Remarks on decay of small solutions to systems of Klein-Gordon equations with dissipative nonlinearities

Transcription

1 Remarks on decay of small solutions to systems of Klein-Gordon equations with dissipative nonlinearities Donghyun Kim (joint work with H. Sunagawa) Department of Mathematics, Graduate School of Science Osaka University November 22, 2013 Donghyun Kim (joint work with H. Sunagawa) 1 / 34

2 Outline 1 Introduction Donghyun Kim (joint work with H. Sunagawa) 2 / 34

3 Outline 1 Introduction Donghyun Kim (joint work with H. Sunagawa) 3 / 34

4 General Setting : Nonlinear Klein-Gordon Equation We consider the following Cauchy problem: (NLKG) { ( + m 2 j )u j = F j(u, u), (t, x) (0, ) R n, j = 1,..., N, u j(0, x) = εf j(x), tu j(0, x) = εg j(x), x R n, j = 1,..., N where = t 2, m j > 0, n N, N N and u = (u j) 1 j N : [0, ) R n R N, u = ( t,xu α j) α =1,1 j N. F = (F j) 1 j N : smooth function of (u, u) and p-th order around the origin, i.e., F j(u, u) = O(( u + u ) p ) as (u, u) 0. ε > 0, f j, g j C0 (R n ). Donghyun Kim (joint work with H. Sunagawa) 4 / 34

5 Our Interests We are interested in SDGE and the large time behavior (asymptotic profile or time decay) for the solution of (NLKG). SDGE (Small Data Global Existence) holds for (NLKG) ε 0 > 0 s.t. ε (0, ε 0]! u sol. of (NLKG) in C ([0, ) R n ) The global solution u tends to a free solution (in Energy sense). u free = (u free j ) 1 j N s.t. lim N t j=1 where u free solves ( + m 2 j)u free j u j(t) u free j (t) Emj = 0 = 0 with (u free j, tu free j ) t=0 H 1 L 2 and the energy is given by u(t) 2 E m = 1 tu(t, x) 2 + xu(t, x) 2 + m 2 u(t, x) 2 dx. 2 R n Donghyun Kim (joint work with H. Sunagawa) 5 / 34

6 Outline 1 Introduction Donghyun Kim (joint work with H. Sunagawa) 6 / 34

7 Why 1-Dimensional? (NLKG) { ( + m 2 j )u j = F j(u, u), (t, x) (0, ) R n, j = 1,..., N, u j(0, x) = εf j(x), tu j(0, x) = εg j(x), Theorem (S. Klainerman, G. Ponce, J. Shatah, 80 s) x R n, j = 1,..., N (F j : p-th order around the origin) If p > 1 + 2, then SDGE holds for (NLKG) and the solution tends to the n free solution. What about p = 1 + 2, i.e., (n, p) = (1, 3), (2, 2)? n From now on, we restrict our interests only on the case of (n, p) = (1, 3). Donghyun Kim (joint work with H. Sunagawa) 7 / 34

8 : N = 1 (Single case) We consider the following single NLKG in 1 space dimension with cubic nonlienarity. ( + 1)u = F (u, tu, xu), (t, x) (0, ) R Theorem (B. Yordanov, 95) F = u 2 t u x SDGE does not hold. Theorem (K. Moriyama, 97 / S. Katayama, 99) For some F, SDGE holds and the solution u tends to the free solution. (For example, F = 3uu 2 t 3uu 2 x u 3, etc.) Donghyun Kim (joint work with H. Sunagawa) 8 / 34

9 : N = 1 (Single case) ( + 1)u = F (u, tu, xu), (t, x) (0, ) R Theorem (J. -M. Delort, 2001) F = u 3 SDGE holds and u(t, x) = 1 ( ) Re a(x/t)e i{(t2 x 2 ) 1/ (1 x/t 2 ) 1/2 + a(x/t) 2 log t} + o(t 1/2 ) t as t, uniformly in x R, where ( ) + = max{, 0}, and a(y) is a suitable C-valued smooth function of y = x/t vanishing when y 1. u decays like O(t 1/2 ) in L but does not behave like the free solution: u free (t, x) = 1 ( ) Re a(x/t)e i(t2 x 2 ) 1/2 + + o(t 1/2 ). t Donghyun Kim (joint work with H. Sunagawa) 9 / 34

10 : N = 1 (Single case) ( + 1)u = F (u, tu, xu), (t, x) (0, ) R Theorem (H. Sunagawa, 2006) F = ( tu) 3 SDGE holds and ( ) t 1/2 Re a(x/t)e i(t2 x 2 ) 1/2 + u(t, x) = + O (t 1/2 (log t) 3/2) a(x/t) 2 (1 x/t 2 ) 1 + log t as t, uniformly in x R where a(y) is a suitable smooth function of y = x/t vanishing when y 1. u decays like O(t 1/2 (log t) 1/2 ) in L Donghyun Kim (joint work with H. Sunagawa) 10 / 34

11 : N 2 We consider the following system of NLKG in 1 space dimension with cubic nonlinearities: ( + m 2 j)u j = F j(u, tu, xu), m j > 0, j = 1,..., N 2 where u = (u j) 1 j N. For example, we consider the 2-component system: (S1) { ( + m 2 1 )u 1 = F 1(u 2, tu 2, xu 2), ( + m 2 2)u 2 = F 2(u 1, tu 1, xu 1). Theorem (H. Sunagawa, 2003) If (m 1 3m 2)(m 1 m 2)(3m 1 m 2) 0 then SDGE holds for (S1) and the solution tends to the free solution. Donghyun Kim (joint work with H. Sunagawa) 11 / 34

12 : N 2 Now we consider the following system of NLKG: { ( + m 2 1 )u 1 = αu 4 2, (S2) ( + m 2 2)u 2 = βu 3 1 where m 1 m 2, α, β R. Theorem (H. Sunagawa, 2005) u 1(t, x) = 1 Re m 1 t u 2(t, x) = 1 m 2 t Re ( a(x/t)e im 1(t 2 x 2 ) 1/2 + ) + O(t 1+δ ), ( (A(x/t)log t + b(x/t))e im 2(t 2 x 2 ) 1/2 + ) + O(t 1+δ ) where δ > 0 small and A(x/t) does not vanish if m 2 = m 1 or m 2 = 3m 1. Here a(y), b(y) are smooth functions which vanish when y 1. u 2 decays no faster than O(t 1/2 log t) in L Donghyun Kim (joint work with H. Sunagawa) 12 / 34

13 : N 2 One more example : ( + 1)U = U 2 U where U is complex-valued in this case. In view of U = u 1 + iu 2, we have { ( + 1)u1 = (u u 2 2)u 1, (S3) ( + 1)u 2 = (u u 2 2)u 2. The complex version of the result by J. -M. Delort(2001), was given by Theorem (H. Sunagawa, 2005) SDGE holds for (S3) and the solution u = (u 1, u 2) satisfies u(t, x) Cε(1 + t) 1/2 for all t 0. Remark) Delort s approach does not work for C-valued case. u decays like O(t 1/2 ) in L in this case. Donghyun Kim (joint work with H. Sunagawa) 13 / 34

14 Our Goal Like this, for the mass resonance case, we cannot expect even O(t 1/2 ) (decay rate of the free solution) of the solution in general. But for some F, we may expect the decay rate O(t 1/2 ) and even o(t 1/2 ) for the solution. We give a structural condition of the cubic nonlinearity F j under which the solution to (NLKG): ( + 1)u j = F j(u, tu, xu), j = 1,..., N admits a global solution and it decays like ( O t 1/2 (log t) 1/2) in L if ε is small enough. Donghyun Kim (joint work with H. Sunagawa) 14 / 34

15 Outline 1 Introduction Donghyun Kim (joint work with H. Sunagawa) 15 / 34

16 Notations Cubic homogeneous part of the nonlinearity Fj cub (ξ, η, ζ) = lim r 3 F j(rξ, rη, rζ), r 0 (ξ, η, ζ) R 3N Upper branch of a hyperbola H + = {ω = (ω 0, ω 1) R 2 : ω 0 > 0, ω 2 0 ω 2 1 = 1} Φ = (Φ j) 1 j N : C N H + C N, Φ j(y, ω) = 1 2πi λ =1 for Y C N and ω H +. Fj cub (Re(Y λ), ω 0 Im(Y λ), ω 1 Im(Y λ)) dλ λ 2 Y, Z C N = N j=1 YjZj, Y = Y, Y C N Donghyun Kim (joint work with H. Sunagawa) 16 / 34

17 Theorem 1 (Small Data Global Existence) Theorem (Small Data Global Existence) Assume there exists a N N positive Hermitian matrix A such that Im Φ(Y, ω), AY C N 0 for all (Y, ω) C N H +. Then there exists ε 0 > 0 such that if ε (0, ε 0], the Cauchy problem (NLKG) admits a unique global classical solution. Moreover, it satisfies t,xu(t, I ) L C(1 + t) 1/2 for all t 0. I 1 Donghyun Kim (joint work with H. Sunagawa) 17 / 34

18 Theorem 2 (Decay Estimates I) Theorem (Decay Estimates I) Assume there exist a N N positive Hermitian matrix A and a constant C > 0 such that Im Φ(Y, ω), AY C N C ω 0 Y 4 for all (Y, ω) C N H +. Then the global solution of (NLKG) satisfies (1 + t) 1/2 u(t, ) L C log(2 + t) for all t 0. Donghyun Kim (joint work with H. Sunagawa) 18 / 34

19 Theorem 3 (Decay Estimates II) Theorem (Decay Estimates II) Assume there exist a N N positive Hermitian matrix A and a constant C > 0 such that Im Φ(Y, ω), AY C N C ω0 Y 3 4 for all (Y, ω) C N H +. Then the global solution of (NLKG) satisfies t,xu(t, I (1 + t) 1/2 ) L C log(2 + t) I 1 for all t 0. Remark) Using finite propagation speed, we obtain L p decay for p [2, ]. Donghyun Kim (joint work with H. Sunagawa) 19 / 34

20 Example 1 (Nonlinear Dissipation) ( + 1)U = µ 1 U 2 U µ 2 tu 2 tu, µ 1 R, µ 2 > 0 (U : Complex-valued) (U = u 1 + iu 2) { ( + 1)u1 = µ 1(u u 2 2)u 1 µ 2{( tu 1) 2 + ( tu 2) 2 } tu 1, ( + 1)u 2 = µ 1(u u 2 2)u 2 µ 2{( tu 1) 2 + ( tu 2) 2 } tu 2 Φ 1(Y, ω) = µ1 ( ) iµ2ω3 0 3 Y 1 2 Y Y 2 2 Y 1 + Y2 2 Y 1 8 Φ 2(Y, ω) = µ1 iµ2ω3 0 8 ( 3 Y 2 2 Y Y 1 2 Y 2 + Y 2 1 Y 2 ) A = diag(1, 1) Im Φ(Y, ω), AY C 2 = µ2ω3 ( 0 2 Y Y Y 1 2 Y Y1 2 + Y2 2 2) 8 Cω0 Y 3 4 Donghyun Kim (joint work with H. Sunagawa) 20 / 34

21 Example 1 : ( + 1)U = µ 1 U 2 U µ 2 t U 2 t U Theorem 3 (Decay Estimates II) t,xu(t, I (1 + t) 1/2 ) L C log(2 + t) I 1 Remark) Using finite propagation speed, we also obtain L p decay: t,xu(t, I (1 + t) (1/2 1/p) ) L p C, p [2, ] log(2 + t) I 1 which yields the energy of the solution U(t, ) E decays like O((log t) 1/2 ) where U(t, ) E = ( 1/2 1 tu(t, x) 2 + xu(t, x) 2 + U(t, x) dx) 2. 2 R Donghyun Kim (joint work with H. Sunagawa) 21 / 34

22 Example 2 : ( + 1)U = U 2 t U ( + 1)U = U 2 tu (U : Complex-valued) (U = u 1 + iu 2) { ( + 1)u1 = (u u 2 2) tu 1, ( Φ 1(Y, ω) = iω0 8 Φ 2(Y, ω) = iω0 8 ( + 1)u 2 = (u u 2 2) tu 2 ) Y 1 2 Y Y 2 2 Y 1 Y2 2 Y 1 ( ) Y 2 2 Y Y 1 2 Y 2 Y1 2 Y 2 Donghyun Kim (joint work with H. Sunagawa) 22 / 34

23 Example 2 : ( + 1)U = U 2 t U A = diag(1, 1) Im Φ(Y, ω), AY C 2 = ω0 ( 4 Y1 2 Y Y1 2 Y2 2 2) 8 Cω 0 Y 4 Theorem 2 (Decay Estimates I) (1 + t) 1/2 U(t, ) L C log(2 + t) Donghyun Kim (joint work with H. Sunagawa) 23 / 34

24 Outline 1 Introduction Donghyun Kim (joint work with H. Sunagawa) 24 / 34

25 Step 1 : Reduction of the Problem (i) Take B > 0 : supp f j supp g j {x R : x B} and let τ 0 > 1 + 2B. Fact : We may treat data on {(t, x) R 2 : (t + 2B) 2 x 2 = τ 2 0, t > 0} (ii) Change of variables : (t, x) (τ, z) by t + 2B = τω 0(z), x = τω 1(z) for x < t + 2B where ω(z) = (ω 0(z), ω 1(z)) H +. (iii) Define v(τ, z) = (v j(τ, z)) 1 j N by u j(t, x) = χ(z) τ v j(τ, z) where χ(z) Ce 3 z. J. -M. Delort, Existence globale et comportement asymptotique pour l équation de Klein - Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm (2001). Donghyun Kim (joint work with H. Sunagawa) 25 / 34

26 Step 1 : Reduction of the Problem According to (i) (iii), (NLKG) is reduced to (NLKG*) where L = 2 τ 1 τ 2 finite { Lvj = G j(τ, z, v, τ v, zv), τ > τ 0, z R, j = 1,, N, (v j, τ v j) τ=τ0 = (ε f j, ε g j), z R, j = 1,, N ( z χ (z) χ(z) z + χ (z) χ(z) 1 ) + 1, 4 G j = χ(z)2 Fj cub (v, ω 0 (z) τ v, ω 1 (z) τ v) + Q j (τ, z, v, τ v, zv), τ Q j = ( ) (τ 2 or τ 3 or τ 4 ) (bounded functions of z) P 3 (v, τ v, zv), P 3 : a homogeneous polynomial of order 3 consisting of (v, τ v, zv), f j, g j : sufficiently smooth functions of z with compact support. Donghyun Kim (joint work with H. Sunagawa) 26 / 34

27 Step 2 : A Priori Estimate First, we set M(T ) = for the solution v to (NLKG*). sup ( v(τ, z) + τ v(τ, z) + 1τ ) zv(τ, z) (τ,z) [τ 0,T ) R Lemma (A Priori Estimate) Under the assumption of Theorem 1, ε 1 > 0, C > 0 s.t. M(T ) ε implies M(T ) Cε for ε (0, ε 1]. Here C is independent of T, ε. Remark) Once this lemma is proved, SDGE follows immediately by the standard continuity arguments. Donghyun Kim (joint work with H. Sunagawa) 27 / 34

28 M(T ) = sup ( v(τ, z) + τ v(τ, z) + 1τ ) zv(τ, z) (τ,z) [τ 0,T ) R Proof of Lemma (i) We introduce the C N -valued function α = (α j) 1 j N by ( α j(τ, z) = e iτ ) v j(τ, z), i τ (τ, z) [τ 0, T ) R for the solution v(τ, z) to (NLKG*). In view of the relations α(τ, z) 2 = v(τ, z) 2 + τ v(τ, z) 2 1 zv(τ, z) Cε, τ it suffices to show that sup α(τ, z) Cε. (τ,z) [τ 0,T ) R Donghyun Kim (joint work with H. Sunagawa) 28 / 34

29 (ii) We note that α j τ = ie iτ χ(z) 2 τ F cub j (v, ω 0(z) τ v, ω 1(z) τ v) + 1 Rj(τ, z) τ 2 Cετ 1/4 = iχ(z)2 Rj(τ, z) Φ j(α, ω(z)) + S j(τ, z) + τ τ 2 where the resonant term is the form of S j(τ, z) = ( ) e ibt (bdd. funct. of z) cubic terms of (α 1, α 1,, α N, α N ) τ finite with b R \ {0}. Donghyun Kim (joint work with H. Sunagawa) 29 / 34

30 (iii) Now we put ν A(Y ) = Y, AY C N for Y C N so that Y, AZ C N ν A(Y )ν A(Z), c 1 Y ν A(Y ) c 2 Y, c j > 0 for Y, Z C N. Then we get τ ( νa(α(τ, z)) 2) = 2 Re τ α(τ, z), Aα(τ, z) C N = 2χ(z)2 Im Φ(α, ω(z)), Aα τ C N 0 2 Re S, Aα C N + 1 τ 2 νa(α)2 + Cε2 τ 3/2 which yields ν A(α(τ, z)) 2 τ Cε Re S, Aα C N dσ τ 0 Cε Re S, Aα + 2 Re R, Aα τ 2 τ + ν A(α(σ, z)) 2 dσ τ 0 σ 2 τ Cε 2 + ν A(α(σ, z)) 2 dσ α(τ, z) Cε τ 0 σ 2 Donghyun Kim (joint work with H. Sunagawa) 30 / 34

31 Step 3 : Proof of the Decay Estimates (i) Proof of Theorem 2 From Step 2, we have α j τ = iχ(z)2 Rj(τ, z) Φ j(α, ω(z)) + S j(τ, z) +. τ τ 2 β j = α j V j ( V j(τ, z) Cε 3 τ 1 ) β j τ = iχ(z)2 Φ j(β, ω(z)) + ρ j(τ, z) τ with ρ j(τ, z) Cετ η, 1 < η < 2. Donghyun Kim (joint work with H. Sunagawa) 31 / 34

32 Now we put Ψ(τ) = ν A(β(τ, z)) 2. By the direct calculations, we get d dτ ( (log(τω0(z))) 2 Ψ(τ) ) = (log(τω 0(z))) 2 dψ dτ 2 log(τω0(z)) (τ) + Ψ(τ) τ C τχ(z) 2 ω + Cε2 (log(τω 0(z))) 2. 0(z) τ η (log(τω 0(z))) 2 Ψ(τ) C log(τω0(z)) χ(z) 2 ω 0(z) + Cε 2 ω 0(z) β(τ, z) CΨ(τ) 1/2 C χ(z)2 ω 0(z) log(τω 0(z)) + Cεω0(z)1/2 log(τω 0(z)) Donghyun Kim (joint work with H. Sunagawa) 32 / 34

33 α j = β j + V j α(τ, z) χ(z)ω 0(z) 1/2 C log(τω0(z)) We remember that our change of variable is u j(t, x) = χ(z) τ Re(α j(τ, z)e iτ ) with t + 2B = τω 0(z), x = τω 1(z) for x < t + 2B. Finally we get u(t, x) α(τ, z) χ(z)ω0(z)1/2 τω0(z) C τω0(z) log(τω 0(z)) C(1 + t) 1/2. log(2 + t) Donghyun Kim (joint work with H. Sunagawa) 33 / 34

34 (ii) Proof of Theorem 3 We remember that the assumption of Theorem 3 is Im Φ(β(τ, z), ω(z)), Aβ(τ, z) C N C ω 0(z) 3 β(τ, z) 4. α(τ, z) χ(z)ω 0(z) 3/2 C log(τω0(z)) Noting that we finally obtain tu j(t, x) = χ(z)ω0(z)3/2 τω0(z) Im(α je iτ ) + (Remainder), tu(t, x) α(τ, z) χ(z)ω0(z)3/2 τω0(z) C(1 + t) 1/2 log(2 + t). Donghyun Kim (joint work with H. Sunagawa) 34 / 34