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

In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in \cite{Garetto2018}. In the case of space dependent coefficients, we prove a representation formula for solutions that allows us to derive results of regularity and propagation of singularities.


Introduction
In this work, we continue the study of non-diagonalisable systems that begun in [GJR18] by proving a result on solution representations and propagation of singularities for a hyperbolic system with x-dependent principal part.Let us consider with the usual notation D t = −i∂ t and D x = −i∂ x .We assume that A(x, D x ) = a ij (x, D x ) m i,j=1 is an m × m matrix of pseudo-differential operators of order 1, i.e., a ij ∈ Ψ 1 1,0 (R n )) and that B(t, x, D x ) = b ij (t, x, D x ) m i,j=1 is an m × m matrix of pseudo-differential operators of order 0, i.e., b ij ∈ C([0, T ], Ψ 0 1,0 (R n )).We also assume that the matrix A is upper triangular and hyperbolic, i.e., A(x, D x ) = Λ(x, D x ) + N(x, D x ) = diag(λ 1 (x, D x ), λ 2 (x, D x ), . . ., λ m (x, D x )) + N(x, D x ) with real eigenvalues λ 1 (x, ξ), λ 2 (x, ξ), . . ., λ m (x, ξ) and We recall that the well-posedness of this kind of systems has been proven in anisotropic Sobolev spaces in [GJR18] 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 [IM84] who studied 3 × 3 systems where the twice iterated Poisson bracket vanishes and Ichinose [Ich82] studied 2 × 2 systems under the same condition.In [Roz83], Rozenblum considered smoothly diagonalisable systems with transversally intersecting characteristics, and derived a formula for the propagation of its singularities.Consequently, the transversality condition was removed in [KR07], 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 [KR07] to non-diagonalisable hyperbolic systems with variable multiplicity.The paper is organised as follows.Section 2 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 Section 3. Section 4 is devoted to propagation of singularities.The paper ends in Section 5 with 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 µ 1,0 (R n ) is the space of symbols of order µ and type (1, 0), i.e., a = a(x, ξ) The set of pseudo-differential operators associated to the symbols in S µ 1,0 (R n ) is denoted by Ψ µ 1,0 (R n ).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 ], Ψ µ 1,0 ) for symbols and operators, respectively.For the sake of simplicity we will adopt the abbreviated notations S µ and Ψ µ for S µ 1,0 (R n ) and Ψ µ 1,0 (R n ), respectively, and CS µ and CΨ m for C([0, T ], S µ ) and C([0, T ], Ψ µ 1,0 ), respectively.With I µ we denote the class of Fourier Integral Operators with amplitude in S µ , i.e., of operators of the type where ϕ is a phase function and f is the Fourier transform of f .This notation is standard and further details can be found in [GJR18] and the references therein, for instance [Hör71].In this paper we will use the short expression integrated Fourier integral operator to denote an operator of the type where the Fourier transform of f = f (y, s) is meant with respect to the variable y.
, where the constant C may depend on χ.Since we only work in R n and no confusion can arise, we drop the indication of R n from 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 1.2.Assumptions on the matrices A(x, D x ) and B(t, x, D x ).In this paper, we make the following assumptions on lower order terms and multiplicities: (H1) (Lower order terms) The entries of the matrix belong to C([0, T ], Ψ 0 ) and are of decreasing order below the diagonal, i.e., for some j, k ∈ {1, . . ., m} and λ j (x, ξ) and λ k (x, ξ) are not identically equal near (x, ξ) then there exists some N ≤ M such that Remark 1.1.In [GJR18], for A depending on t as well and under the condition (H1) on B, we proved that for any s ∈ R, u 0 k ∈ H s+k−1 , k = 1, . . ., m, and f k ∈ C([0, T ], H s+k−1 ), k = 1, . . ., m, the Cauchy problem (1.1) has a unique anisotropic Sobolev solution u with components u k ∈ C [0, T ], H s+k−1 , k = 1, . . ., m.In this paper we do the microlocal analysis of solutions in the case of A depending only on x.
We are now ready to state the main result of our paper.This is a representation formula for the solution u which shows how this depends on initial data and righthand 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 where A(x, D x ) is an upper-triangular matrix of pseudo-differential operators of order 1 and B(t, x, D x ) is a matrix of pseudo-differential operators of order 0, continuous with respect to t.Let u 0 and f have components u 0 j and f j , respectively, with u 0 j ∈ H s+j−1 (R n ) 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., u j ∈ C([0, T ], H s+j−1 ) for j = 1, . . ., m; (ii) for any N ∈ N, the components u j , j = 1, . . ., m, of the solution u are given by where R j,l , S j,l ∈ L(H s , C([0, T ], H s+N −l+j )) and the operators H l−j j,l , K l−j j,l ∈ L(C([0, T ], H s ), C([0, T ], H s−l+j )) are integrated Fourier Integral Operators of order l − j.

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 [GJR18].
For each eigenvalue λ j (x, ξ) of A(x, ξ), we will be denoting by G 0 j θ the solution to and by G j g the solution to The b jj are the diagonal elements of the lower order term B(t, x, D x ) in (1.1).The operators G 0 j and G j can be locally represented by a Fourier integral operator and an integrated Fourier integral operator, respectively, i.e., and with ϕ j (t, s, x, ξ) solving the eikonal equation and ϕ j (t, x, ξ) = ϕ j (t, 0, x, ξ).Note that the amplitudes C j in (2.2) have asymptotic expansions +∞ k=0 C j,−k where the element C j,−k (s, x, ξ) is of order −k, k ∈ N, and satisfies transport equations with initial data at t = s.By construction, c j (t, x, ξ) = C j (t, 0, x, ξ).In the above construction of propagators for hyperbolic equations, we have c j ∈ S 0 , so that G 0 j ∈ I 0 .Further, to simplify the analysis of the regularising part in (1.3) we introduce the notation i.e., the integrated Fourier integral operator G j can now be written as (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 Theorem 1.3.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).
2.1.1.Integrated Fourier integral operators.The composition of integrated Fourier integral operators like in (2.2) was studied in [KR07].Their result, that we recall in the sequel, is crucial for our proof and is a generalisation of a previous result by Rozenblum in [Roz83].Let t 1 , t 2 , . . ., t l ∈ [0, T ], t = (t 1 , t 2 , . . ., t l ) and let H(t) be the operator where λ i are pseudo-differential operators of order 1.By [Hör71], H(t) is a (parameter dependent) Fourier integral operator and its canonical relation where and the Φ t j are the transformations corresponding to a shift by t along the trajectories of the Hamiltonian flow defined by the λ j .
Theorem 2.1 (Thm 2.1 in [KR07]).With the above notation, assume that not all λ i s are identical to each other and let (H2) be satisfied for the λ j in (2.5).Further, suppose that D(t) ∈ Ψ 0 .Then, the operator Remark 2.2.If the global estimate in the definition of the symbols classes S m 1,0 in [GJR18] is replaced with an estimate that holds locally on every compact set, then the the conclusions of Theorem 2. ).
Theorem 2.1 allows us to investigate the composition G i G j .

The composition
Let G i and G j be two operators as in (2.2).Then, we can write with E i , E j as in (2.4).If we now iterate this k times, we obtain where t = dt 1 . . .dt 2k−1 .This is an operator of the same type as Q l above so we can apply Theorem 2.1 and obtain the following corollary.
Remark 2.4.The same conclusion as in Corollary 2.3 holds true if we have a product that contains a collection G 0 j 's as long as there is at least one integrated version G j present.

Solution representations
This section is devoted to the proof of Theorem 1.3.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 × m case.We adopt the notations introduced in Section 2.
3.1.The 2 × 2 case.Let us consider the system where T , and with the operators and We suppose that all entries of A(x, D x ) belong to Ψ 1 1,0 and all entries of B(t, x, D x ) belong to CΨ 0 1,0 .As detailed in Subsection 2.2 in [GJR18], we obtain the equations are of order 0 under the assumption (H1) made on the lower order terms; here in particular b 21 ∈ Ψ −1 .From (3.3), we have (3.4) 3.2.Inversion of the operator L 1 .Adopting the notations introduced in [GJR18] we introduce the operator is of order 0 and and from the Sobolev mapping properties of Fourier Integral Operators (see Lemma 1 in [GJR18]) 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 L 1 is 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 Corollary 2.3 we get that for every N ∈ N the operator L −1 can be written as a finite sum of powers of the operator G 0 1 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 ∈ N such that It is important to remark here that the estimates needed to ensure the small norm of the operator G 0 1 , 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 −1 1 as written above to both sides of the equality ).We obtain the following representation for u 1 , where Denoting R 1 G 0 1 and R 1 G 1 ((a 12 + b 12 )G 0 2 by R 1,1 and R 1,2 respectively, and R 1 G 1 and R 1 G 1 ((a 12 + b 12 )G 2 by S 1,1 and S 1,2 , respectively, we have that where • the operators H l−1 1,l and K l−1 1,l are of order l−1 and therefore map C([0, T ], . The same argument is true for u 2 .We have in this way obtained the representation formula stated in Theorem 1.3.

3.4.
The m × m case.We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3.Throughout this proof we refer to the proof of Theorem 1 in [GJR18], i.e.Theorem 1.4 in this paper.Theorem 1.4 proves 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 for i = 1, . . ., m, where G j−i i,j and G 1 i,j are operators of order j − i and 1, respectively and We begin by substituting We get Since all the operators above are of order ≤ 0 we conclude that the operator is invertible on a sufficiently small interval [0, T ] and, therefore, with U 0 m−1 and G j−m+1 m−1 defined by the right-hand side of (3.5).In particular, where G 1 m−1,m = D 1 m−1,m is an integrated Fourier integral operator with symbol of order 1.Note that we choose the notation D 1 m−1,m in order to have simpler notations for the compositions of operators in the computations below.Substituting u m and u m−1 into u m−2 and making use of (3.6) we find a similar formula to (3.6) for u m−2 (see (24) in [GJR18]) with U 0 m−2 defined as follows: Hence, by implementing (3.7) in (3.8) we have By collecting the terms U 0 m−1 and U 0 m we conclude that U 0 m−2 can be written as , where the operators D 1 m−2,m−1 and D 2 m−2,m are of order 1 and 2, respectively.By iterating the same argument we prove that for every j = 1, . . ., m − 1, where k − j is the order of the operator D k−j j,k .For a precise construction of the operators D k−j j,k we refer the reader to the proof of Theorem 1 in [GJR18].Since ) we conclude that We now argue as in the case 2 × 2 and we apply Corollary 2.3 which holds thanks to the hypotheses (H1) and (H2).We obtain that for every N there exists M such that L −1 1 = M j=0 (G 0 1 ) j modulo some regularising operator R 1 mapping C([0, T ], H s ) into C([0, T ], H s+N ) and therefore By collecting all the terms with u 0 k and all the terms with f k , k = 1, . . ., m we see that where H l−1 1,l and K l−1 1,l have order l−1 and R 1,l and S 1,l are regularising.This is due to the fact (G 0 1 ) j is an operator of order 0 as well as G 0 k and G k , and D k−1 1,k is an operator of order k − 1.The regularising operator R 1 generates R 1,l and S 1,l .These last two operators map C([0, T ], H s ) into C([0, T ], H s+N −l+1 ).We have therefore proven the second assertion of this theorem for j = 1.Following the proof of Theorem 1 in [GJR18] we now have that where the operator G 0 2 is of zero order and its definition involves invertible operators L m−1 , L m−2 , . . ., L 2 and G −1 2 is of order −1.Hence, by inverting the operator L 2 = I − G 0 2 on a sufficiently small interval [0, T ] we have By definition of U 0 2 and by the representation formula for u 1 obtained above we can write Note that the operators above are of order l − 2. By arguing as for u 1 and by writing L −1 2 as a finite number of powers of G −1 2 plus a regularising operator we arrive at the formula with the desired order and regularising properties.We conclude the proof by iterating the same scheme.From formula (28) in [GJR18] we obtain for j > 2 the following expression for u j : where the operator involved are of order l − j.Writing L −1 j by Neumann series we can conclude that where H l−j j,l and K l−j j,l have order l − j and R j,l and S j,l map C([0, T ], H s ) into C([0, T ], H s+N −l+j ).
Remark 3.1.Note that in [GJR18], we defined S m by global estimates on R n × R n .If one replaces that definition with a locally over every compact sets version then Theorem 1.3 still holds true with the spaces L(H s , C([0, T ], H s+N )) and )), respectively.
3.5.Regularity results.We conclude this section with some regularity results in L p and Hölder spaces.These are obtained by arguing as in [KR07] Theorem 2.2 and Theorem 3.1.
Theorem 3.2.Let 1 < p < ∞ and α = (n − 1) 1 p − 1 2 .Let A(x, D x ) be an m×m upper-triangular matrix of pseudo-differential operators of order 1 and suppose that the eigenvalues λ i (x, ξ) ∈ S 1 of A(x, ξ) are real and satisfy (H2).Assume further, that B(t, x, D x ) is an m × m matrix of pseudo-differential operators of order 0 satisfying (H1).Then, for any compactly supported u 0 ∈ L p α ∩ L 2 comp , the solution u = u(t, x) of the Cauchy problem (1.1) satisfies u(t, •) ∈ L p loc , for all t ∈ [0, T ].Moreover, there is a positive constant C T such that Local estimates can be obtained in other spaces as well, for s ∈ R and α as above.In detail, assuming u 0 below is compactly supported, we have

Propagation of singularities
We now want to analyse the solution u under a microlocal point of view.In particular we want to see how its wavefront set is related to the wavefront set of the initial data.Thanks to the assumptions (H1) and (H2) and the representation formula in Theorem 1.3 we are able to extend the result of propagation of singularities in [KR07] to systems with not diagonalisable principal part (in upper-triangular form).For the sake of the reader we recall below some basic notions of microlocal analysis which can be found in [Hör90] and [Hör71].
is defined via its complement as follows: (x 0 , ξ 0 ) belongs to (WF(u)) c if and only if there exists a χ ∈ C ∞ 0 (R n ) with χ(x 0 ) = 0 and a conic neighbourhood Γ of ξ 0 such that for every N ∈ N there exists a positive constant C N,χ such that Let us now discuss the propagation of singularities for operators Q l from Theorem 2.1, given by Similar analysis was done in [KR07].It is clear that singularities propagate along broken Hamiltonian flows.Let We recall from the definition of H(t) that its canonical relation and the Φ t j are the transformations corresponding to a shift by t along the trajectories of the Hamiltonian flow defined by the λ j .Also, recall that t 1 , t 2 , . . ., t l ∈ [0, T ], t = (t 1 , t 2 , . . ., t l ), and H(t) is the operator where λ i are pseudo-differential operators of order 1.Let Φ J (t, x, ξ) be the corresponding broken Hamiltonian flow.It means that points follow bicharacteristics of λ j 1 until meeting the characteristic of λ j 2 , and then continue along the bicharacteristic of λ j 2 , etc.In this procedure the singularities may accumulate if wave front sets for different broken trajectories project to the same point of X.We can rewrite (4.1) as where t = (t 1 , . . ., t l ) ranges over the simplex with Z(t j ) found from (4.2).It is then possible to treat it as a standard Fourier integral operator with the change of variables t = ζ|ξ| −1 .Let K be a cone in e iϕ(x,y,θ) a(x, y, θ)u(y)dydθ be a Fourier integral operator with integration over the cone K with respect to θ.Let K j be K or a face of K. Let ϕ j (x, y, θ j ) = ϕ| K j , θ j ∈ K j .Let Λ j ⊂ T * X × T * X be a Lagrangian manifold with boundary: Then we have the following statement on the propagation of singularities, see [KR07].
Consequently, combining these observations with Theorem 1.3 we obtain the following property.
Corollary 4.3.Let n ≥ 1, m ≥ 2, and let where A(x, D x ) is an upper-triangular matrix of pseudo-differential operators of order 1 and B(t, x, D x ) is a matrix of pseudo-differential operators of order 0, continuous with respect to t. Recall that under condition (H1) and (H2), for any N ∈ N, the components u j , j = 1, . . ., m, of the solution u are given by where R j,l , S j,l ∈ L(H s , C([0, T ], H s+N −l+j )) and the operators Consequently, up to any Sobolev order (depending on N), the wave front set of u j is given by with each of the wave front sets for terms in the right hand side of (4.4) given by the propagation along the broken Hamiltonian flow as in Theorem 4.2.
We conclude the paper by presenting some applications of Theorem 1.3 and Theorem 1.4.

Application: Higher order hyperbolic equations
In this section we want to study the well-posedness of the Cauchy problem where each A m−j (t, x, D x ) is a scalar differential operator of order m − j with continuous and bounded coefficients depending on t and x.As usual, D t = 1 i ∂ t and denote the principal part of the operator A m−j , We assume that the problem above is hyperbolic, i.e., the characteristic equation has m real valued roots λ 1 , λ 2 , • • • , λ m .In addition we work under the hypothesis that the roots λ i , i = 1, . . ., n are symbols of order 1, i.e., λ i ∈ C([0, T ], S 1 ), for all i = 1, . . ., n.For this reason we assume that (H0) the coefficients of the equation above are continuous in t and smooth in x, with bounded derivatives of any order α ∈ N n 0 with respect to x.We will make use first of Theorem 1.4 and then of Theorem 1.3.5.1.Well-posedness in Sobolev spaces.We begin by reducing the m-order partial differential equation in (5.1) into a first order system of pseudo-differential equations.Let D x be the pseudo-differential operator with symbol ξ .The transformation with k = 1, ..., m, makes the Cauchy problem (5.1) equivalent to the following system with initial condition (5. 3) The matrix in (5.2) can be written as A + B with It is clear that the eigenvalues of the symbol matrix A(t, x, ξ) are the roots λ j (t, ξ), j = 1, ..., m.
We want to apply Theorem 1.4 to our Cauchy problem.This means to find under which hypotheses the equation in (5.1) can be reduced into a first order system with upper-triangular principal part and lower order terms of suitable order as in (H1).
where each A m−j (t, x, D x ) is a differential operator of order m − j with continuous and bounded coefficients depending on t and x as in (H0).Let A (m−j) denote the principal part of the operator A m−j .Assume that the roots of the corresponding characteristic polynomial are real valued symbols and that A m−j+1 (t, x, ξ) ∈ C([0, T ], S 0 ) for all j = 1, . . ., m − 1.If f ∈ C([0, T ], H s+m−1 ) and g k ∈ H s+m−1 (R n ) for all k = 1, . . ., m then the Cauchy problem (5.1) has a unique solution u ∈ C m−1 ([0, T ], H s+m−1 ).
Proof.We consider the associated reduced system with principal part given by the matrix The operators b (j) are of order 1 so since we assume that A m−j+1 (t, x, ξ) ∈ C([0, T ], S 0 ) for j = 1, . . ., m − 1 it follows that b (j) ≡ 0 for j = 1, . . ., m − 1.This means that the Sylvester matrix A is actually upper-triangular and that the matrix B of the lower order terms is of the following type: T ], S 0 ) for j = 1, . . ., m − 1 we have that b j is a pseudo-differential operator of order j − m for j = 1, . . ., m.We are therefore under the assumptions of Theorem 1.4 for the matrices A and B. Since of the reduced Cauchy problem belong to the space H s+k−1 (R n ) for all k = 1, . . ., m.Thus, by Theorem 1.4 there exists a unique solution u(t, x) to the Cauchy problem under consideration such that for k = 1, . . ., m.By Sobolev mapping properties of pseudo-differential operators it follows that u ∈ C m−1 ([0, T ], H s+m−1 ).
(i) The equation , where x ∈ R and a 1 is real valued falls into the class of equations considered in the previous theorem.Indeed, the characteristic polynomial τ 2 − a 1 (t, x)τ ξ has two real roots and A 2 = a 2 (t, x) is an operator of order 0. (ii) Let us now consider the second order Cauchy problem    D 2 t u = a 2 (t)D 2 x u + b 1 (t)D x u + b 2 (t)D t u + b 3 (t)u + f (t, x), u(0, x) = g 0 (x), D t u(0, x) = g 1 (x), where (t, x) ∈ [0, T ] × R, the equation coefficients are continuous and a ∈ C 1 with a(t) ≥ 0. Making use of the standard reduction into first order system of pseudo-differential equations we have that the Cauchy problem above is equivalent to (5.4) where U = (u 1 , u 2 ) T = ( D x u, D t u) T and U(0, x) = U 0 = ( D x g 0 , g 1 ) T .In the sequel we will denote the right-hand side of the system above with A(t, D x )U + B(t, D x ) + F, where A and B are defined by operators of order 1 and 0, respectively.The principal part matrix A(x, ξ) = 0 ξ a 2 (t)ξ 2 ξ −1 0 is not upper triangular and it is not diagonalisable because of the zeros of the coefficient a. However it can be reduced into upper-triangular form.We refer here the reader to [GR13] and to the appendix in [GJR18].The matrix A has λ 1 (t, ξ) = −a(t)ξ and λ 2 (t, ξ) = a(t)ξ as eigenvalues.It follows that h (1) = 1 a(t)ξ ξ −1 and h (2) = 1 −a(t)ξ ξ −1 are eigenvectors corresponding to λ 1 and λ 2 , respectively.Choose now h (1) (the argument is analogous with h (2) ).The matrix T 1 = (h (1) , e 2 ) = 1 0 a(t)ξ ξ −1 1 is invertible.Its inverse is Note that the operator T −1 1 (t, D x )A(t, D x )T 1 (t, D x ) can be therefore written as λ 1 (t, D x ) D x 0 λ 2 (t, D x ) .
We can now use this transformation to reduce the system D t U = AU into upper-triangular form.More precisely, for U = T 1 V , we have that the system Under the assumptions that a ∈ C 1 ([0, T ]) we easily see that the eigenvalues λ 1 and λ 2 belong to C([0, T ], S 1 ).In addition, condition (H1) is fulfilled if the symbol above is of order −1.This is the case when b 1 (t) + b 2 (t)a(t) + D t a(t) = 0, for all t ∈ [0, T ] (for instance when b 2 ≡ 0 and b 1 = −D t a).Since T −1 is a matrix of pseudo-differential operators of order 0 we have that V (0, x) = V 0 = T −1 1 (0, D x )U 0 has the same regularity properties of U 0 .We can therefore apply Theorem 1.3 to this system and obtain the following result.