Strauss conjecture on asymptotically Euclidean manifolds

Strauss conjecture on asymptotically Euclidean manifolds Xin Yu (Joint with Chengbo Wang) Department of Mathematics, Johns Hopkins University Baltimore, Maryland 21218 xinyu@jhu.edu Mar 12-Mar 13, 2010

The Problem We consider the wave equations on asymptocially Euclidean manifolds (M, g) ( ) { g u = ( 2 t g )u = F (u) on R + M u(0, ) = f, t u(0, ) = g F (u) u p when u is small. g = 1 ij det g i det gg ij j is the Laplace-Beltrami operator. Assumptions on the metric g 1 2 α N n α x (g ij δ ij ) = O( x α ρ ), (H1) with δ ij = δ ij being the Kronecker delta function. g is non-trapping. (H2)

Goals For small data, we want to set up: Note Global existence result (Strauss Conjecture) for n = 3, 4 and p > p c. where p c is the larger root of the equation (n 1)p 2 (n + 1)p 2 = 0. Local existence result for n = 3 and p < p c with almost sharp life span T ɛ = Cɛ p(p 1) p 2 2p 1 +ɛ. p c = 1 + 2 for n = 3, p c = 2 for n = 4.

Earlier Work in Minkowski space R + R n 79 John: n=3, global sol n for p > 1 + 2, almost global sol n for p < 1 + 2; 81 Struss Conjecture: n 2, global sol n iff p > p c, where p c is the larger root of (n 1)p c (n + 1)p c 2 = 0. 81 Glassey: Verify for n = 2; 87 Sideris: Blow up for p < p c ; 95 Zhou: Verify for n = 4; 99 Georgiev, Lindblad, Sogge and 01 Tataru: n 3 and p > p c.

Earlier Work (continued) On more general domains. Perturbed by obtacles 1 08 D.M.S.Z: Nontrapping, g =, n = 4, p > p c ; 2 08 H.M.S.S.Z: Nontrapping, n = 3, 4, p > p c ; 3 09 Yu: Trapping (Limited), n = 3, 4, p > p c ; n = 3, p < p c. 10 Han and Zhou: Star-shaped obstacle and n 3: Blow up when p < p c with an upper bound of life span. Asymptotically Euclidean metric 09 Sogge and Wang: n = 3, p > p c under symmetric metric.

Our Result (Global existence part) Theorem Suppose (H1) and (H2) hold with ρ > 2. Also assume 2 u i uf i (u) u p. i=1 If n = 3, 4, p c < p < 1 + 4/(n 1), then there is a global solution (Z α u(t, ), t Z α u(t, )) Ḣ s Ḣ s 1, α 2, with small data and s = s c ɛ.

Sample proof in Minkowski space Iteration method Let u 1 0, u k solves { ( 2 t g )u k (t, x) = F p (u k 1 (t, x)), (t, x) R + Ω u k (0, ) = f, t u k (0, ) = g. Continuity argument. Guaranteed by the Strichartz estimates, x ( n 2 +1 γ)/p u L p t Lp r L 2 ω (f, g) (Ḣγ,Ḣγ 1 ) + x n 2 +1 γ F L 1 t L 1 r L2 ω for 1/2 1/p < γ < n/2 1/p, and energy estimates, u L t Ḣ γ x f Ḣ γ + g Ḣγ 1.

Our proof for the case p > p c Set up the argument.define the norm X : Set u(t, ) X = u L sγ ( x <R) + x ( n 2 +1 γ)/p u L p r L 2 ω( x >R) M k = ( Z α u L k t Ḣ γ (R + R n ) + t Z α u L k t Ḣ γ 1 (R + R n ) + Z α u L p t X ). GOAL: Show M k < Cɛ if Z α (f, g) ( Ḣ γ,ḣγ 1 ) < ɛ.

Proof for p > p c, continued Key Ingredients. KSS and Strichartz Estimates x 1 2 s ɛ Z α u L 2 t L 2+ x n 2 n+1 p s ɛ Z α u x L p Energy Estimates t Lp x L2+η ω ( Z α f Ḣs + Z α ) g Ḣs 1, ( ) Z α u L t Ḣ + Z α u s L t Ḣ + Z α u s 1 L p t Lqs x ( x 1) where q s = 2n/(n 2s). ( Z α f Ḣs + Z α ) g Ḣs 1, ({ x >1})

Transformation on the Equation Set P = g g g 1. We will prove the estimates if u is the solution of ( 2 + P)u = F, so that u(t) = cos(tp 1 2 )f +P 1 2 sin(tp 1 2 )g + t 0 P 1 2 sin((t s)p 1 2 )F (s)ds. Equivalence: if v solves ( 2 t g )v(t, x) = G(t, x), we have relation u = gv, F = gg.

Proof of the estimates with order 0 KSS estimates: 08 Bony, Häfner. Strichartz estimates: Interpolation between KSS estimates and angular Sobolev inequality, x n 2 α e itp1/2 f (x) eitp1/2 L t, x L 2+η f (x) ω L t Ḣ f x α Ḣ ; (1) x α Energy estimates: Equivalence of P s/2 and s with s [0, 1]; Local Energy decay (By interpolation between KSS estimates), βu L 2 f t H s Ḣs + g Ḣs 1.

KSS and Energy estimates with higher order derivatives Z α =, use relation between and P 1/2. 1 u Ḣs P s/2 u L 2 x, for s [ 1, 1]; 2 3/2 µ 1 < µ 2 µ 3 3/2, then x µ 3 l u L 2 (R d ) x µ 2 P 1/2 u n L 2 (R d ) x µ 3 l u L 2 (R d ). Z α = 2, use relation between 2 and P. 1 For s [0, 1], we have 2 x f Ḣs Pf Ḣs + f Ḣs. Pf Ḣs x α f Ḣs. 2 For 0 < µ 3/2 and k 2, we have l=1 x µ 2 x u L 2 x x µ u L 2 x + x µ Pu L 2 x.

KSS and Energy estimates with higher order derivatives (continued) When Z α = Ω or Z α = Ω 2, then Z α u solves ( t 2 + P)Z α u = [P, Z α ]u, with initial data (Z α f, Z α g). Commutator terms [P, Ω]u = r 2 α α u. [P, Ω 2 ]u = r 2 α α u. α 3 where r i C is such that x α r j (x) = O ( x ρ j α ), α,

KSS and Energy estimates with higher order derivatives (continued) Techniques to handle commutator terms Let w solve the wave equation with f = g = 0, x 1/2 s ɛ w L 2 t L 2 x x (1/2)+ɛ F L 2 t Ḣ s 1 ; w L t Ḣx s x 1/2+ɛ F L 2 t Ḣx s 1. Fractional Lebniz rule. For any s ( n/2, 0) (0, n/2), fg Ḣs f L Ḣ s,n/ s g Ḣs. For any s [0, 1], ɛ > 0 and α = N, we have x (1/2) ɛ x α u L 2 t Ḣ f s 1 ḢN+s 1 Ḣs + g ḢN+s 2. Ḣs 1 α =N

Weighted Strichartz estimates with higher order derivatives x n 2 n+1 p s ɛ Z α u L p t Lp x L2+η ω ({ x >1}) ( Z α f Ḣs + Z α g Ḣs 1) Interpolation between p = 2 and p = p = 2: KSS estimates; p = : x n 2 s Z α u L t, x L 2+η ω Z α u L t Ḣ s x ( Z α f Ḣs + Z α g Ḣs 1)

Local Energy Decay with higher order derivatives Interpolation between s = 0 and s = 1. s = 0, φz α u L 2 t,x x 1/2 ɛ x Z α 1 u L 2 t,x ( Z α u 0 Ḣ1 + Z α u 1 L 2) s = 1, α k 1 α k ( Z α u 0 L 2 + Z α ) u 1 Ḣ 1. φz α u L 2 t Ḣ 1 φ x Z α u L 2 t,x + φ Z α u L 2 t,x x 1/2 ɛ x Z α u L 2 t,x + x 3/2 ɛ Z α u L 2 t,x ( Z α u 0 Ḣ1 + Z α ) u 1 L 2. α k

Our result: Local existence part Theorem Suppose (H1) and (H2) hold with ρ > 2. Also assume 2 u i uf i (u) u p. i=1 If n = 3, 2 p < p c = 1 + 2, then there is an almost global solution (Z α u(t, ), t Z α u(t, )) Ḣ s Ḣ s 1, α 2 with almost sharp life span, T = c δ p(p 1) p 2 +ɛ 2p 1. with small data and s = s d = 1/2 1/p. Idea of Proof. The local result and life span follows if we use the local in time KSS estimates for 0 < µ < 1/2 instead of the KSS estimates for µ > 1/2.

Local in time KSS estimates For 0 < µ < 1/2, x µ Z α 1/2 µ+ɛ u L 2 T L 2 T x ( Z α f L 2 + Z α ) g Ḣ 1. Proof. Away from the origin, use the KSS estimates for small perturbation equations. (1 + T ) 2a x 1/2+a ( u + u / x ) 2 L 2 ([0,T ] R n ) T u (0, ) 2 L + (u + u/ x )( F + ( h + h x )/ u )dxdt 2 x α k 0 Near the origin, use the local energy estimates, φz α u L p ( Z α f t Ḣs Ḣs + Z α ) g Ḣs 1. α k

Further Problem Morawetz est: x 1/2 s e itd f L 2 t,x f Ḣs, 0 < s < n 1 2. Existence theorem for quasilinear wave equations on Asymptotically Euclidean manifolds, with null condition assumed. High dimension existence results for semilinear wave equation.