Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, II : microlocal analysis

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ScienceDirect

J. Differential Equations 269 (2020) 7881–7905

www.elsevier.com/locate/jde

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, II: Microlocal analysis

^✩

Claudia Garetto

^a,^∗

, Christian Jäh

^b

, Michael Ruzhansky

^c,d

aDepartmentofMathematicalSciences,LoughboroughUniversity,Loughborough,Leicestershire,LE113TU, UnitedKingdom

bMathematischesInstitut,Georg-AugustUniversitätGöttingen,Bunsenstraße3-5,D-37037Göttingen,Germany cDepartmentofMathematics:Analysis,LogicandDiscreteMathematics,GhentUniversity,Belgium dSchoolofMathematicalSciences,QueenMaryUniversityLondon,MileEnd,London,E14NS,UnitedKingdom

Received 2April2020;accepted 30May2020

Abstract

Inthispaperwecontinuethestudyofnon-diagonalisablehyperbolicsystemswithvariablemultiplicity startedbytheauthorsin[1].Inthecaseofspacedependentcoefficients,weprovearepresentationformula forsolutionsthatallowsustoderiveresultsofregularityandpropagationofsingularities.

MSC:primary35L45;secondary46E35

Keywords:Hyperbolicsystems;Fourierintegraloperators;Microlocalanalysis;Propagationofsingularities

Contents

1. Introduction . . . 7882 1.1. Notationsandpreliminarynotions . . . 7883 1.2. AssumptionsonthematricesA(x,Dx)andB(t,x,Dx) . . . 7884

✩ TheauthorsweresupportedinpartsbytheFWOOdysseusProjectG.0H94.18N:AnalysisandPartialDifferential Equations,theLeverhulmeTrustGrantRPG-2017-151andbyEPSRCGrantEP/R003025/1.

* Correspondingauthor.

E-mailaddresses:[email protected](C. Garetto),[email protected](C. Jäh), [email protected],[email protected](M. Ruzhansky).

https://doi.org/10.1016/j.jde.2020.05.038

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2. Auxiliaryresults . . . 7885

2.1. CompositionofFIOsandregularisingeffect . . . 7886

3. Solutionrepresentations . . . 7888

3.1. The2×2 case . . . 7888

3.2. InversionoftheoperatorL₁. . . 7889

3.3. Representationformulas . . . 7890

3.4. Them×mcase . . . 7891

3.5. Regularityresults . . . 7894

4. Propagationofsingularities . . . 7895

5. Application:higherorderhyperbolicequations . . . 7897

5.1. Well-posednessinSobolevspaces . . . 7898

5.2. Representationformulaforthesolution . . . 7902

References . . . 7905

1. Introduction In this work, we continue the study of non-diagonalisable systems that begun in [1] by proving a result on solution representations and propagation of singularities for a hyperbolic system with x-dependent principal part. Let us consider D_tu=A(x, D_x)u+B(t, x, D_x)u+f (t, x), (t, x)∈ [0, T] ×Rⁿ, u|t=0=u0, x∈Rⁿ, (1.1) with the usual notation Dt= −i∂tand Dx= −i∂x. We assume that A(x, D_x) = a_ij(x, D_x)m i,j=1 is an m ×mmatrix of pseudo-differential operators of order 1, i.e., a_ij ∈¹_1,0(Rⁿ))and that B(t, x, Dx) = bij(t, x, Dx)m i,j=1is an m ×mmatrix of pseudo-differential operators of order 0, i.e., b_ij ∈C([0, T], ⁰_1,0(Rⁿ)). We also assume that the matrix A is upper triangular and hyperbolic, i.e., A(x, D_x)=(x, D_x)+N (x, D_x)=diag(λ1(x, D_x), λ₂(x, D_x), . . . , λ_m(x, D_x))+N (x, D_x) with real eigenvalues λ1(x, ξ ), λ2(x, ξ ), . . . , λm(x, ξ )and N (x, Dx)= ⎛ ⎜⎜ ⎜⎜ ⎜⎝ 0 a12(x, Dx) a13(x, Dx) · · · a1m(x, Dx) 0 0 a23(x, Dx) · · · a2m(x, Dx) ... ... ... · · · ... 0 0 0 . . . a_m₋_1m(x, D_x) 0 0 0 . . . 0

⎞

⎟⎟

⎟⎠ .

We recall that the well-posedness of this kind of systems has been proven in anisotropic Sobolev spaces in [1] under specific assumptions on the lower order terms.

Propagation of singularities for systems with vanishing iterated Poisson brackets has been studied by several authors as Iwasaki and Morimoto [6] who studied 3 ×3 systems where the twice iterated Poisson bracket vanishes and Ichinose [5] studied 2 ×2 systems under the

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same condition. In [8], Rozenblum considered smoothly diagonalisable systems with transver- sally intersecting characteristics, and derived a formula for the propagation of its singularities.

Consequently, the transversality condition was removed in [7], replaced by a weaker condition of intersection of finite order at points of multiplicity, with propagation of singularities result as well. Here we extend the results of [7] to non-diagonalisable hyperbolic systems with variable multiplicity. The paper is organised as follows. Section2 collects some important notions of Fourier integral operators and related integral operators relevant to our problem. The main well-posed result and corresponding representation formula is proven in Section3. Section4is devoted to propagation of singularities. The paper ends in Section5with an application of our results to higher order hyperbolic equations with multiplicities.

1.1. Notations and preliminary notions

For the convenience of the reader we recall here some notations and preliminary notions that we will use throughout the paper.

Let μ ∈R. We recall that S_1,0^μ (Rⁿ)is the space of symbols of order μand type (1, 0), i.e., a=a(x, ξ ) ∈C^∞(Rⁿ×Rⁿ)belongs to S_1,0^μ (Rⁿ)if there exist constants Cα,β>0 such that

∀α, β∈N₀ⁿ : |∂_x^α∂_ξ^βa(x, ξ )| ≤C_α,βξ^m^−|^β^| ∀(x, ξ )∈Rⁿ×Rⁿ,

with ξ =(1 + |ξ|²)^1/2.

The set of pseudo-differential operators associated to the symbols in S_1,0^μ (Rⁿ)is denoted by ^μ_1,0(Rⁿ). If the symbol has an extra (continuous) dependence on t∈ [0, T]we will use the notations C([0, T], S^μ)and C([0, T], ^μ_1,0)for symbols and operators, respectively. For the sake of simplicity we will adopt the abbreviated notations S^μand ^μfor S_1,0^μ (Rⁿ)and ^μ_1,0(Rⁿ), respectively, and CS^μand C^mfor C([0, T], S^μ)and C([0, T], ^μ_1,0), respectively.

With I^μwe denote the class of Fourier Integral Operators with amplitude in S^μ, i.e., of operators of the type

I_ϕ(a)(f )(t, x)=

Rⁿ

e^{iϕ(t,x,ξ )}a(t, x, ξ )f (ξ )dξ,

where ϕ is a phase function and fˆis the Fourier transform of f. This notation is standard and further details can be found in [1] and the references therein, for instance [3]. In this paper we will use the short expression integrated Fourier integral operatorto denote an operator of the type

t 0

Rⁿ

eiϕ(t,s,x,ξ )a(t, s, x, ξ )f (ξ, s)dξ ds,

where the Fourier transform of f=f (y, s)is meant with respect to the variable y.

By L^pα(Rⁿ), we denote the Sobolev space (I−)⁻^α²L^p(Rⁿ). As usual, we set H^s=L²_s. By · _L^p

loc(Rⁿ) we denote any localisation of the L^p(Rⁿ)-norm, i.e. the estimate f _L^p

loc(Rⁿ)≤C means that for all χ∈C₀^∞(Rⁿ), we have the estimate χf _Lp(Rⁿ)≤C, where the constant C

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may depend on χ. Since we only work in Rⁿand no confusion can arise, we drop the indication of Rⁿfrom here on.

Finally, we recall that the Poisson bracket of two differentiable functions f =f (x, ξ )and g=g(x, ξ )is defined as

{f , g} = n i=1

∂f

∂ξ_i

∂g

∂x_i − ∂f

∂x_i

∂g

∂ξ_i.

1.2. Assumptions on the matrices A(x, D_x)and B(t, x, Dx)

In this paper, we make the following assumptions on lower order terms and multiplicities:

(H1) (Lower order terms) The entries of the matrix B(t, x, D_x) = [b_ij(t, x, D))]^m_i,j₌₁belong to C([0, T], ⁰)and are of decreasing order below the diagonal, i.e.,

bij∈C([0, T], ^j⁻ⁱ) fori > j . (1.2) (H2) (Multiplicities) There exists M∈N such that if λj(x, ξ ) =λk(x, ξ ) for some j, k∈ {1, . . . , m}and λ_j(x, ξ )and λk(x, ξ )are not identically equal near (x, ξ )then there exists some N≤Msuch that

λ_j(x, ξ )=λ_k(x, ξ ) ⇒ H_λ^N

j(λ_k):= {λ_j,{λ_j, . . . ,{λ_j, λ_k}}. . .} =0, where the Poisson bracket {·, ·}in H_λ^N

j is iterated Ntimes.

Remark 1.1. In [1], for Adepending on tas well and under the condition (H1) on B, we proved that for any s∈R, u⁰_k ∈H^s⁺^k⁻¹, k=1, . . . , m, and fk ∈C([0, T], H^s⁺^k⁻¹), k=1, . . . , m, the Cauchy problem (1.1) has a unique anisotropic Sobolev solution uwith components uk∈ C

[0, T], H^s⁺^k⁻¹

, k=1, . . . , m. In this paper we do the microlocal analysis of solutions in the case of Adepending only on x.

Remark 1.2. The condition (H2) was introduced in [7, p.3]. For 1 < p <∞ and α=(n − 1)|1/p−1/2|, it was proved in [7] that when the matrix A(x, D_x)is smoothly microlocally diagonalisable, with smooth eigenspaces and real eigenvalues λ_j, j =1, . . . , m, fulfilling the condition (H2) then for every compactly supported initial data u0∈L^pα ∩L²_comp the Cauchy problem (1.1) has a unique solution usuch that u(t, ·) ∈L^p_locfor every t∈ [0, T]. Previously, a similar result had been proved in L²by Rozenblum in [8], in the special case of (H2) with N=1.

We are now ready to state the main result of our paper. This is a representation formula for the solution uwhich shows how this depends on initial data and right-hand side. This dependence is given in terms of integral operators (of Fourier type) modulo regularising operators of order N, i.e. mapping H^s into H^s⁺^N, for any s∈R.

Theorem 1.3. Let n ≥1, m ≥2, and let

Dtu=A(x, Dx)u+B(t, x, Dx)u+f (t, x), (t, x)∈ [0, T] ×Rⁿ, u|t=0=u0(x), x∈Rⁿ,

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where A(x, D_x)is an upper-triangular matrix of pseudo-differential operators of order 1and B(t, x, D_x) is a matrix of pseudo-differential operators of order 0, continuous with respect to t. Let u₀ and f have components u⁰_j and f_j, respectively, with u⁰_j ∈H^s⁺^j⁻¹(Rⁿ) and f_j ∈C([0, T], H^s⁺^j⁻¹)for j =1, . . . , m. Then, under condition (H1) and (H2), we have the following:

(i) the Cauchy problem above has a unique anisotropic Sobolev solution u, i.e., uj ∈ C([0, T], H^s⁺^j⁻¹)for j=1, . . . , m;

(ii) for any N∈N, the components uj, j =1, . . . , m, of the solution uare given by

uj(t, x)= m l=1

H^l_j,l⁻^j(t )+Rj,l(t )

u⁰_l +

K^l_j,l⁻^j(t )+Sj,l(t )

fl, (1.3)

where R_j,l, S_j,l ∈ L(H^s, C([0, T], H^s⁺^N⁻^l⁺^j)) and the operators H^l_j,l⁻^j, K_j,l^l⁻^j ∈ L(C([0, T], H^s), C([0, T], H^s⁻^l⁺^j)) are integrated Fourier Integral Operators of order l−j.

For the convenience of the reader we recall here Theorem 1 in [1] which proves already assertion (i) in Theorem1.3.

Theorem 1.4 (Theorem 1 in [1]). Let n ≥1, m ≥2, and let

D_tu=A(t, x, D_x)u+B(t, x, D_x)u+f (t, x), (t, x)∈ [0, T] ×Rⁿ,

u|t=0=u₀(x), x∈Rⁿ, (1.4)

where A(t, x, D_x) =(a_ij(t, x, D_x))^m_i,j₌₁ is an upper-triangular matrix of pseudo-differential operators of order 1 and B(t, x, Dx) =(bij(t, x, Dx))^m_i,j₌₁ is a matrix of pseudo-differential operators of order 0, continuous with respect to t. Assume that (H1) holds. Then, given u⁰_j∈ H^s⁺^j⁻¹(Rⁿ)and f_j ∈C([0, T], H^s⁺^j⁻¹)for j =1, . . . , m, the Cauchy problem (1.4)has a unique anisotropic Sobolev solution u, i.e., uj∈C([0, T], H^s⁺^j⁻¹)for j =1, . . . , m.

2. Auxiliary results

This section contains some auxiliary results on Fourier integral operators and related integral operators that we will use throughout the paper. For the convenience of the reader, we begin by recalling some notations introduced in [1].

For each eigenvalue λj(x, ξ )of A(x, ξ ), we will be denoting by G⁰_jθthe solution to Dtw=λj(x, Dx)w+bjj(t, x, Dx)w,

w(0, x)=θ (x), and by G_jgthe solution to

Dtw=λj(x, Dx)w+bjj(t, x, Dx)w+g(t, x), w(0, x)=0.

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The bjj are the diagonal elements of the lower order term B(t, x, D_x)in (1.1). The operators G⁰_j and Gj can be locally represented by a Fourier integral operator and an integrated Fourier integral operator, respectively, i.e.,

G⁰_jθ (t, x)=

Rⁿ

e^iϕ^j^{(t,x,ξ )}c_j(t, x, ξ )θ (ξ )dξ (2.1)

and

G_jg(t, x)= t 0

Rⁿ

e^iϕ^j^{(t,s,x,ξ )}C_j(t, s, x, ξ )g(s, ξ )dξ ds, (2.2)

with ϕj(t, s, x, ξ )solving the eikonal equation

∂tϕj=λj(x,∇xϕj(t, s, x, ξ )), ϕj(s, s, x, ξ )=x·ξ,

and ϕ_j(t, x, ξ ) =ϕ_j(t, 0, x, ξ ). Note that the amplitudes C_j in (2.2) have asymptotic expan- sions ^+∞

k=0

Cj,−k where the element Cj,−k(s, x, ξ )is of order −k, k∈N, and satisfies transport equations with initial data at t =s. By construction, cj(t, x, ξ ) =C_j(t, 0, x, ξ ). In the above construction of propagators for hyperbolic equations, we have cj∈S⁰, so that G⁰_j∈I⁰.

Further, to simplify the analysis of the regularising part in (1.3) we introduce the notation

Ej(t, s)g(s, x)=

Rⁿ

e^iϕ^j^{(t,s,x,ξ )}C_j(s, x, ξ )g(s, ξ )dξ, (2.3)

i.e., the integrated Fourier integral operator Gj can now be written as

G_jg(t, x)= t 0

Ej(t, s)g(s, x)ds. (2.4)

2.1. Composition of FIOs and regularising effect

In this section, we state and prove auxiliary results that are crucial to the proof of the solution representation formula stated in Theorem1.3. In particular, we investigate the mapping properties of compositions and powers of Fourier integral operators and integrated Fourier integral operators as in (2.1) and (2.2). This will be useful when analysing the regularising part of our representation formula (1.3).

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2.1.1. Integrated Fourier integral operators

The composition of integrated Fourier integral operators like in (2.2) was studied in [7]. Their result, that we recall in the sequel, is crucial for our proof and is a generalisation of a previous result by Rozenblum in [8].

Let t1, t₂, . . . , t_l∈ [0, T], t=(t₁, t₂, . . . , t_l)and let H (t)be the operator

H (t )=eîλ^j¹^t¹eîλ^j²^(t²⁻^t¹⁾·. . .·eîλ^jl^(t^l⁻^t^l⁻¹⁾e⁻îλ^jl^t^l, (2.5) where λi are pseudo-differential operators of order 1.

By [3], H (t )is a (parameter dependent) Fourier integral operator and its canonical relation ^t⊆T^∗Rⁿ×T^∗Rⁿis given by

^t=

(x, p, y, ξ ): (x, p)=^t(y, ξ )

,

where

^t=^t_j¹

1◦ · · · ◦^t_j^l⁻^t^l−1

l ◦⁻_j^t^l

l+1

and the ^t_j are the transformations corresponding to a shift by t along the trajectories of the Hamiltonian flow defined by the λj.

Theorem 2.1 (Thm 2.1 in [7]). With the above notation, assume that not all λis are identical to each other and let (H2) be satisfied for the λj in (2.5). Further, suppose that D(t) ∈⁰. Then, the operator

Ql= t 0

t₁

0

. . .

t_l₋₁

0

D(t )H (t ) dtl. . . dt1

belongs to L(H^s, H^s⁺^{N (l)}), where N (l) → +∞as l→ +∞.

Remark 2.2. If the global estimate in the definition of the symbols classes S_1,0^m in [1] is replaced with an estimate that holds locally on every compact set, then the conclusions of Theorem2.1 hold true if L(H^s, H^s⁺^{N (l)})is replaced by L(H_comp^s , H_loc^s⁺^{N (l)}).

Theorem2.1allows us to investigate the composition GiGj. 2.1.2. The composition GiG_j

Let Giand Gj be two operators as in (2.2). Then, we can write

G_iG_ju= t 0

t1

0

Ei(t, t₁)Ej(t₁, t₂)u dt₂dt₁,

with Ei, Ej as in (2.4). If we now iterate this ktimes, we obtain

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(G_iG_j)^k= t 0

t₁

0

· · ·

t2k−1

0

Ei(t, t₁)Ej(t₁, t₂)·. . .·Ei(t_2k₋₃, t_2k₋₂)Ej(t_2k₋₂, t_2k₋₁) dt,

where t =dt1. . . dt2k−1. This is an operator of the same type as Ql above so we can apply Theorem2.1and obtain the following corollary.

Corollary 2.3. Let conditions (H1) and (H2) be satisfied. Let j∈ {1, . . . , m}, s∈R. Then, for all N∈N, k∈ {1, . . . , m}, there exists M∈Nsuch that

_k

i=1

G_{σ (i)} M

∈L

C([0, T], H^s), C([0, T], H^s⁺^N)

,

where σ is an element of the symmetric group over {1, . . . , m}.

Remark 2.4. The same conclusion as in Corollary2.3holds true if we have a product that contains a collection G⁰_j’s as long as there is at least one integrated version Gjpresent.

3. Solution representations

This section is devoted to the proof of Theorem1.3. For the sake of simplicity and for the advantage of the reader we give first a detailed explanatory proof for 2 ×2 systems and we then pass to consider the m ×mcase. We adopt the notations introduced in Section2.

3.1. The 2 ×2case

Let us consider the system

D_tu=A(x, D_x)u+B(t, x, D_x)u+f (t, x), (t, x)∈ [0, T] ×Rⁿ,

u|t=0=u₀, x∈Rⁿ, (3.1)

where u0(x) =

u⁰₁(x), u⁰₂(x)T

, f (t, x) =

f1(t, x), f2(t, x)T

, and with the operators A(x, Dx) and B(t, x, Dx)given by

A(x, D_x)=

λ₁(x, D_x) a₁₂(x, D_x) 0 λ₂(x, D_x)

(3.2) and

B(t, x, D_x)=

b₁₁(t, x, D_x) b₁₂(t, x, D_x) b₂₁(t, x, D_x) b₂₂(t, x, D_x)

.

We suppose that all entries of A(x, D_x)belong to ¹_1,0and all entries of B(t, x, D_x)belong to C⁰_1,0.

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As detailed in Subsection 2.2 in [1], we obtain the equations

u1=G⁰₁(u⁰₁)+G1(f1)+G1(a12u2)+G1(b12u2)

=G⁰₁(u⁰₁)+G₁(f₁)+G₁((a₁₂+b₁₂)u₂), u₂=G⁰₂(u⁰₂)+G₂(f₂)+G₂(b₂₁u₁), and with that

u₁=G⁰₁(u⁰₁)+G₁((a₁₂+b₁₂)G⁰₂(u⁰₂))+G₁(f₁)+G₁((a₁₂+b₁₂)G₂(f₂)) +G₁((a₁₂+b₁₂)G₂(b₂₁u₁)),

u₂=G⁰₂(u⁰₂)+G₂(b₂₁G⁰₁(u¹₀))+G₂(b₂₁G₁(f₁))+G₂(f₂) +G₂(b₂₁G₁((a₁₂+b₁₂)u₂)).

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We note that the operators G1◦(a₁₂+b₁₂) ◦G₂◦b₂₁and G2◦b₂₁◦G₁◦(a₁₂+b₁₂)are of order 0 under the assumption (H1) made on the lower order terms; here in particular b21∈⁻¹. From (3.3), we have

u1−G1((a12+b12)G2(b21u1))

=G⁰₁(u⁰₁)+G1((a12+b12)G⁰₂(u⁰₂))+G1(f1)+G1((a12+b12)G2(f2))

and

u2−G2(b21G1((a12+b12)u2))

=G⁰₂(u⁰₂)+G2(b21G⁰₁(u¹₀))+G2(b21G1(f1))+G2(f2). (3.4) 3.2. Inversion of the operator L1

Adopting the notations introduced in [1] we introduce the operator L₁:=I−G₁◦(a₁₂+b₁₂)◦G₂◦b₂₁=I−G₁⁰

Note that G₁⁰=G₁◦(a₁₂+b₁₂) ◦G₂◦b₂₁is of order 0 and from the Sobolev mapping properties of Fourier Integral Operators (see Lemma 1 in [1]) the norm of this operator can be estimated by a constant times the length of the time interval [0, T]. So it can be made as small as wanted by a suitable choice of T. It follows that L1is invertible for T small enough and its inverse can be written as sum of a Neumann series. More precisely, under the assumptions (H1) and (H2) from Corollary2.3we get that for every N∈Nthe operator L⁻¹can be written as a finite sum of powers of the operator G₁⁰ modulo some regularising operator mapping C([0, T], H^s)into C([0, T], H^s⁺^N)), i.e., for every N∈N, there exists M∈Nsuch that

L⁻₁¹=

+∞

k=0

G₁⁰k

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= M k=0

G1⁰

k

modL(C([0, T], H^s), C([0, T], H^s⁺^N)).

It is important to remark here that the estimates needed to ensure the small norm of the operator G₁⁰, do not depend on the initial data and therefore one can repeat the same argument covering the original interval [0, T].

3.3. Representation formulas

We now apply the operator L⁻₁¹as written above to both sides of the equality L1u1=G⁰₁(u⁰₁)+G1((a12+b12)G⁰₂(u⁰₂))+G1(f1)+G1((a12+b12)G2(f2)).

We obtain the following representation for u1, where

R1=L⁻₁¹− M k=0

G⁰₁_k

is a regularising operator, i.e., R1∈L(C([0, T], H^s), C([0, T], H^s⁺^N)):

u₁= M k=0

G₁⁰k

G⁰₁(u⁰₁)

=H¹⁻¹1,1u⁰₁

+ M k=0

G₁⁰k

G₁((a₁₂+b₁₂)G⁰₂(u⁰₂))

=H1,2²⁻¹u⁰₂

+ M k=0

G₁⁰k

G₁(f₁)

=K1,1¹⁻¹f₁

+ M k=0

G₁⁰k

G₁((a₁₂+b₁₂)G₂(f₂))

=K²1,2⁻¹f₂

+R₁G⁰₁(u⁰₁)+R₁G₁((a₁₂+b₁₂)G⁰₂(u⁰₂)) +R₁G₁(f₁)+R₁G₁((a₁₂+b₁₂)G₂(f₂)).

Denoting R1G⁰₁ and R1G1((a12 +b12)G⁰₂ by R1,1 and R1,2 respectively, and R1G1 and R1G1((a12+b12)G2by S1,1and S1,2, respectively, we have that

u₁= 2

l=1

H^l_1,l⁻^j(t )+R_1,l(t )

u⁰_l +

K^l_1,l⁻^j(t )+S_1,l(t )

f_l,

where

• the operators H_1,l^l⁻¹ and K_1,l^l⁻¹ are of order l −1 and therefore map C([0, T], H^s) into C([0, T], H^s⁻^l⁺¹),

• R_1,1and S1,1map H^sinto C([0, T], H^s⁺^N),

• R_1,2and S1,2map H^sinto C([0, T], H^s⁺^N⁻¹).

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This means that H¹_1,l⁻^l, K¹_1,l⁻^l∈ L(C([0, T], H^s), C([0, T], H^s⁻^l⁺¹)) and R_1,l, S_1,l ∈L(H^s, C([0, T], H^s⁺^N⁻^l⁺¹)). The same argument is true for u₂. We have in this way obtained the representation formula stated in Theorem1.3.

3.4. The m ×mcase

We are now ready to prove Theorem1.3.

Proof of Theorem1.3. Throughout this proof we refer to the proof of Theorem 1 in [1], i.e.

Theorem1.4in this paper. Theorem1.4proves the well-posedness of this Cauchy problem in anisotropic Sobolev spaces. It remains to prove the representation formula for the components of the solution u. We begin by observing that under our hypotheses we can write

u_i=U_i⁰+

j <i

G^j_i,j⁻ⁱ(u_j)+

i<j≤m

G¹_i,j(u_j),

for i=1, . . . , m, where G^j_i,j⁻ⁱ and G¹_i,j are operators of order j−iand 1, respectively and U_i⁰=G⁰_iu⁰_j+G_i(f_i).

We begin by substituting

u_m=U_m⁰+

j <m

G^j_m,j⁻^m(u_j),

into

um−1=U_m⁰₋₁+

j <m−1

G^j_m⁻₋^m_1,j⁺¹(uj)+G¹_m₋_1,m(um).

We get

u_m₋₁=U_m⁰₋₁+

j <m−1

G^j_m⁻₋^m_1,j⁺¹(u_j)+G¹_m₋_1,mU_m⁰+

j <m

G¹_m₋_1,mG^j_m,j⁻^m(u_j)

=(U_m⁰₋₁+G¹_m₋_1,mU_m⁰)+

j <m−1

(G^j_m⁻₋^m_1,j⁺¹(u_j)+G¹_m₋_1,mG^j_m,j⁻^m(u_j))

+G¹_m₋_1,mG⁻_m,m¹ ₋₁um−1.

Since all the operators above are of order ≤0 we conclude that the operator L_m₋₁=I−G¹_m₋_1,mG⁻_m,m¹ ₋₁:=I−G⁰_m₋₁ is invertible on a sufficiently small interval [0, T]and, therefore,

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u_m₋₁−G¹_m₋_1,mG⁻_m,m¹ ₋₁u_m₋₁=(U_m⁰₋₁+G¹_m₋_1,mU_m⁰)

+

j <m−1

(G^j_m⁻₋^m_1,j⁺¹(u_j)+G¹_m₋_1,mG^j_m,j⁻^m(u_j)), (3.5)

yields

u_m₋₁=L⁻_m¹₋₁U_m⁰₋₁+L⁻_m¹₋₁

j <m−1

G^j_m⁻₋^m₁⁺¹u_j, (3.6)

with U_m⁰₋₁and G^j_m⁻₋^m₁⁺¹defined by the right-hand side of (3.5). In particular,

U_m⁰₋₁=U_m⁰₋₁+D¹_m₋_1,mU_m⁰, (3.7) where G¹_m₋_1,m=D_m¹₋_1,mis an integrated Fourier integral operator with symbol of order 1. Note that we choose the notation D_m¹₋_1,min order to have simpler notations for the compositions of operators in the computations below. Substituting um and um−1 into um−2 and making use of (3.6) we find a similar formula to (3.6) for um−2(see (24) in [1]) with U_m⁰₋₂defined as follows:

U_m⁰₋₂=U_m⁰₋₂+G¹_m₋_2,m₋₁L⁻_m¹₋₁U_m⁰₋₁

+G¹_m₋_2,mU_m⁰+G¹_m₋_2,mG⁻_m,m¹ ₋₁L⁻_m¹₋₁U_m⁰₋₁. (3.8) Hence, by implementing (3.7) in (3.8) we have

U_m⁰₋₂=U_m⁰₋₂+G¹_m₋_2,m₋₁L⁻_m¹₋₁(U_m⁰₋₁+D¹_m₋_1,mU_m⁰)

+G¹_m₋_2,mU_m⁰+G¹_m₋_2,mG⁻_m,m¹ ₋₁L⁻_m¹₋₁(U_m⁰₋₁+D_m¹₋_1,mU_m⁰).

By collecting the terms U_m⁰₋₁and U_m⁰ we conclude that U_m⁰₋₂can be written as U_m⁰₋₂=U_m⁰₋₂+D_m¹₋_2,m₋₁U_m⁰₋₁+D²_m₋_2,mU_m⁰,

where the operators D_m¹₋_2,m₋₁and D_m²₋_2,mare of order 1 and 2, respectively. By iterating the same argument we prove that for every j=1, . . . , m −1,

U_j⁰=U_j⁰+

k>j

D^k_j,k⁻^jU_k⁰,

where k−j is the order of the operator D^k_j,k⁻^j. For a precise construction of the operators D_j,k^k⁻^j we refer the reader to the proof of Theorem 1 in [1]. Since

u₁=U₁⁰+G₁⁰u₁,

where the operator I−G₁⁰is invertible with inverse L⁻₁¹(see [1]) we conclude that