Journal of Mathematical Analysis and Applications A note on weakly hyperbolic equations with analytic principal part ✩✩

Well-posedness In this note we show how to include low order terms in the C ∞ well-posedness results for weakly hyperbolic equations with analytic time-dependent coeﬃcients. This is achieved by doing a different reduction to a system from the previously used one. We ﬁnd the Levi conditions such that the C ∞ well-posedness continues to hold.


Introduction
In this note we study the Cauchy problem M(t, D t , D x )u(t, x) = 0, (t, x) ∈ (δ, T + δ) × R n , coefficients are in the Hölder class, a ν, j ∈ C α , 0 < α < 1, it was shown by the authors in [5,Remark 8] that the Cauchy problem (1.1) is well-posed in Gevrey classes G s (R n ) provided that 1 s < 1 + α 1−α (if α = 1, it is sufficient to assume a ν, j of the principal part, |ν| = j, are analytic. There are also previous works on second and higher order equations using the quasisymmetrisers, see e.g. [4]. In [6], the assumptions on (1.1) have been formulated in terms of the characteristic roots while the Levi conditions on lower order terms can be expressed in terms of the coefficients of the operator M. In addition, the quasisymmetriser has been used in [6] while here we use the symmetriser. We refer to [5] and [6] for a review of the existing literature for this problem.
Recently, Jannelli and Taglialatela [10] treated Eq. (1.1) with analytic coefficients, without lower order terms, proving the C ∞ well-posedness under assumptions that can be expressed entirely in terms of the coefficients of the operator M. The purpose of this note is to show how to extend this to include lower order terms with Levi conditions still formulated in terms of the coefficients of M. This will be achieved in this paper by doing a different reduction of (1.1) to the first order system which will allow us to include the lower order terms in the energy. Indeed, in the case of the homogeneous operators, the reduction used in [10], analogous to the one used by Colombini and Kinoshita in [2], was done to a system which is homogeneous of order one in ξ . Such a reduction cannot be used in the present context because the lower order terms introduce singularities in the symbols at ξ = 0. Instead, by employing a reduction to a pseudo-differential system, also used in [5], we are able to avoid such singularities. Thus, the analysis in this note is based on a reduction from [5] combined with a number of results from [10], with a subsequent treatment of lower order terms in the energy under Levi conditions introduced below. An interesting feature is that the C ∞ well-posedness holds for analytic coefficients in the principal part and lower order terms which are only continuous. We also give a result for bounded (and possibly discontinuous) lower order terms.
Our main point in this note is to demonstrate that by applying a different reduction to a system one can handle lower order terms in the energy estimates. For specific equations (e.g. second order equations) this result can be improved by other methods, but this lies outside our scope here.
In [1], Colombini, de Giorgi and Spagnolo, and in [3], Colombini and Spagnolo gave examples of second order equations with time-dependent coefficients which are not distributionally well-posed. In this paper, we also prove the distributional well-posedness of (1.1) in our setting.
In Section 2 we introduce the notations and recall the results of [10]. In Section 3 we give our results. In Section 4 we give the proofs, and in Section 5 we analyse the meaning of the assumptions on both the principal part and lower order terms, and compare the obtained results with those in [6].

Preliminaries
We begin by recalling the theorem proved in [10] for the Cauchy problem where t 0 ∈ (δ, T + δ), and is homogeneous of order m. This requires some preliminary notions which are collected in the sequel. Let be the companion matrix of P (t, τ , ξ/|ξ |). By construction the matrix A 0 (t, ξ) is homogeneous of oder zero in ξ , and the eigenvalues of A 0 (t, ξ)|ξ | are the characteristic roots τ 1 (t, ξ), . . . , τ m (t, ξ) of P (t, τ , ξ).
For the moment let us fix t ∈ (δ, T + δ) and ξ so that P is a polynomial in τ with constant coefficients. In [ Let Q j be the principal j × j minor of Q obtained by removing the first m − j rows and columns of Q and let j its determinant. When j = m we use the notations Q and instead of Q m and m . The hyperbolicity of P can be seen at the level of the symmetriser Q and of its minors as stated in the following proposition (see [9]). Clearly, when t and ξ vary in (δ, T + δ) and R n , respectively, r becomes a function r (t, ξ) homogeneous of degree 0 in ξ and analytic in t. When r is not identically zero one can define the function , which is homogeneous of degree 0 in ξ as well, analytic on the interval (δ, T + δ). In addition, the following property holds for and : if t → (t, ξ) vanishes of order 2k at a point t then t → (t, ξ) vanishes of order 2k − 2 at t . Note that estimating the quotient ∂ t Q V , V / Q V , V is equivalent to estimating the roots of the generalised Hamilton- Q co is the cofactor matrix of Q . We recall that the cofactor of Q is the matrix with entries q co the determinant of the submatrix obtained from Q by removing the i-th row and the j-th column. Finally, from the known identity valid for the roots λ j , j = 1, . . . ,m, of the generalised Hamilton-Cayley polynomial, we see that d 2 plays a fundamental role when one wants to defined as above the check function of Q . Replacing Q with Q j in the definition of d 2 we define the check function ψ j (t, ξ) of Q j . Clearly, ψ j (t, ξ) is homogeneous of order zero in ξ . Note that when m = 1 the check function ψ is set to be identically zero. We are now ready to state the C ∞ well-posedness theorem of Jannelli and Taglialatela given in [10].
Since the purpose of this note is to describe the possibility of adding lower order terms to L we will avoid long technicalities and will focus on the non-degenerate case, i.e., the case with (·, ξ) ≡ 0 is not identically zero in (δ, T + δ).
We skip the treatment of the Cauchy problem (2.1) in the degenerate case since it is lengthy but remark that the analysis can be carried out in this case as well along the lines of the analysis of the general case in [6]. In the present context, it would make use of r (·, ξ) and the corresponding check function ψ r (t, ξ), where r = r(ξ ) is the greatest integer such that r (·, ξ) ≡ 0 in (δ, T + δ). For more details on these for the case of homogeneous L, see Theorem 1 and Section 3 in [10]. [10].) Let L(t, ∂ t , ∂ x ) as in (2.1) be a weakly hyperbolic homogeneous operator with analytic coefficients in (δ, T +δ).

Theorem 2.2. (See
Let P (t, τ , ξ) be the characteristic polynomial and A 0 (t, ξ) the companion matrix of P (t, τ , ξ/|ξ |). Let Q (t, ξ) be its symmetriser and holds for all t ∈ [a, b]. Then, the Cauchy problem We can write condition (2.5) in a different way by introducing the set Σ(ξ) = {t 1 , . . . , t N(ξ ) : (t j , ξ) = 0} and the function Note that by the analyticity of in t it follows that the function N(ξ ) is locally bounded (see the proof of Lemma 5 in [6]). Using again the fact that (t, ξ) is analytic in t and homogeneous of order 0 in ξ one can prove that there exist constants c 1 > 0 and c 2 > 0 independent of ξ such that for all t ∈ (δ, T + δ) and ξ = 0. Hence, (2.5) can be reformulated as follows: there exists a constant C > 0 independent of ξ such that for all t ∈ [a, b] and ξ = 0. This extends to any space dimension the one-dimensional observation of Jannelli and Taglialatela made in [10, p. 1000].

Results
We are now ready to study the Cauchy problem (1.1) or, in other words, to add lower order terms to the equation in (2.1). First, we describe the reduction of (1.1) to a system since we will be making use of the symmetriser of the corresponding companion matrix. We rewrite the equation First of all we perform the standard reduction to a system of pseudo-differential equations as in [5] by setting where D x is the pseudo-differential operator with symbol ξ = (1 + |ξ | 2 ) 1 2 . This transformation makes the mth-order equation above equivalent to the first order system for l = 1, . . . ,m, and generate the column vector U 0 (x). In the following theorem we use functions ψ and which have been defined at the beginning of Section 2.
Then the Cauchy problem (1.1) is C ∞ well-posed in [a, b] with initial data at t 0 = a, and it is also well-posed in D (R n ).
One of the features of this result is that we can allow lower order terms to be complex-valued. In Remark 4.6 we will comment on a small simplification of the Levi conditions (ii) if the matrix B is real. In the case when lower order terms are discontinuous but still bounded, we have the following counterpart of the result above. The proof is similar to that of Theorem 4 in [6].
The same distributional conclusion as we had in Theorem 3.1 also holds in Theorem 3.2, with the solution provided that the Cauchy data are all in D (R n ).

Proofs
Note that the eigenvalues of the matrix A 1 = ξ A(t, ξ) are the roots of the characteristic polynomial P (t, τ , ξ) and that the entries of the matrix A are related to the entries of the matrix A 0 in (2.2) by the formula a j (t, ξ) ξ m− j+1 = h m− j+1 (t, ξ)|ξ | m− j+1 . Applying the Fourier transform to the system (3.1) we obtain the system Note that by performing a reduction to a system of pseudo-differential equations the symmetriser Q (t, ξ), defined as in [6,Section 3], is a matrix of 0-order symbols which can be expressed in terms of the rescaled roots τ j (t, ξ) ξ −1 (or eigenvalues of the matrix A). More precisely, the entries q ij of the symmetriser Q (t, ξ) are polynomials in τ 1 ξ −1 , . . . , τ m ξ −1 .
Making use of the concept of the Bezout matrix associated to the couple of polynomials (P , ∂ τ P ) it is also possible to express the entries of the symmetriser in terms of the coefficients h j , j = 1, . . . ,m. For further details we refer the reader to [10, p. 998]. We begin by proving some basic properties of the symmetriser which will be employed to prove the C ∞ well-posedness.

The symmetriser Q
It is useful to make a comparison between the symmetriser Q of the matrix A and the symmetriser Q 0 with homogeneous entries of order 0 employed by Jannelli and Taglialatela in [10]. These two matrices are both symmetric with polynomial entries in τ 1 ξ −1 , . . . , τ m ξ −1 and τ 1 |ξ | −1 , . . . , τ m |ξ | −1 , respectively, as defined in [6, Section 3]. By construc- 0 (t, ξ), (4.2) where and 0 are the determinants of Q and Q 0 , respectively, and 0 is expressed in terms of the 0-homogeneous roots τ i /|ξ |. The following lemma on symmetric positive semi-definite matrices will be in the sequel applied to Q .
holds for all t ∈ [a, b] and V ∈ C m .

Lemma 4.2. Let Q (t, ξ) be the symmetriser of the weakly hyperbolic matrix A(t, ξ) defined above. Then, there exist two positive
constants c 1 and c 2 such that holds for all t ∈ [a, b], ξ ∈ R n and V ∈ C m .
Let I be a closed interval in R. We recall (see also [10, pp. 1003-1004]) that if B(t) and C (t) are two real symmetric m × m matrices, C (t) is nonnegative and det C (t) has only isolated zeros then if and only if the roots λ i of the generalised Hamilton-Cayley polynomial we conclude that (4.3) holds if and only if d 2 is bounded. We are now ready to prove the following lemma.
by (2.4) and the definition of d 0 , d 1 and d 2 we get Applying

The hyperbolic energy
Let us work on any subinterval holds for all 1 i, j m, t ∈ [a, b] and ξ ∈ R n , then (4.8) holds.

Proof.
We begin by observing that the matrix Q B − B * Q has entries d ij = q im b j − b i q jm . It follows that and by the hypothesis (4.10) we get for some constant c > 0, uniformly in t ∈ [a, b] and ξ ∈ R n . Finally, combining this estimate with the bound from below of Lemma 4.2 we obtain (4.8). 2 Remark 4.6. Note that when the lower order terms are real-valued then the matrix Q B − B * Q is skew-symmetric. This means that d ij = q im b j − b i q jm is identically zero when i = j and d ij = −d ji . It follows that the Levi conditions (4.8) can be rewritten as (4.11) for 1 i < j m, t ∈ [a, b] and ξ ∈ R n .
We are now ready to prove Theorem 3.1.
Proof of Theorem 3.1. First of all, by the finite speed of propagation for hyperbolic equations we can always assume that the Cauchy data in (1.1) are compactly supported. We refer to the Kovalevskian energy and the hyperbolic energy introduced above. We note that in the energies in consideration we can assume |ξ | 1 since the continuity of V (t, ξ) in ξ implies that both energies are bounded for |ξ | 1. In particular, the Levi condition (4.10) for |ξ | 1 yields the energy estimate (4.9) for |ξ | for |ξ | 1. At this point setting ε = e −1 ξ −1 we have that there exist constants C > 0 and κ ∈ N such that V (b, ξ) C ξ pq+κ V (a, ξ) , (4.14) for |ξ | 1. This proves the C ∞ well-posedness of the Cauchy problem (1.1). Similarly, (4.14) implies the well-posedness of (1.1) in D (R n ). 2 We notice that it is enough to put Levi conditions on terms up to order −(m − 2) for the C ∞ well-posedness. More precisely, let h := [ m− 1 2 ]. Then, it is enough to put Levi conditions on terms up to order −(2h − 1) in order to get the C ∞ well-posedness. Indeed, if Now, by setting the Levi condition and applying the bound from below for the symmetriser in Lemma 4.2, we obtain This will allow us to proceed with the energy estimate (4.9) even under the new Levi condition (4.15). The Levi conditions (4.10) and (4.15) seem unrelated. We simply note that, in the case of second order equations with principal part D tt u − a(t)D xx , a 0, and arbitrary lower order terms, (4.15) is a condition on both the lower order terms b 1 and b 2 whereas (4.10) leaves b 2 free. Finally, as suggested by the referee, one could slightly generalise the Levi condition (4.10) as follows: (q im b j − b i q jm )(t, ξ) c (t, ξ) + ∂ t (t, ξ) . (4.16) Indeed, (4.16) implies Q (t, ξ)B(t, ξ) − B * (t, ξ)Q (t, ξ) V (t, ξ), V (t, ξ) c 1 + |∂ t (t, ξ)| E(t, ξ) and therefore the inequality (4.9) holds. For the second order equation D tt u − t 2μ D xx u + it ν D x u = 0, μ, ν > 0, the condition (4.16) forces