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Initial-Boundary Problems for Semilinear Hyperbolic Systems with Singular Coefficients

We use the framework of Colombeau algebras of generalized functions to study existence and uniqueness of global generalized solutions to mixed non-local problems for a semilinear hyperbolic system. Coefficients of the system as well as initial and boundary data are allowed to be strongly singular, as the Dirac delta function and derivatives thereof. To obtain the existence-uniqueness result we prove a criterion of invertibility in the full version of the Colombeau algebras.

Our goal is to find global solutions to problem (1)-(3) when the data Λ, A, B, C, D, and H are allowed to be strongly singular (at least of the Dirac delta type). This entails multiplication of distributions in (1) and (3). Indeed, since initial singularities expand from ∂Π into Π along characteristic curves of (1), one can expect that solutions within Π are at least as singular as they are on ∂Π. Furthermore, since the characteristics of (1) are singular themselves, we also meet the problem of composition of two singular functions (for instance, the composition of singular initial data with singular characteristic curves). It is known [3] that even if F is a regular function, but S is a singular one, then F (S(x)) is not welldefined in D ′ . Finally, it should be noted that such three ingredients as singularities, nonlinear operations, and differentiation, cannot be presented unrestrictedly within D ′ . All this makes impossible to use the framework of the distribution theory for our purpose. Nevertheless, such a differential-algebraic structure as an algebra of generalized functions is able to deal with the above problems in a quite reasonable way. We here use the Colombeau version G of an algebra, which is defined on any domain in R n as well as on its closure, is a sheaf, and admits restrictions to the coordinate planes.
We hence assume that entries of A are generalized functions in the Colombeau algebra G[−L, L], entries of B, C, and H are from G(R + ), and entries of Λ and D are from G(Π).
Another advantage of using Colombeau algebra of generalized functions lies in the fact that in a variety of important cases the division by generalized functions, in particular the division by discontinuous functions and measures, is defined in G. The latter, of course, is impossible in D ′ . We completely describe the cases when the division is possible by obtaining a criterion of invertibility in G(Ω).
The plan of our exposition is as follows. Section 2 presents some preliminaries. In Section 3 we extend the criterion of invertibility from the simplified version of Colombeau algebra G s (Ω) (see [4]) to its full version G(Ω). The main result of the paper is given in Section 4, where we prove the global existence-uniqueness theorem within G(Ω).
A novelty of the paper is that it treats singular coefficients in (1) in the context of mixed problems for a quite wide range of boundary conditions which can be classical as well as nonclassical (nonseparable and integral).
Existence-uniqueness results within Colombeau algebras for two-dimensional hyperbolic problems with discontinuous coefficients were studied in [6,9,11,12]. Note that the discontinuity implies global boundedness estimates on the coefficients within Colombeau algebra G, thereby avoiding the negative effect of infinite propagation speed. At the present paper we do not impose the assumption of global boundedness on coefficients of (1), thereby allowing them to be strongly singular. In [10] the authors use the Colombeau algebra of tempered generalized functions G τ to succeed with strongly singular coefficients in Cauchy problems for hyperbolic systems.

Preliminaries
In this section we summarize the relevant material on Colombeau algebras of generalized functions.
Let Ω ⊂ R n be a domain in R n . We denote by G(Ω) and G(Ω) the full version of Colombeau algebra of generalized functions over Ω and Ω, respectively. To define G(Ω) and G(Ω), we introduce the mollifier spaces in order to parametrize the regularizing sequences of generalized functions. For q ∈ N 0 denote We now introduce the algebra of moderate elements E M (Ω) in the following way. Define Now E M (Ω), is defined to be a subalgebra of E(Ω) consisting of elements u ∈ E(Ω) with the following property: The ideal N (Ω) consists of all u ∈ E M (Ω) such that ∀K ⊂ Ω compact , ∀α ∈ N n 0 , ∃N ∈ N such that ∀q ≥ N, ∀ϕ ∈ A q (R n ), Finally, This is an associative, commutative differential algebra. The algebra G(Ω) on open set is constructed in the same manner (with Ω in place of Ω in the definition above). Note that G(Ω) admits a canonical embedding of D ′ (Ω). We will use the notation U = [(u(ϕ, x)) ϕ∈A 0 (R n ) ] for elements U of Colombeau algebra G(Ω) with u(ϕ, x) to be a representative of U.
To reduce information from generalized functions to the level of distributions, we use the notion of an associated distribution. We say that U ∈ G(Ω) admits f ∈ D ′ (Ω) as associated distribution (or U is associated to f ), denoted by U ≈ f , if for all ψ ∈ D(Ω) there exists N ∈ N such that

Criterion of invertibility in the full version of Colombeau algebra of generalized functions
In spite of the fact that G(Ω) is not a field, the division by singular distributions (in particular, by discontinuous functions and measures) is sometimes possible. It is given by the following criterion of (multiplicative) invertibility.
Let Ω ⊂ R n be an arbitrary subdomain in R n .
Theorem 1 Let U ∈ G(Ω) (U ∈ G(Ω)). Then the following two conditions are equivalent: Note that the criterion of invertibility for the simplified version of Colombeau algebra G s (Ω), where Ω is open, was proved in [4].
Proof. We use the argument similar to that presented in [4]. We prove the desired assertion for an arbitrary fixed open set Ω (the proof for the closed set Ω is similar).
Fix an arbitrary compact set K ⊂ Ω. We first prove that there exists p ∈ N such that for all ϕ ∈ A p (R n ) there is η > 0 with v(ϕ ε , x) = 0 for all x ∈ K and 0 < ε < η. Assume, to the contrary, that the latter is not true. This means that for each p ∈ N there exist ϕ ∈ A p (R n ), a sequence ε n ց 0, and a sequence x n ∈ K such that v(ϕ εn , x n ) = 0 for all n ≥ 1. Hence 0 = u(ϕ εn , x n )v(ϕ εn , x n ) = 1 + n(ϕ εn , x n ) and finally n(ϕ εn , x n ) = −1 for all n ≥ 1, a contradiction to the fact that N ∈ N (Ω).
is true for all sufficiently small ε. Since K is an arbitrary compact subset of Ω, the desired assertion follows.
(ii) ⇒ (i). Consider a covering (K i ) i∈N of Ω by compact sets K i such that This fact is true due to the sheaf properties of G(Ω). Set Fix an arbitrary i ∈ N. By assumption, there exists p ∈ N such that for all ϕ ∈ A p (R n ) there is a constant η(ϕ) > 0 such that the expression 1/u(ϕ ε , x) exists for all 0 < ε < η(ϕ) and for all for sufficiently small ε and from the moderateness of ∂ α u(ϕ, x). Proceeding similarly with the higher-order derivatives of From the definition of Colombeau generalized functions and the construction of ✷ We now take into account the definition of Colombeau generalized numbers and the fact that an element U ∈ G(Ω) is a constant iff there is r ∈ C such that U −r = 0 in G(Ω). The following corollary provides a criterion of invertibility of Colombeau generalized numbers within the full version of Colombeau algebras. For the same result within the simplified version of Colombeau algebras see [4].
Corollary 2 Let r ∈ C. Then the following two conditions are equivalent: (i) r is invertible in C, i.e., there exists s ∈ C with rs = 1 in C.
This example shows that within G(Ω) the division by the derivatives of the deltafunction is possible.
Then the multiplicative inverse of U is unique.
Proof. We prove the desired assertion for an open set Ω (the proof for the closed set Ω is similar).
Assume, to the contrary, that U possesses two multiplicative inverses V 1 , V 2 ∈ G(Ω). This implies the equality We conclude from Theorem 1, specifically from the local invertibility estimate, that U ∈ N (Ω), hence that V 1 − V 2 ∈ N (Ω), and finally that V 1 = V 2 in G(Ω), a contradiction to our assumption. ✷

Existence and uniqueness of Colombeau generalized solutions
In this section we develop the results of [8] to the case of singular coefficients in (1). Simultaniously, we consider less restrictive conditions on the initial data in (2) and (3). To prove a general global existence and uniqueness result in Colombeau algebra of generalized functions, we need the following definition of generalized functions of a less restrictive growth if comparing with 1/ε-growth (see the definition of E M ).
Note that this definition generalizes Definition 2 from [8].
We now make several assumptions on the initial data of problem (1)-(3). Let γ(ε) and γ 1 (ε) be positive functions from (0, 1) to (0, ∞) having the properties for each N ∈ N. Assume that 1. The mapping U → f (x, t, U) and all its derivatives are polynomially bounded, uniformly over (x, t) varying in compact subsets of Π.
2. The mapping U → ∇ U f (x, t, U) is globally bounded, uniformly over (x, t) varying in compact subsets of Π.
4. Λ i and D ij for i ≤ n and j ≤ n are locally of γ-growth on Π.
5. B ij and C ij for i ≤ n and j ≤ n are locally of γ-growth on [0, ∞).
6. ∂ x Λ i for i ≤ n are locally of γ 1 -growth on Π.

The determinant of the matrix
Let U ∈ G(Ω) and a smooth function g(x) be slowly increasing at the infinity. By the definition of G(Ω), we have g(U) ∈ G(Ω). Due to this fact and Assumption 1, f (x, t, U) is a well-defined element of G(Ω). Condition 2 is, in fact, sufficient and is imposed to ensure the global classical solvability of problem (1)-(3) with smooth initial data. We need Assumption 7 to transform the initial problem into an equivalent integral-operator form. Assumption 8 ensures the compatibility of (2) and (3) of any desired order. The hyperbolicity of system (1) is ensured by Assumption 3.
The point of Assumption 4 is that it allows us to consider Λ i , B ij (t), C ij (t), and D ij (x, t) being discontinuous functions, the delta functions, and the derivatives thereof. An illustration of this fact is given by Example 3 if one takes γ(l(ϕ)) in place of 1/l(ϕ), where γ is specified by γ(p) = 1 2 log log log (1/p). If one takes in addition γ 1 (p) = log log (1/p), the same example shows that Assumptions 4 and 5 on Λ do not contradict one another.
We are prepared to state the main result of the paper. Proof. We first transform problem (1)-(3) into an equivalent integral-operator form. Note that all algebraic operations as well as operation of integration over finite intervals will be carried out on the level of representatives. Denote by ω i (τ ; x, t) the i-th characteristic of (1) passing through a point (x, t) ∈ Π, i.e., ξ = ω i (τ ; x, t) is the solution to the Cauchy problem: The smallest value of τ ≥ 0 at which the characteristic ξ = ω i (τ ; x, t) intersects ∂Π will be denoted by t i (x, t). By Assumption 7 and Theorem 1, det R(t) has an inverse with entries in G(Π). Using in addition Proposition 4, we conclude that there exists a unique element (det R) −1 ∈ G(Π) such that det R (det R) −1 = 1. This means that the local part of boundary conditions (3) is solvable with respect to those components of U whose characteristics move into Π. Using this fact and integrating each equation of (1) along the corresponding characteristic curve, we obtain the following integral-operator form of (1)- (3): where It is easy to see that problems (1)-(3) and (5) are equivalent in G(Ω). Given T > 0, denote In [8] we proved that problem (1)-(3) with smooth initial data has a unique smooth solution in Π T , whatsoever T > 0. For this purpose we used the contraction mapping principle and obtained local smooth solution. In parallel, we obtained local a priori estimates for the latter. To obtain global smooth solution, we used finite iteration of the local a priori estimates. We also derived global apriori estimates for this solution.
To prove the existence of a generalized solution to the problem under consideration, let us rewrite just mentioned estimates from [8], with taking care of the norm of Λ i as well as of the norms of the elements of R. Notice that the assumptions imposed on Λ i and R here differ from those imposed in [8]. To be precise, in [8] Λ i and R ij for all i, j ≤ n are assumed to be, respectively, smooth and Colombeau generalized functions locally of bounded growth. Referring the reader to [8] for details, we now write down the final a priori estimates for a global smooth solution U in a suitable for our purposes form. Set With this notation, we have where t(m) ≤ min{L/E Λ,max (0, 0), 1/q m }, P 1,m is a polynomial of degree 8⌈T /t(m)⌉ with positive constant coefficients not depending on ε, and P 2,m is a polynomial whose degree depends on m but neither on T nor on t(m) (and, therefore, not depending on ε) with positive constant coefficients depending on f and not depending on ε.
The classical smooth solution to problem (1)-(3) satisfying estimates (6) in Π T for all m ∈ N 0 can be constructed by the sequential approximation method. This solution will serve to build up a representative of the Colombeau solution. We now construct the latter. Accordingly to the assumptions of the theorem, we consider all the initial data as elements of the corresponding Colombeau algebras. We choose representatives λ, a, b, c, d, and h of Λ, A, B, C, D, and H, respectively, with the properties required in the theorem. Hence a representative of R is therewith defined. We will denote it by r. Let φ = ϕ ⊗ ϕ ∈ A 0 (R 2 ). Consider a prospective representative u = u(φ, x, t) of U which is the classical smooth solution to problem (1)-(3) with initial data λ(φ, x, t), a(ϕ, x) and boundary data b(ϕ, t), c(ϕ, t), d(φ, x, t), h(ϕ, t). It remains to show the moderatness of u, i.e. that u ∈ E M . To do so, we will obtain moderate growth estimates of u(φ ε , x, t) in terms of the regularization parameter ε.
Let ε be small enough and φ ∈ A N (R 2 ) with N chosen so large that the following conditions are true: a) the moderation property holds for a(ϕ ε , x) and h(ϕ ε , t); b) the local-γ-invertibility estimate (see Definition 6) holds for λ i (φ ε , x, t) and r(ϕ ε , t).
The moderate estimates on t as well as on mixed derivatives follow immediately from (1) by successive differentiation. This finishes the existence part of the proof.
The proof of the uniqueness part follows the same scheme. The only difference is that now we consider problem (1)-(3) with right hand sides of (2) and (3) in N . The analysis is even simplier since by [2], it is sufficient to check negligibility at order zero. The proof is complete. ✷

Remark 8
To prove the theorem, we used an integral-operator form (5) of the problem under consideration. Considering (5) with respect to a Colombeau function U ∈ G(Π T ), we see that the right hand side of (5) includes compositions of generalized functions. Specifically, we have compositions of the singular initial and boundary data as well as the function U with the singular characteristic curves. Note that the Colombeau algebra G is invariant under superposition with smooth polynomially bounded maps. In spite of the fact that the latter is not the case for the compositions involved by (5), all terms in (5) are well-defined in the Colombeau sense. To show this, consider system (5) with U replaced by u(φ, x, t), where the latter is a representative of the Colombeau solution stated in the theorem. From the proof it follows that, given (x, t) ∈ Π T , the domain of dependence for u(φ ε , x, t) is included in a compact subset of Π T which is independent of ε > 0 and ϕ ∈ A 0 (R).