Fronts for Periodic KPP Equations

Transcription

1 Fronts for Periodic KPP Equations Steffen Heinze Bioquant University of Heidelberg September 15th 2010

2 ontent Fronts and Asymptotic Spreading Variational Formulation for the Speed Qualitative onsequences Generalizations

3 KPP in periodic media onsider the KPP equation in a periodic medium: (KPP) t u(t, x) = (A(x) u(t, x)) + f (x, u(t, x)) t > 0, x IR n u(0, x) = u 0 (x) 0 A, b, f are 1 and 1-periodic in each direction x i A(x) IR n n positive definite, not necessarily symmetric f (x, 0) = f (x, 1) = 0 f (x, u)/u is decreasing for u > 0 u f (x, 0) =: µ(x) > 0 for 0 < u < 1 e.g. f (x, u) = µ(x)u(1 u) lassical case n = 1: u t = Au xx + µu(1 u)

4 Travelling waves A travelling wave solution in direction e IR n, e = 1 with speed c satisfies with ξ = ct + e x IR: u(t, x) = U(ξ, x), U(ξ, x) is periodic w.r.t. x i (TW) c ξ U(ξ, x) = ( + e ξ )(A(x)( + e ξ )U(ξ, x)) + f (x, U(ξ, x)) U(, x) = 0, U(, x) = 1 We need the following linear operator. With µ(x) = u f (x, 0) and λ 0 let (L λ φ)(x) = (A(x) φ(x)) + µ(x)φ(x) φ(x)e λe x is 1-periodic Due to the last condition L λ is not selfadjoint! Let k(λ) be the principal eigenvalue with corresponding eigenfunction φ(x) > 0.

5 Theorem (Berestycki, Hamel 02) A travelling wave exist iff k(λ) c c(e) := min λ>0 λ Explanation: Let (U(ξ, x), c) be a travelling wave. U(ξ, x) e λξ v(x), ξ λ > 0, v periodic Then φ(x) := e λe x v(x) > 0 satisfies Lφ(x) = λcφ(x) Hence c = k(λ) λ. onstruction of upper and lower solutions shows, that every c in the range of k(λ) λ occurs. How does c(e) depend on A, µ, e?

6 Asymptotic Spreading onsider an initial value 0 u 0 (x) 1 for KPP with compact, nonempty support and let u(t, x) be the solution. Theorem (Weinberger 02, Beresytcki, Hamel, Nadin 08)) For c(f ) w(e) = min f e>0 f e the following holds lim u(t, x + wte) = 0, x t u(t, x + wte) = 1, x lim t IRn, w > w(e) IRn, 0 w < w(e) w(e) is called the spreading speed in direction e. We will see that the formula for w(e) can be inverted:. c(e) = sup(e f )w(f ) f

7 Variational Formulation of the Speed There exists a variational formulation of k(λ) using the maximum principle: k(λ) = (L λ φ)(x) inf sup 0<φe λe x per 2 x φ(x) Difficult to use for qualitative analysis. Seek an integral variational principle. The adjoint operator of L λ is (L λ ψ) = (A(x)T ψ(x)) + µ(x)ψ(x) ψ(x)e λe x is 1-periodic Observe that k(λ) is a critical value of: ( ψa φ+µφψ) dx critical with constraint φψ dx = 1

8 Goal: Transform s.t. convex and concave part are seperated: φ(x) = α(x)e λρ(x), ψ(x) = α(x)e λρ(x) G(α, ρ, λ) := = α(x), ρ(x) e x are 1-periodic ( ( α λα ρ)a( α + λα ρ) + µα 2) (λ 2 α 2 ρa s ρ + 2λα ρa a α αa s α + µα 2) critical point with constriant α 2 = 1 where A s := 1 2 (A + AT ), A a := 1 2 (A AT )

9 Now the following saddle point property for k(λ) follows: Theorem (Donsker-Varadhan 76, Holland 78) k(λ) = sup inf G(α, ρ, λ) = inf sup G(α, ρ, λ) α ρ ρ α α, ρ e x per 1 (IR n ), α > 0, α 2 dx = 1 The Euler Lagrange equations are: (A s α) + λ 2 α ρa s ρ + λα A a ρ + µα = k(λ)α λ (α 2 A s ρ) + (A a α 2 ) = 0 This is a selfadjoint problem for α coupled to a Poisson equation for ρ.

10 Saddle point principle for the speed k(λ) Goal: Eliminate λ in c(e) = inf λ>0 λ. Idea: onsider ( = 2 if (µα 2 αa s α) G(α, ρ, λ) J(α, ρ) := inf λ>0 λ 1/2 α 2 ρa s ρ) + (µα 2 αa s α) 0 and otherwise. ρa a α 2, Let (α λ, ρ λ ) be the saddle point of G(α, ρ, λ). and let λ > 0 be the unique minimizer of k(λ)/λ. Have to show, that J(α λ, ρ λ ) > holds.

11 This follows from 0 = d dλ k(λ) λ λ = αλ 2 ρ λ A s ρ λ 1 λ 2 (µαλ 2 α λ A s α λ ) Since J is convex, w.r.t. λ, ρ and concave w.r.t. α we obtain: Theorem: (Saddle point principle) c(e) = sup inf J(α, ρ) = inf sup J(α, ρ) α ρ ρ α α, ρ e x per 1 (IR n ), α > 0, α 2 dx = 1

12 Dual variational principle Goal: Dualize the infimum into a supremum: For simplicity assume A a = 0. Let v(x) be a nontrivial periodic divergence free 1 vector field. ( 2 ( e v) = 2 v ρ) va 1 v α 2 α 2 ρa ρ and equality holds iff v = γα 2 A ρ for some γ IR. This is the Euler Lagrange equation for ρ. Theorem (Maximization principle) c(e) 2 /4 = sup α,v! 2 v e α, v per 1 (IR n ), α > 0, (µα 2 αa α) va 1 v α 2 α 2 dx = 1, v = 0, v 0

13 For n = 1 this simplyfies to: c(e) 2 /4 = sup α 1 (µα 2 Aα 2 ) Aα 2

14 Qualitative onsequences Smooth Dependence The principle eigenvalue k(λ) is simple. Hence it depends smoothly on parameters. One can show that d dλ k(λ) λ λ = 0 implies d 2 d 2 λ k(λ) λ λ > 0. Hence the minimal speed also depends smoothly on parameters. Suppose that the functional J depends on a parameter t and (α t, ρ t ) is the saddle point. Then d dt c(t) = tj(α t, ρ t, t)

15 Dependence on the direction Extend the definition of c(e) and w(e) as a homogeneous function to all e IR n of degree 1 and degree -1 respectively. Theorem c(e) and w(e) satisfy: 1. c(e), w(e) > 0, e 0 2. c(se) = sc(e), s 0 3. c(e + f ) c(e) + c(f ) 4. c(e) is the support function of a convex body. 5. 1/w(e) is the support function of the polar convex body, e f i.e. 1/w(e) = sup c(f ) f 0 6. c(e) = sup e f w(f ) f 0

16 Dependence on the period onsider the symmetric case A a = 0. For L > 0 replace in (KPP) A(x) and f (x, u) by A(x/L) and f (x/l, u). A rescaling of x gives: c(l) 2 /4 = sup inf (µα 2 1 α ρ L 2 αas α) Let (α L, ρ L ) be the saddle point of J. Theorem d dl c(l) = 2 L 3 α L A α L 0 α 2 ρa s ρ holds. Equality holds for some L 0 > 0 or equvalently for all L > iff ρa ρ ρa ρ + µ µ = 2 where ρ solves (A ρ) = 0, ρ(x) e x is 1-periodic

17 Remark: There exist nonconstant media, s.t. the speed is independent of the period and the direction. (test case for numerics)

18 Homogenization For simplicity consider the case A a = 0 only. Theorem The following limit exists lim c(l) =: c 0 = 2 µea h e L 0 and equals the minimal speed of the homogenized equation. Proof: hoosing α = 1 in the variational principle gives: c(l) 2 /4 µ inf ρa ρ = 2 µea h e ρ

19 Let (α L, ρ L ) be the critical point of J(α, ρ, L). We know: (µαl 2 1 L 2 α LA α L ) 0 This implies α L 1 in Hper 1. We have for every ρ: c(l) 2 /4 = J(α L, ρ L ) J(α L, ρ) αl 2 ρa ρ µ ρa ρ µαl 2 Minimizing over ρ completes the proof.

20 Large period limit Theorem Suppose A a = 0. Then lim c(l) = c = sup L α,v ( v e) 2 µα2 va 1 v α 2 In 1-D we have c = sup α ( 1 1 µα Aα 2 ) 1 The supremum can be evaluated, conjectured by Hamel, Fayard, Roques 10.

21 Theorem Let µ = sup µ(x) and assume Then 1 µ (a(µ µ)) 1/2 2 0 c = inf η>µ η η µ a ( µ µ a ) 1/2 This holds e.g. if µ is piecewise 2. If µ is constant this gives c = 2 µ ( 1 0 ) 1 1 a

22 Proof: The lower is obtained by choosing α = (a(η µ)) 1/4 in the variational principle and maximizing over η. The upper bound follows from ( ) 2 ψ µ a α 2 1 (η µ) aα 2 and minimizing over η. Equality of these bounds holds iff the condition above holds.

23 Dependence on µ Theorem If A is constant, then µ γ(µ) := c(µ) 2 is increasing and convex. In particular if µ 2 (x) = µ 1 (x + a) holds, then c( µ 1 + µ 2 ) c(µ 1 ) 2 follows, i.e. fragmentation decreases the minimal speed. Proof: Let (α, v) be the maximizer of J (α, v, (µ 1 + µ 2 )/2) where J is the functional of the dual variational principle. c( µ 1 + µ 2 2 ) 2 = J(α, v, µ 1 + µ 2 ) 2 2 = 1 2 J(α, v, µ 1) J(α, v, µ 2) 2 c(µ 1) 2 + c(µ 2 ) 2 2

24 Optimization over µ Theorem Suppose that A is constant and let µ > 0 be given. Then sup 1R µ=µ 0 c(µ) = c(µδ x0 ) where δ x0 is the Dirac functional at x 0. In 1-D this is finite. Proof: The convexity implies for every periodic ν with integral one: c(µ ν) c(µ) The variational formula can be extended to measures. hoosing µ = µδ x0 completes the proof. Related rearrangement results: Nadim (2010)

25 Generalizations Less regular data A, f, µ, e.g. µ could be a characterstic function. f (x, u) < 0 for u > M instead of f (x, 1) = 0. Then U + (x) is a periodic function instead of U + = 1. Instead of µ(x) 0, assume k(0) > 0, e.g. µ > 0 is enough. Perforated domains. Space and time periodic data.