Hyperbolic second order equations with non-regular time dependent coefficients

In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means to assume that the coefficients are less regular than H\"older. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a `very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.


Introduction
In this paper we study equations of the type where the coefficients are real and a i ≥ 0 for all i = 1, . . . , n. It follows that this equation is hyperbolic (but not necessarily strictly hyperbolic). This kind of equations appears in many physical phenomena where discontinuous or singular entities are involved, for instance in the wave propagation in a layered medium. An example is the wave equation where the coefficients a i are Heaviside or Delta functions. In particular, when n = 2, we can have the equation x 2 u(t, x) = 0, where 0 < t 0 < t 1 ≤ t 2 ≤ T , H t 0 ,t 1 is the jump function with H t 0 ,t 1 (t) = 0 for t < t 0 and t > t 1 and H t 0 ,t 1 (t) = 1 for t 0 ≤ t ≤ t 1 , and δ t 2 is the Delta function concentrated at t 2 . In this paper we will use the expression a real-valued distribution for a distribution u ∈ D(R) such that u(ϕ) ∈ R for all real-valued test functions ϕ. Similarly, we will write u ≥ 0 if u(ϕ) ≥ 0 for all non-negative test functions ϕ ≥ 0. This is clearly the case of the coefficients above.
As usual, we will often rewrite the equation (1) using the notation D t = −i∂ t and D x i = −i∂ x i . The well-posedness of the corresponding Cauchy problem has been studied by many authors in the case of regular coefficients. If the coefficients a i and b i are sufficiently regular we can refer to the fundamental paper by Colombini, de Giorgi and Spagnolo [CDGS79], showing that even if the coefficients are smooth, the well-posedness of the Cauchy problem (3) can be expected to hold only in Gevrey spaces. In fact, the famous example of Colombini and Spagnolo [CS82] shows that even if all b i = 0 and all a i are smooth, the Cauchy problem (3) may not be distributionally well-posed due to multiplicities. On the other hand, if the operator (1) is strictly hyperbolic, it was shown in [CJS87] that the Cauchy problem (3) may be still distributionally not well-posed if the coefficients are less regular, e.g. only Hölder. These examples, already for the second order equations with time-dependent coefficients as in (3), show the following by now well-known qualitative facts: • if the coefficients are smooth and the equation is strictly hyperbolic, the Cauchy problem (3) is distributionally well-posed (of course, much more is known, but it is less important for our purposes here); • if the coefficients are smooth but the equation has multiplicities, then the Cauchy problem (3) may be not distributionally well-posed. However, it becomes well-posed in the appropriate classes of ultradistributions (depending on additional properties of coefficients or characteristic roots); • if the equation is strictly hyperbolic but the coefficients are only Hölder continuous, the Cauchy problem (3) may be not distributionally well-posed. However, it becomes well-posed in the appropriate classes of ultradistributions; • if the coefficients of the equation are continuous (and not Hölder continuous), there may be no ultradistributional well-posedness. However, it may become well-posed in the space of Fourier hyperfunctions.
As we see in the above statements, if we want to continue having a well-posedness result, the reduction in regularity assumptions on the coefficients leads to the necessity to weaken the notion of solution to the Cauchy problem and to enlarge the allowed class of solutions. A threshold between distributional and ultradistributional well-posedness for equations with time-dependent coefficients (on the level of C ∞ and Gevrey well-posedness) in terms of the regularity of coefficients has been discussed by Colombini, del Santo and Reissig [CDSR03]. We note that for x-dependent coefficients the situation becomes even much more subtle: for example, while very general Gevrey well-posedness results are available for Gevrey coefficients (see, e.g. Bronshtein [Bro80] or Nishitani [Nis83]), the C ∞ well-posedness of second order equations with smooth coefficients is heavily dependent on the geometry of characteristics (see, e.g. [BPP12,PP09]).
Again, most of such results can be translated into distributional or ultradistributional well-posedness, but still for equations with smooth or Gevrey coefficients.
The aim of this paper is to analyse the Cauchy problem (3) under much weaker regularity assumptions on coefficients. The general goal of reducing the regularity of coefficients for evolution partial differential equations has both mathematical and physical motivations, and has been thoroughly discussed by Gelfand [Gel63], to which we refer also for the philosophical discussion of this topic.
Before we proceed with our approach, let us mention that the Cauchy problem (3) for operators with irregular coefficients has history and motivations from specific applied sciences. For example, problems of this type appear in geophysical applications with delta-like sources and discontinuous or more irregular media (for example, fractal-type media occurs naturally in the upper crust of the Earth or in fractured rocks), see [MB99], and especially [HdH01] for a more detailed discussion and further references in geophysics and in tomography. Such problems have been treated using microlocal constructions in the Colombeau algebras, see e.g. Hörmann and de Hoop [HdH01,HdH02]. If the coefficients are measurable such equations often fall in the scope of problems which can be handled by semigroup methods, as in Kato [Kat95]. However, to the best of our knowledge, there are no approaches to problems with irregularities like those in (2), providing both a well-posedness statement and a relation to 'classical' solutions.
In this paper, we will look at the Cauchy problem (3) in different settings, the most general being that the coefficients a i and b i are distributions. In this case, in view of the famous Schwartz impossibility result on multiplication of distributions [Sch54], the first question that already arises is how to interpret the equation (1) when u is a distribution as well. And, a related question for our purposes, is how to interpret the notion of a solution to the Cauchy problem (3). In view of the discussion above, it appears natural that in order to obtain solutions in this setting, one should weaken the notion of a solution to the Cauchy problem since ultradistributions or hyperfunctions may not be sufficient for such purpose.
The aim and the main results of this paper are to show that • one can introduce the notion of 'very weak solutions' to the Cauchy problem (3), based on regularising coefficients and the Cauchy data with certain adaptation of Friedrichs mollifiers. Then, one can show that very weak solutions exist even if the coefficients and the Cauchy data are (compactly supported) distributions (Theorem 2.6); • if the coefficients are sufficiently regular, namely, if they are in the class C 2 , the very weak solutions all coincide in a certain sense, and are related to (coincide with) other known solutions. More precisely, if the Cauchy data are Gevrey ultradifferentiable functions, any very weak solution (for any regularisation of the coefficients) converges in the strong sense to the classical solution in the limit of the regularisation parameter. If the Cauchy data are distributions, any very weak solution (for any regularisation of the coefficients) converges in the ultradistributional sense to the ultradistributional solution in the limit of the regularisation parameter. See Theorem 2.7 for a precise formulation.
The appearance of the class C 2 is due to the fact that since we do not assume that the equation is strictly hyperbolic, the C 2 -regularity of coefficients does guarantee that the characteristic roots of (1) are Lipschitz, and hence we know that the Gevrey or ultradistributional well-posedness holds. In the case the equation is strictly hyperbolic, the assertions above still hold if the coefficients are e.g. Lipschitz. Some further refinements are possible given precise relations between regularities of coefficients and roots of a hyperbolic polynomial (Bronshtein's theorem [Bro79] and its refinements as in [COP12]).
The idea of considering regularisations of coefficients or solutions of hyperbolic partial differential equations in different senses is of course natural. For example, after regularising (e.g. non-Lipschitz, Hölder, etc.) coefficients with a parameter ε, relating ε to some frequency zones in the energy estimate often yields the Gevrey or even C ∞ well-posedness (see e.g. [CDGS79,CDSK02], and other papers). It is not always possible to relate ε to frequency zones in which case families of solutions can be considered as a whole: for example, for hyperbolic equations with discontinuous coefficients regularised families have been already considered by Hurd and Sattinger [HS68], with a subsequent analysis of limits of these regularisations in L 2 as ε → 0.
The purpose of this paper is to carry out a thorough analysis of appearing families of solutions and, by formulating a naturally associated notion of 'very weak' solution, to relate it (as ε → 0) to known classical, distributional or ultradistributional solutions.
In the next section we provide more specifics to the above statements. In particular, we briefly review the relevant ultradistributional well-posedness results, and put the notion of a very weak solution to a wider context.
In what concerns the literature review for second order Cauchy problems (3), we will only give very specific references relevant to our subsequent purposes: for 'regular' coefficients much is known, for sharp results see e.g. already Colombini, de Giorgi, Spagnolo [CDGS79], Nishitani [Nis83]. Also, we do not discuss other interesting phenomena on the borderline of the existence of strong solutions (e.g. irregularity in t can be sometimes compensated by favourable behaviour in x, see e.g. Cicognani and Colombini [CC13]).

Main results
As we mentioned in the introduction, already when the coefficients are regular, there are several types of assumptions where we can expect qualitatively different results. On one hand, for very regular data, we may have well-posedness in the spaces of smooth, Gevrey, or analytic functions. At the duality level, this corresponds to the well-posedness in spaces of distributions, ultradistributions, or Fourier hyperfunctions.
We start by recalling the known results for coefficients which are regular: in [GR13], extending the one-dimensional result of Kinoshita and Spagnolo in [KS06], we have obtained the following well-posedness result: (i) If the coefficients a j , b j , j = 1, . . . , n, belong to C k ([0, T ]) for some k ≥ 2 and g j ∈ γ s (R n ) for j = 1, 2 then there exists a unique solution u ∈ C 2 ([0, T ]; γ s (R n )) of the Cauchy problem (3) provided that (ii) if the coefficients are of class C ∞ on [0, T ] then the Cauchy problem (3) is well-posed in any Gevrey space; (iii) under the hypotheses of (i), if the initial data g j are Gevrey Beurling ultradistributions in E ′ (s) (R n ) for j = 1, 2 then there exists a unique solution u ∈ C 2 ([0, T ]; D ′ (s) (R n )) of the Cauchy problem (3) provided that (iv) under the hypotheses of (ii) the Cauchy problem (3) is well-posed in any space of ultradistributions; (v) finally if the coefficients are analytic on [0, T ] then the Cauchy problem (3) is C ∞ and distributionally well-posed.
For the sake of the reader we briefly recall the definitions of the spaces γ s (R n ) and γ (s) (R n ) of (Roumieu) Gevrey functions and (Beurling) Gevrey functions, respectively. These are intermediate classes between analytic functions (s = 1) and smooth functions. In the sequel, N 0 = {0, 1, 2, . . . }.
Definition 2.2. Let s ≥ 1. We say that f ∈ C ∞ (R n ) belongs to the Gevrey (Roumieu) class γ s (R n ) if for every compact set K ⊂ R n there exists a constant C > 0 such that for all α ∈ N n 0 we have the estimate sup We say that f ∈ C ∞ (R n ) belongs to the Gevrey (Beurling) class γ (s) (R n ) if for every compact set K ⊂ R n and for every A > 0 there exists a constant C > 0 such that for all α ∈ N n 0 we have the estimate sup Let now γ (s) c (R n ) be the space of Beurling Gevrey functions with compact support. Its dual is the corresponding space D ′ (s) (R n ) of ultradistributions and E ′ (s) (R n ) is the subspace of compactly supported ultradistributions. We refer to [GR12] for relevant properties and Fourier characterisations of these spaces of ultradifferentiable functions and ultradistributions.
(iii) If the roots are distinct then Gevrey and ultradistributional well-posedness hold provided that respectively.
It is our purpose in this paper to prove well-posedness of the Cauchy problem (3) when the coefficients are less than Hölder.
The first main idea now is to start from distributional coefficients a i and b i , i = 1, . . . , n, to regularise them by convolution with a suitable mollifier ψ obtaining families of smooth functions (a i,ε ) ε and (b i,ε ) ε , namely where ψ ω(ε) (t) = ω(ε) −1 ψ(t/ω(ε)) and ω(ε) is a positive function converging to 0 as ε → 0. It turns out that the nets (a i,ε ) ε and (b i,ε ) ε are C ∞ -moderate, in the sense that their C ∞ -seminorms can be estimated by a negative power of ε (see (22)). More precisely, we will make use of the following notions of moderateness.
In the sequel, the notation K ⋐ R n means that K is a compact set in R n .
is C ∞ -moderate if for all K ⋐ R n and for all α ∈ N n 0 there exist N ∈ N 0 and c > 0 such that sup exists a constant c K > 0 and there exists N ∈ N 0 such that if for all K ⋐ R n there exist N ∈ N 0 , c > 0 and, for all k ∈ N 0 there exist N k > 0 and c k > 0 such that x ∈ K and ε ∈ (0, 1]. We note that the conditions of moderateness are natural in the sense that regularisations of distributions or ultradistributions are moderate, namely we can think that by the structure theorems for distributions, while also the regularisations of the compactly supported Gevrey ultradistributions can be shown to be Gevrey-moderate. Thus, while a solution to a Cauchy problems may not exist in the space on the left hand side of an inclusion like the one in (5), it may still exist (in a certain appropriate sense) in the space on its right hand side. The moderateness assumption will be enough for our purposes. However, we note that regularisation with standard Friedrichs mollifiers will not be sufficient, hence the introduction of a family ω(ε) in the above regularisations.
We can now introduce a notion of a 'very weak solution' for the Cauchy problem (3).
is a very weak solution of order s of the Cauchy problem (3) if there exist (i) C ∞ -moderate regularisations a i,ε and b i,ε of the coefficients a i and b i , respectively, for i = 1, . . . , n, (ii) γ s -moderate regularisations g 0,ε and g 1,ε of the initial data g 0 and g 1 , respectively, such that (u ε ) ε solves the regularised problem for all ε ∈ (0, 1], and is C ∞ ([0, T ]; γ s (R n ))-moderate.
The main results of this paper can be summarised as the following solvability statement complemented by the uniqueness and consistency in Theorem 2.7.
Theorem 2.6. Let the coefficients a i , b i of the Cauchy problem (3) be distributions with compact support included in [0, T ], such that a i , b i are real-valued and a i ≥ 0 for all i = 1, . . . , n. Let the Cauchy data g 0 , g 1 be compactly supported distributions. Then, the Cauchy problem (3) has a very weak solution of order s, for all s > 1.
In fact, Theorem 2.6 will be refined according to the regularity of the initial data. More precisely, we will distinguish between the following cases: Case 1: distributional coefficients and Gevrey initial data; Case 2: distributional coefficients and smooth initial data; Case 3: distributional coefficients and distributional initial data.
The uniqueness and consistency result for very weak solutions of the Cauchy problem (3) is as follows. We distinguish between Gevrey Cauchy data and the general distributional Cauchy data: Theorem 2.7. Assume that the real-valued coefficients a i and b i are compactly supported, belong to C k ([0, T ]) with k ≥ 2 and that a i ≥ 0 for all i = 1, . . . , n. Let 1 < s < 1 + k 2 . • Let g 0 , g 1 ∈ γ s c (R n ). Then any very weak solution (u ε ) ε converges in the space C([0, T ]; γ s (R n )) as ε → 0 to the unique classical solution in C 2 ([0, T ], γ s (R n )).
In particular, this limit exists and does not depend on the C ∞ -moderate regularisation of the coefficients. • Let g 0 , g 1 ∈ E ′ (R n ). Then any very weak solution (u ε ) ε converges in the space . In particular, this limit exists and does not depend on the C ∞ -moderate regularisation of coefficients a i and b i and the Gevrey-moderate regularisation of the initial data g 0 , g 1 .
In Theorem 2.7, we assume that 1 < s < 1+ k 2 in order to make sure that the unique classical or ultradistributional solutions exist, provided by Theorem 2.1. Theorem 2.7 will follow from Theorem 7.1.
The proof of Theorem 2.6 relies on classical techniques for weakly hyperbolic equations (quasi-symmestriser, energy estimates, Gevrey-wellposedness, etc.) and ideas from generalised function theory (regularisation). In particular, proving the existence of a very weak solution coincides, by fixing the mollifiers, with proving well-posedness of the corresponding Cauchy problem in a suitable space of Colombeau type. This space will be chosen according to the regularity of the initial data. So, the proof of Theorem 2.6 will follow from the well-posedness results in Theorems 4.7, 5.3 and 6.3.
We note that the proof of Theorem 2.6 actually provides us with a description of possible regularisations, in particular, of functions ω(ε) used in the regularisation of coefficients in (4). Indeed, ω(ε) will be of the type c(log(ε −1 )) −r or of the type c(log(ε −1 )) −r 1 ε −r 2 , for c > 0 and r, r 1 , r 2 > 0.
We note that the idea of considering regularisations of coefficients and solutions of partial differential equations in different senses has been seen in the literature. For example, after regularising (e.g. non-Lipschitz, Hölder) coefficients with a parameter ǫ, relating ǫ to some frequency zone in the energy estimate often yields the Gevrey or even C ∞ well-posedness (see e.g. Colombini, del Santo and Kinoshita [CDSK02] and other papers). For less regularity, e.g. for hyperbolic equations with discontinuous coefficients regularised families have been already considered by Hurd and Sattinger [HS68], with a subsequent analysis of limits of these regularisations in L 2 as ε → 0. An interesting result of well-posedness has been obtained for discontinuous and in general distributional coefficients in the Colombeau context by Lafon and Oberguggenberger [LO91]. In their paper they proved that first order symmetric systems of differential equations with Colombeau coefficients and Colombeau initial data have a unique Colombeau solution under suitable logarithmic type assumptions on the principal part. This result, while it can be easily extended to pseudo-differential systems, cannot be directly applied to our equation, since the system to which would can reduce our equations is in general, non-symmetric and non-strictly hyperbolic.
It will be useful also to us to use the developed machinery of Colombeau algebras in the proofs. Especially, this will provide an easy-to-get refinement of the uniqueness part of the corresponding statements. However, we need to work in algebras of generalised functions based on regularisations with Gevrey functions since smooth solutions do not have to exist due to multiplicities.
As mentioned above, we will employ quasi-symmetriser techniques, or more precisely, a parametrised version of the quasi-symmetriser seen in [GR13]. This is the topic of the next section.

Parameter dependent quasi-symmetriser
In this paper, we will be applying the standard reduction of a scalar second order equation to the 2 × 2 system: setting we transform the equation We now assume that the equation coefficients are distributions with compact support contained in [0, T ]. Since the formulation of (1) might be impossible due to issues related to the product of distributions, we replace (1) with a regularised equation. In other words, we regularise every a i and b i by convolution with a mollifier in C ∞ c (R n ) and get nets of smooth functions as coefficients. More precisely, let ψ ∈ C ∞ c (R), ψ ≥ 0 with ψ = 1 and let ω(ε) be a positive function converging to 0 as ε → 0.
. By the structure theorem for compactly supported distributions, we have that there exists L ∈ N 0 and c > 0 such that for all i = 1, . . . , n. Regularising the equation (1) means equivalently to regularise the system (6) as Note that λ 1,ε ξ and λ 2,ε ξ are the roots of the characteristic polynomial and fulfil the inequality employed by Kinoshita and Spagnolo in [KS06] to obtain Gevrey well-posedness for the corresponding Cauchy problem. It is clear that the regularised equation (1) and the corresponding first order system have solutions (u ε ) ε and (U ε ) ε , respectively, depending on the parameter ε ∈ (0, 1]. By Fourier transformation in x the system where V ε (t, ξ) = (F U ε (t, ·))(ξ). Finally, by regularising the initial data as well if needed (for instance in Case 3), we transform the Cauchy problem (3) into The well-posedness of this regularised Cauchy problem will be obtained by constructing a quasi-symmetriser for the matrix A ε and the corresponding energy. Before proceeding with the technical details we recall some general basic facts. For more details see [DS98,KS06].
where the matrices Q (vii) There exists a constant C m such that We finally recall that a family {Q α } of nonnegative Hermitian matrices is called nearly diagonal if there exists a positive constant c 0 such that Lemma 3.2. Let {Q α } be a family of nonnegative Hermitian m × m matrices such that det Q α > 0 and det Q α ≥ c q α,11 q α,22 · · · q α,mm for a certain constant c > 0 independent of α. Then, Lemma 3.2 is employed to prove that the family Q Then the family of matrices {Q (m) δ (λ) : 0 < δ ≤ 1, λ ∈ S M } is nearly diagonal. We conclude this section with a result on nearly diagonal matrices depending on three parameters (i.e. δ, t, ξ) which will be crucial in the next section. Note that this is a straightforward extension of Lemma 2 in [KS06] valid for two parameter (i.e. δ, t) dependent matrices.
be a nearly diagonal family of coercive Hermitian matrices of class C k in t, k ≥ 1. Then, there exists a constant C T > 0 such that for any non-zero continuous function V : 3.2. The quasi-symmetriser of the matrix A ε . We now focus on the matrix A ε corresponding to the Cauchy problem we are studying. It is clear that we will get a family of quasi-symmetrisers (Q (2) δ (λ ε )) ε , where λ ε = (λ 1,ε , λ 2,ε ). More precisely, by direct computations we get where λ 1,ε and λ 2,ε are defined as in (7). Thus, Note that from the formula (7), λ 1,ε and λ 2,ε are nets of smooth functions fulfilling the estimate , ξ ∈ R n and ε ∈ (0, 1]. Finally, since λ 1,ε and λ 2,ε in (7) are roots of a second order equation, they fulfil , so the condition on the roots used in [KS06] and in [GR13] is trivially fulfilled with constant M = 2 (see (6) in [GR13]).
Note that . Adopting the notations of [KS06] we then have that the bound from below (19) in [KS06] is fulfilled with c 0 = 1 8 . This means that the family of matrices A careful analysis of the proof of Lemma 2 in [KS06] allows us to extend Lemma 3.4 to the family of quasi-symmetrisers (Q in Lemma 2 is in our case equal to (1/8) −(1−1/k) .

Lemma 3.6. Let {Q
(2) δ,ε (t, ξ) : 0 < δ ≤ 1, 0 < ε ≤ 1, 0 ≤ t ≤ T, ξ ∈ R n } be the nearly diagonal family of quasi-symmetrisers introduced above. Then, for any continuous for all ξ ∈ R n , δ ∈ (0, 1] and ε ∈ (0, 1]. We are now ready to prove the well-posedness of the Cauchy problem (3). This will consist of two parts: (i) choice of the framework, (ii) energy estimates. We begin by considering Case 1: distributional coefficients and Gevrey initial data 4. Case 1: well-posedness for Gevrey initial data We want to prove the well-posedness of the Cauchy problem (3) when the coefficients of the equation are distributions with compact support and the initial data are compactly supported Gevrey functions. This will be achieved in a suitable algebra of Colombeau type containing the usual Gevrey classes as subalgebras. We start by developing these objects. 4.1. Gevrey-moderate families. We begin by investigating the convolution of a compactly supported Gevrey function with a mollifier ϕ ∈ S (R n ) with ϕ(x) dx = 1 and x α ϕ(x) dx = 0 for all α = 0 and ϕ ε (x) := ε −n ϕ(x/ε). The following holds: Proposition 4.1. Let σ > 1. Let u ∈ γ σ c (R n ) and let ϕ be a mollifier as above. Then (i) there exists c > 0 such that x ∈ R n and ε ∈ (0, 1]; (ii) there exists c > 0 and for all q ∈ N 0 a constant c q > 0 such that for all ξ ∈ R n and ε ∈ (0, 1]. Proof. (i) By convolution with the mollifier ϕ ε and straightforward estimates we obtain for all α ∈ N n 0 , x ∈ R n and ε ∈ (0, 1]. (ii) Analogously, by Taylor expansion and the properties of the mollifier ϕ (in particular since x α ϕ(x) dx = 0 for all α = 0) we get for any q ∈ N 0 the following estimate: Note that the estimate above holds for all α ∈ N n 0 and q ∈ N 0 uniformly in x ∈ R n and ε ∈ (0, 1]. (iii) By Fourier transform we get that u * ϕ ε (ξ) = u(ξ) ϕ ε (ξ) = u(ξ) ϕ(εξ) and therefore since u ∈ γ σ c (R n ) and ϕ ∈ S (R n ) the third assertion is trivial.
In Definition 2.4 we introduced the notion of a moderate net, i.e., a net of functions (f ε ) ε ∈ γ σ (R n ) (0,1] is γ s -moderate if for all K ⋐ R n there exists a constant c K > 0 and there exists N ∈ N 0 such that for all α ∈ N n 0 , x ∈ K and ε ∈ (0, 1]. Analogously one can talk of γ σ -negligible nets. Definition 4.2. Let σ ≥ 1. We say that (u ε ) ε is γ σ -negligible if for all K ⋐ R n and for all q ∈ N 0 there exists a constant c q,K > 0 such that for all α ∈ N n 0 , x ∈ K and ε ∈ (0, 1]. We can now prove the following proposition. (i) If (u ε ) ε is γ σ -moderate and there exists K ⋐ R n such that supp u ε ⊆ K for all ε ∈ (0, 1] then there exist c, c ′ > 0 and N ∈ N 0 such that for all ξ ∈ R n . (ii) If (u ε ) ε is γ σ -negligible and there exists K ⋐ R n such that supp u ε ⊆ K for all ε ∈ (0, 1] then there exists c > 0 and for all q > 0 there exists c q > 0 such that Proof. (i) By elementary properties of the Fourier transform and since supp u ε ⊆ K for all ε we have that for all α ∈ N n 0 and ξ ∈ R n . Let us now write ξ 2M | u ε (ξ)| 2 as where c α > 0. Hence, from (19) we have and, therefore, from α! ≤ |α| |α| , we conclude It is clear that this last estimate implies uniformly in ε ∈ (0, 1] and ξ ∈ R n . Note that by direct computations on the binomial coefficients one can see that the constant c ′ M is of the type C ′M so for some suitable constant C > 0. It follows that and therefore introducing a suitable constant ν > 0 (depending on σ) we have that for all ξ ∈ R n and ε > 0. Concluding, recognising the Taylor series of an exponential in the previous formula, we arrive at for a suitable constants c, c ′ > 0 as desired.
(ii) The proof in (i) can be repeated for a γ σ c -negligible net (u ε ) ε . From the assumption of negligibility it is immediate to see that the estimate | u ε (ξ)| ≤ c q ε q e −cε 1 σ ξ 1 σ , holds uniformly in ξ and ε.
(iii) If (u ε ) ε is a net of tempered distributions satisfying (i) then by the Fourier characterisation of Gevrey functions (u ε ) ε is a net of Gevrey functions of order σ. More precisely, Assume now that |ξ| ≥ 1. Hence Note that there exists a constant c σ > 0 such that Finally combining (20) with (21) we conclude that there exists a constant C > 0 such that for all x ∈ R n and ε ∈ (0, 1]. (iv) If (u ε ) ε is a net of tempered distributions satisfying (ii) then by calculations analogous to the ones above (replacing −N with q) we have that for all ε ∈ (0, 1] and x ∈ R n . Making use of the previous definitions of γ σ -moderate and negligible net (see Definition (4.2) and the paragraph above),we introduce the quotient space We now investigate the relationship between G σ (R n ) and the classical Colombeau algebra We recall that a net (u ε ) ε is C ∞ -moderate is for all K ⋐ R n and all α ∈ N n 0 there exist c > 0 and N ∈ N 0 such that for all x ∈ K and ε ∈ (0, 1]. A net (u ε ) ε is C ∞ -negligible is for all K ⋐ R n , all α ∈ N n 0 and all q ∈ N 0 there exists c > 0 such that uniformly in x ∈ K and ε ∈ (0, 1]. For the general analysis of G(R n ) we refer to e.g. Oberguggenberger [Obe92].
Proposition 4.4. For all σ ≥ 1, Proof. To prove that G σ (R n ) is a subalgebra of G(R n ) we need to prove that γ σmoderate and γ σ -negligible nets are elements of E M (R n ) and N (R n ), respectively and that if a γ σ -moderate net belongs to N (R n ) then it is automatically γ σ -negligible. The first two implications are clear from the definition of γ σ -moderate and γ σ -negligible net. Finally, if (u ε ) ε is γ σ -moderate and belongs to N (R n ) then for all K ⋐ R n we have Choosing q = 2q ′ + N and by simple estimates we get The quotient space G σ (R n ) is a sheaf. This means that one can introduce a notion of restriction and a notion of support. More precisely, x ∈ R n \ supp u if there exists an open neighbourhood V of x such that u| V = 0 in G σ (V ). Define G σ c (R n ) as the algebra of compactly supported generalised functions in G σ (R n ). Making use of the previous arguments on γ σ -moderate and -negligible nets we can prove the following proposition.
(i) If u ∈ G σ (R n ) has compact support then it has a representative (u ε ) ε and a compact set K such that supp u ε ⊆ K uniformly in ε.
We begin by recalling that if u ∈ G(R n ) has compact support then it has a representative (u ε ) ε with supp u ε contained in a compact set K uniformly with respect to ε. In other words, there exists ψ ∈ C ∞ c (R n ) identically one on a neighbourhood of supp u such that ψu = u in G(R n ). It follows that if u ∈ G σ (R n ) has compact support then ψu = u in G σ (R n ). Indeed, This means that (ψu ε ) ε is γ σ c -moderate. Since ψu ε − u ε is γ σ -moderate and belongs to N (R n ) as well, we conclude that (ψu ε − u ε ) ε is γ σ -negligible.
An analogous version of Proposition 4.5 can be proven for G σ (R n ) and γ σ (R n ), but it goes beyond the purpose of this paper.
In this paper we will also make use of the following factor space.
Note that the estimates in Definition 4.6 express the usual Colombeau properties in t and the new Gevrey-Colombeau features in x and that Moreover, in G([0, T ]; G σ (R n )) one can make use, at the level of representatives, of the characterisations by Fourier transform seen above (uniformly in t ∈ [0, T ]).

4.2.
Energy estimate and well-posedness. Let us define the energy It follows that , provided that V = 0. Hence, we can rewrite (24) as In the following we take any fixed integer k ≥ 2. Writing now from the bound from below in Proposition 3.5(i), Lemma 3.6 and the estimates on the roots λ i,ε (t, ξ), i = 1, 2, we have that uniformly in all the variables and parameters. Combining now (27) with the estimate on |((Q δ,ε )(t, ξ)V (t, ξ), V (t, ξ))| above, by Gronwall lemma we obtain Making use of the estimates in Proposition 3.5(i), of the definition of Q (2) δ,ε and of the fact that ω(ε) −1 ≥ 1, we obtain This implies, for M = (3L + k)/k, or equivalently for a suitable constant C > 0. We begin by assuming that the initial data are in γ s (R n ). This means that Since our solution is depending on the parameter ε from now on we will adopt the notation V ε . Note that when the initial data are in γ s c (R n ) we do not need any regularisation to embed them in the algebra G s (R n ), due to Proposition 4.1(ii). Hence, and by simple estimates If s < σ, the condition or, in other words, to the condition Assume now that ω(ε) −1 is moderate, i.e. ω(ε) −1 ≤ cε −r for some r ≥ 0. Hence, there exists N ∈ N 0 such that under the assumption (31) the estimate (30) yields which proves that the net U ε = F −1 (V ε 1 ξ ≥Rε ) is γ s -moderate. It remains to estimate V ε (t, ξ) when ξ ≤ R ε . Going back to (30) we have that if ξ ≤ R ε then At this point, choosing ω(ε) −M R ε 1 σ of logarithmic type, i.e., we can conclude that there exists N ∈ N 0 and c ′ , C ′ > 0 such that for all ε ∈ (0, 1], t ∈ [0, T ] and ξ ≤ R ε . This together with (32) and Proposition 4.3(iii) shows that the net (U ε (t, ·)) ε is γ s -moderate on R n for 1 < s < σ = 1 + k 2 .
We are now ready to state and prove the following well-posedness theorem.
Theorem 4.7. Let where the coefficients a i and b i are real-valued distributions with compact support contained in [0, T ] and a i is non-negative for all i = 1, . . . , n. Let g 0 and g 1 belong to γ s c (R n ) with s > 1. Then there exists a suitable embedding of the coefficients a i 's and b i 's into G([0, T ]) such that he Cauchy problem above has a unique solution u ∈ G([0, T ]; G s (R n )).
Proof. We begin by writing the equation ). This means that we replace the coefficients a i and b i with the equivalence classes of (a i,ε ) ε and (b i,ε ) ε in G([0, T ]) as in Section 3.
Since the initial data are in G s (R n ) they can be imbedded in G s (R n ) as they are, i.e.
[(g 0 )] ∈ G s c (R n ) and [(g 1 )] ∈ G s c (R n ). Existence. We argue now at the level of representatives and we transform the equation to the first order system (6). From the theory of weakly hyperbolic equations and in particular from [GR13,KS06] we know that that the Cauchy problem with initial data g 0 , g 1 ∈ γ s c (R n ), has a net of (classical) solutions (u ε ) ε ∈ C 2 ([0, T ] : γ s (R n )). More precisely, we know that given s > 1 and for k ≥ 2 there exists a solution (u ε ) ε ∈ C 2 ([0, T ] : γ s (R n )) provided that So, in the arguments which follow we assume s and k in this relation and we perform the embedding of the coefficients a i and b i with a logarithmic scale of the type ω −1 (ε) = c(log(ε −1 )) r , c ≥ 0, as in (33), where r depends on s and k.
It is our task to show that this net is moderate. From the energy estimates in Subsection 4.2 at the Fourier transform level we have that the net (u ε ) ε (or better the corresponding (U ε ) ε ) is γ s (R n )-moderate with respect to x with s as above. Since this moderateness estimate is uniform in t and the coefficients of the equation are smooth and moderate in t ∈ [0, T ] as well, by induction on the t-derivatives and arguing as in [LO91] we can easily conclude that (u ε ) ε is C ∞ ([0, T ]; γ σ (R n ))-moderate for Hence, (u ε ) ε generates a solution u ∈ G([0, T ]; G s (R n )) to our Cauchy problem.
Uniqueness. Assume now that the Cauchy problem has another solution v ∈ G([0, T ]; G s (R n )). At the level of representatives this means ; γ s (R n ))-negligible and (n 0,ε ) ε and (n 1,ε ) ε are both compactly supported and γ s (R n )-negligible. The corresponding first order system is where w 1,ε and w 2,ε are obtained via the transformation This system will be studied after Fourier transform, as a system of the type These kind of systems and the corresponding weakly hyperbolic equations (with right hand-side) have been investigated in [GR12] under even less regular assumptions on the coefficients (Hölder). In particular, see Theorem 3 in [GR12], Gevrey wellposedness results have been obtained for The proof of Theorem 3 in [GR12] can be easily adapted to our situation by inserting everywhere a multiplicative factor ω(ε) −L coming from the regularisation of the coefficients and by replacing e −ρ(t) ξ 1 s with e −ρ(t)ε 1 s ξ 1 s in the formula (4.1) defining V in [GR12]. The estimate (4.9) in [GR12] is therefore transformed into where N ∈ N 0 depends on the equation or better on the regularity of the coefficients and κ 1 , κ 2 > 0 can be chosen small enough. It follows that since the initial data V ε (0, ξ) and the right-hand side F ε (t, ξ) are negligible then (V ε ) ε is negligible as well in the suitable function spaces, or in other words, (u ε − v ε ) ε is C([0, T ], γ s (R n ))negligible. From the equation itself and the fact that the coefficients are nets of smooth functions one can deduce that the net (u ε − v ε ) ε is smooth in t as well and more precisely that it is C ∞ ([0, T ], γ s (R n ))-negligible. This proves that u = v in G([0, T ]; G s (R n )).

Case 2: well-posedness for smooth initial data
We now work under the assumption that the initial data g 0 and g 1 are not Gevrey but still smooth. More precisely, g 0 , g 1 ∈ C ∞ c (R n ). By convolution with a mollifier ϕ ε as in Case 1 we get a net of smooth functions. It is our aim to find for a function u ∈ C ∞ c (R n ) a new regularisation of the type u * ρ ε such that the corresponding net is Gevrey. This will allow us to embed the initial data g 0 and g 1 in an algebra of Gevrey-Colombeau type and to proceed with the well-posedness of the Cauchy problem (3). We begin with the following regularisation inspired by [BB09].
In the sequel ι denotes the map By embedding of C ∞ (R n ) into the Colombeau algebra G(R n ) it follows that u = 0. This shows that the map ι is injective.
Hence, by the properties of χ and by the vanishing moments of ϕ, for any integer q > 1 we get This proves that the net (u * φ ε − u * ρ ε )(x) is γ σ -negligible.
Concluding, we can state that the algebra G σ c (R n ) contains not only γ σ c (R n ) but also C ∞ c (R n ) as a subalgebra. This is obtained by modifying the embedding from u * ϕ ε in Section 4 to u * ρ ε . 5.2. Energy estimates and well-posedness. We now take initial data g 0 , g 1 in C ∞ c (R n ) and we embed them in G s c (R n ) as g 0 * ρ ε and g 1 * ρ ε . By repeating the transformation into first order system and the energy estimates of Case 1 at the Fourier transform level we arrive at (29), i.e.
for a suitable constant C > 0 and M = (3L + k)/k. Since Recall that s > 1 and that k is any fixed integer with k ≥ 2. Now, if s < σ, the following inequalities are equivalent: or, in other words, As in the previous case we take ω(ε) −1 moderate. Under the assumption (42) the estimate (41) implies, for some N ∈ N 0 , This shows that the net U ε = F −1 (V ε 1 ξ ≥Rε ) is γ s -moderate. We still have to estimate V ε (t, ξ) when ξ ≤ R ε . Going back to (41) we have that if ξ ≤ R ε then we can conclude that there exists N ∈ N 0 and c ′ , C ′ > 0 such that for all ε ∈ (0, 1], t ∈ [0, T ] and ξ ≤ R ε . Combing this last estimate with (43) we can conclude, by Proposition 4.3(iii), that, as in the previous case, the net (U ε (t, ·)) ε is γ s -moderate on R n for 1 < s < σ = 1 + k 2 .
We are now ready to state the following well-posedness theorem.
Theorem 5.3. Let where the coefficients a i and b i are real valued distributions with compact support contained in [0, T ] and a i is non-negative for all i = 1, . . . , n. Let g 0 and g 1 belong to C ∞ c (R n ). Then, for all s > 1 there exists a suitable embedding of the coefficients a i and b i into G([0, T ]) such that the Cauchy problem above has a unique solution u ∈ G([0, T ]; G s (R n )).
The energy estimates of Subsection 5.2 and the same arguments of Case 1 show the existence of a solution u ∈ G([0, T ]; G s (R n )). The uniqueness of the solution u is obtained as in the proof of Theorem 4.7. 6. Case 3: well-posedness for distributional initial data We pass now to consider distributional initial data, i.e. g 0 , g 1 ∈ E ′ (R n ), and to investigate their convolution with the mollifier ρ ε .
It follows that the net (u * ρ ε ) ε is γ σ c -moderate and therefore from Proposition 4.3 we have that there exists c > 0 and N ∈ N 0 such that for all ξ ∈ R n and ε small enough (from the proof, ε ∈ (0, 1/2]).
Remark 6.2. Starting from Proposition 6.1 and arguing as for the embedding of E ′ (R n ) into G(R n ) one can easily prove that E ′ (R n ) → G σ c (R n ) : u → [(u * ρ ε ) ε ] is an embedding of E ′ (R n ) into G σ c (R n ). 6.1. Energy estimates and well-posedness. Let us now consider the Cauchy problem a i (t)D 2 x i u(t, x) = 0, u(0, x) = g 0 , D t u(0, x) = g 1 , with g 0 , g 1 ∈ E ′ (R n ). We embed coefficients and initial data in the corresponding Colombeau algebras and we transform the equation into a first order system similarly to Case 1 and 2. In, particular from Proposition 6.1 we have in this case that the initial data V ε (0, ξ) fulfils |V ε (0, ξ)| ≤ ε −N C ′ 0 e −C 0 ε 1 s ξ 1 s , for some N ∈ N 0 . This modifies the estimates of Case 2 only by a multiplying factor ε −N . So for R ε as in (42) we get that there exists N ′ ∈ N 0 such that for all ε, t ∈ (0, T ] and ξ ∈ R n . This result allows us to state the following wellposedness theorem.
Theorem 6.3. Let where the coefficients a i and b i are real valued distributions with compact support contained in [0, T ] and a i is non-negative for all i = 1, . . . , n. Then, the conclusion of Theorem 5.3 holds for initial data g 0 and g 1 in E ′ (R n ) as well.

Consistency with the classical well-posedness results
We conclude this paper by showing that when the coefficients are regular enough and the initial data are Gevrey then the very weak solution coincides with the classical and ultradistributional ones obtained in [GR13,KS06].
Theorem 7.1. Let where the real-valued coefficients a i and b i are compactly supported, belong to C k ([0, T ]) with k ≥ 2 and a i ≥ 0 for all i = 1, . . . , n. Let g 0 and g 1 belong to γ s c (R n ) with s > 1. Then (i) there exists an embedding of the coefficients a i 's and b i 's, i = 1, . . . , n, into G([0, T ]), such that the Cauchy problem above has a unique solution u ∈ G([0, T ]; G s (R n )) provided that 1 < s < 1 + k 2 ; (ii) any representative (u ε ) ε of u converges in C([0, T ]; γ s (R n )) as ε → 0 to the unique classical solution in C 2 ([0, T ], γ s (R n )) of the Cauchy problem (50); (iii) if the initial data g 0 and g 1 belong to E ′ (R n ) then any representative (u ε ) ε of u converges in C([0, T ]; D ′ (s) (R n )) to the ultradistributional solution in C 2 ([0, T ], D ′ (s) (R n )) of the Cauchy problem (50).
Proof. (i) From Section 4 (Case 1) we know that by embedding coefficients and initial data in the corresponding Colombeau algebras the Cauchy problem has a unique solution u ∈ G([0, T ]; G s (R n )). It also has a unique classical solution u ∈ C 2 ([0, T ], γ s (R n )).
(ii) We now want to compare u with u. By definition of classical solution we know that Since the initial data do not need to be regularised because they are already Gevrey there exists a representative (u ε ) ε of u such that for suitable embeddings of the coefficients a i and b i . Noting that the nets (a i,ε − a i ) ε and (b i,ε − b i ) ε are converging to 0 in C([0, T ] × R n ) for i = 1, . . . , n we can rewrite (51) as where n ε ∈ C([0, T ]; γ s (R n )) and converges to 0 in this space. From (53) and (52) we get that u − u ε solves the Cauchy problem By the energy estimates of Case 1 and arguing as in the uniqueness proof of Theorem 4.7 to deal with the right-hand side we arrive after reduction to a system and by application of the Fourier transform to estimate |( V − V ε )(t, ξ)| as in (34), in terms of ( V − V ε )(0, ξ) and the right-hand side n ε (t, x). In particular, since the coefficients are regular enough (of class C k , k ≥ 2), the term ω(ε) −N disappears in (34) and we simply get (54) |( V − V ε )(t, ξ)| ≤ c 1 ξ N e κ 1 ε 1 s ξ 1 s |( V − V ε )(0, ξ)| + c 2 ξ N e κ 2 ε 1 s ξ 1 s | n ε (t, ξ)| Since ( V − V ε )(0, ξ) = 0 and n ε → 0 in C([0, T ]; γ s (R n )) we conclude that u ε → u in C([0, T ]; γ s (R n )). Moreover, since any other representative of u will differ from (u ε ) ε by a C ∞ ([0, T ]; γ s (R n ))-negligible net, the limit is the same for any representative of u.