup { u(t, x) ϕ(x 1 ct), x Ω} t 0. The proof amounts to proving the stability of the bistable wave. All the other results (propagation, complete or partial, blocking) are preserved. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

9 Case study One dimensional problem: tu xxu = f (u) (1 + g(x)), t R, x R (2) where g is a bounded perturbation that satisfies the assumption there exists κ > 0, g(x) e κx for all x R. (3) Theorem (P.) There exists a positive constant ϖ which depends only on f such that if g satisfies (3) and g > ϖ, then there exists a function u = u (t, x) defined for t R, x R solution of (2) which satisfies u (t,.) ϕ(. ct) L (R) 0. (4) t The constant ϖ is given by ϖ = ρ 1 where ρ f 1 is the spectral gap of the linearised operator associated with the travelling wave. Remark: other results for this type of equation concerning transition fronts (see Zlato s). Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

10 Strategy of the proof (Berestycki, Hamel, Matano) Define a sequence (u n) where the functions u n are solutions of { tu n(t, x) u n(t, x) = f (u n) t > n, x R u n( n, x) = ϕ(x 1 + cn). (5) The sequence (u n) converges locally uniformly to an entire solution. In the strong cylinder-like assumption, control with sub and super-solutions. Aim Stability of the bistable wave. Proposition There exist N 0, K, γ such that for all n > N 0, the solution u n of (5) satisfies u n(t, x) ϕ(x 1 ct) Ke γct, t [ n, N 0 [. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

11 Material Moving framework: ξ = x 1 ct + cn. The equation becomes { tu c ξ u ξξ u = f (u) (1 + g(ξ + ct cn)), t > 0, ξ R u(0, ξ) = ϕ(ξ). (6) Spectral decomposition: consider the linear operator L := c ξ + ξξ + f (ϕ). Then there exists X 1 R(L) such that BUC(R) = X 1 N (L), with N (L) = N (L 2 ) = ϕ R. Projection on N (L) : Pψ(ξ) = e, ψ ϕ (ξ) = 1 ( Λ e cz ϕ (z)ψ(z)dz R ) ϕ (ξ). We denote Q = I P the projection on R(L). The operator L satisfies on R(L), for any ρ < ρ 1, e tl R(L) Ce ρt. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

12 Splitting of the problem Lyapunov-Schmidt decomposition There exists ε, as soon as u(t) ϕ ε, one can decompose u(t, ξ) = ϕ(ξ + χ(t)) + v(t, ξ) with continuous χ and e, v(t) = 0. A "linear" parabolic equation on R(L) and an ODE on N (L). Use the coercivity of L on R(L). Bootstrap argument. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

13 Equation on R(L) { v t Lv (g(ξ + ct cn)f (ϕ)v e, gf (ϕ)v ϕ ) = Q [R + g(ξ + ct cn)f (ϕ)] v(0, ξ) = 0 where R = O(v 2 + χ 2 ). If n is large enough, g(ξ + ct cn)f (ϕ) is small. The "important" linear part is L + g(ξ + ct cn)f (ϕ). coercive if g > ρ 1 f. Formally, a Duhamel s formula gives the estimate: Lemma 1 There exist γ > α > 0, as soon as v(t), χ(t) < e α(ct cn), then v(t) < Ce γ(ct cn). Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

14 Equation on N (L) { χ (t) = e, R + e, g(ξ + ct cn)f (ϕ)v + g(ξ + ct cn)f (ϕ) χ(0) = 0. Once again, g(ξ + ct cn)f (ϕ) is small when n is large. The second term controlled by lemma 1. Lemma 2 As soon as v(t), χ(t) < e α(ct cn), χ(t) < Ce γ(ct cn). Conclusion For large n, uniform control of u n(t) ϕ L (R). It gives the behaviour of the entire solution as t. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

15 In a 2-dimensional cylinder-like domain To use the spectral decomposition, we need to be in a cylinder. Assumption There exists a C 2,α diffeomorphism { Ω Ω Φ : (x, y) (x, z) = (x, Φ(x, y)) such that there exist κ, C > 0, for all M R, ( ) Φ Id C min{e κm, 1}. C 2,α Change of variables: Ω {x<m} (ξ, z) = (x ct, Φ(x, y)), u(t, x, y) = ũ(t, ξ, z). y y Ω x Ω x Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

16 Equation in a straight cylinder For all n, we study the stability of ( ) ũ t cũ ξ ũ f (ũ) ũ z Φ 2ũ ξz Φ x + ũ zz 1 Φ 2 x Φ 2 y = 0 t > 0, (ξ, z) Ω ) ũ z (Φ y + Φ2 x Φ Φ y + ũ x ξ Φ y = 0 t > 0, (ξ, z) Ω ũ(0, ξ, z) = ϕ(ξ + cn). (7) Lemma Equation in a cylinder. Some more linear terms, with small coefficients only for x = ξ + ct 1. There exist α 1 and a positive function ψ such that (t, x, y) ψ(y)e α 1(x ct) is a super-solution of the problem (1) on the set Ω {x > ct}. Parabolic estimates: control of ũ ξ, ũ z where Φ is large control of the linear terms. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

17 Splitting of the problem Lyapunov-Schmidt decomposition There exists ε, as soon as ũ(t) ϕ(. + cn) ε, one can decompose ũ(t, ξ, z) = ϕ(ξ + cn + χ(t, z)) + v(t, ξ, z) with continuous χ and e, v(t,., z) = 0, for all z ω. The system becomes χ tϕ χ zzϕ + v t Lv v zz R 1 = 0 χ zϕ + v z = R 2 ũ(0, ξ, z) = ϕ(ξ + cn) (ξ, z) Ω (ξ, z) Ω (8) with R 1 (t) + R 2 (t) Ce α 1(ct cn) for large n, as soon as u ϕ remains "small." Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

18 Equation on R(L) tv Lv zzv = Q [R 1 ] zv = Q [R 2 ] v(0, ξ, z) = 0. t > 0, (ξ, z) Ω t > 0, (ξ, z) Ω The operators L and zz commute ; we can apply the Duhamel s formula. Coercivity of e tl and boundedness of the heat operator gives Lemma 1 There exist γ > β > 0, as soon as v(t) C 2,α + χ(t) C 2,α < e β(ct cn), then v(t) C 2,α < Ce γ(ct cn). (9) Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

19 Equation on N (L) tχ zzχ = e, R 2, zχ = e, R 3, χ(0, z) = 0. Once again, boundedness of the heat operator, t > 0, z ω t > 0, z ω (10) Lemma 2 There exist γ > β > 0, as soon as v(t) C 2,α + χ(t) C 2,α < e β(ct cn), then χ(t) C 2,α < Ce γ(ct cn). We can apply lemma 1 and lemma 2 in a bootstrap argument, which concludes the proof. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

20 Remarks and questions Uniqueness follows easily using Berestycki-Hamel-Matano in a cylinder-like domain, not in the case study. The spreading properties given by Berestycki-Bouhours-Chapuisat are preserved. We ask for an exponential convergence, but at an arbitrary rate. What about weaker convergence? in the case study, transition front? Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19

21 Thank you for your attention. Antoine Pauthier (IMT) ANR Nonlocal 3 juin / 19