Hyperbolic problems with totally characteristic boundary

We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u + \mathcal A(t,x,y,xD_x,D_y) u = f(t,x,y), \quad (t,x,y)\in(0,T)\times\mathbb R_+\times\mathbb R^d, \] where $\mathcal A(t,x,y,xD_x,D_y)$ is a first-order differential operator with coefficients smooth up to $x=0$ and the derivative with respect to $x$ appears in the combination $xD_x$. No boundary conditions are required in such a situation and corresponding initial-boundary value problems are effectively Cauchy problems. We introduce a certain scale of Sobolev spaces with asymptotics and show that the Cauchy problem for the operator $\partial_t + \mathcal A(t,x,y,xD_x,D_y)$ is well-posed in that scale. More specifically, solutions $u$ exhibit formal asymptotic expansions of the form \[ u(t,x,y) \sim \sum_{(p,k)} \frac{(-1)^k}{k!} x^{-p} \log^k \!x \, u_{pk}(t,y) \quad \text{as $x\to+0$} \] where $(p,k)\in\mathbb C\times\mathbb N_0$ and $\Re p\to-\infty$ as $|p|\to\infty$, provided that the right-hand side $f$ and the initial data $u|_{t=0}$ admit asymptotic expansions as $x \to +0$ of a similar form, with the singular exponents $p$ and their multiplicities unchanged. In fact, the coefficient $u_{pk}$ are, in general, not regular enough to write the terms appearing in the asymptotic expansions as tensor products. This circumstance requires an additional analysis of the function spaces. In addition, we demonstrate that the coefficients $u_{pk}$ solve certain explicitly known first-order symmetrizable hyperbolic systems in the lateral boundary. Especially, it follows that the Cauchy problem for the operator $\partial_t+\mathcal A(t,x,y,xD_x,D_y)$ is well-posed in the scale of standard Sobolev spaces $H^s((0,T)\times\mathbb R_+^{1+d})$.


INTRODUCTION
Due to their importance in the physical and engineering sciences, the investigation of hyperbolic initial-boundary problems has a long-standing history.Depending on the hyperbolic differential operators under study, the main questions concern the correct number and kind of boundary conditions to be imposed and well-posedness of the resulting initial-boundary problems in suitable scales of function spaces.See BENZONI-GAVAGE AND SERRE [1] for a recent account.In case of a non-characteristic boundary, it is known that the weak Lopatinskii condition is necessary for well-posedness, while the uniform Lopatinskii condition has been shown by LOPATINSKII [13], KREISS [10], and SAKAMOTO [20] to be necessary and sufficient in order to obtain the strongest possible regularity results, comparable to those one has for the pure Cauchy problem.See also CHAZ-ARAIN AND PIRIOU [3].The understanding of the characteristic case is considerably less complete.There are many works contributing to the uniformly characteristic case, especially for first-order systems when the differential operators under study are symmetric hyperbolic, see e.g.MAJDA AND OSHER [14], OHKUBO [17], RAUCH [18], or SECCHI [24].In these results, one often has more regularity in directions tangent to the boundary than in directions transverse to it.By contrast, for a totally characteristic boundary, one has the same regularity in all directions, as observed already by SAKAMOTO [21].
This observation was our point of departure.We investigate symmetrizable hyperbolic first-order differential systems in space-time cylinders (0, T ) × Ω, where Ω ⊆ R n is a C ∞ domain (or, more general, Ω is a C ∞ manifold with non-empty boundary) and where the lateral boundary (0, T )×∂Ω is totally characteristic.The main technical innovation is to regard the differential operators under study as cone-degenerate with respect to the spatial variables, which in turn is possible due to the totally characteristic boundary.As a consequence, using a suitable calculus for cone-degenerate pseudodifferential operators (detailed below), we construct symmetrizers for the systems under consideration and, as a result of the symmetrization process, are able to establish well-posedness in so-called Sobolev spaces H s,δ P,θ (Ω) with asymptotics (also detailed below).Here, the asymptotics alluded to are discrete conormal asymptotics, given by an asymptotic type P .Special cases include the standard Sobolev spaces H s (Ω) and H s 0 (Ω).Another relevant case is when the function spaces carry no asymptotic information at all, i.e., when θ = 0. Then the asymptotic type P is redundant and H s,δ P,0 (Ω) = K s,δ (Ω) is a weighted Sobolev space.It turns out that in the situation considered no boundary conditions are required.One main contribution of this paper is the revelation that the boundary traces of the solutions themselves satisfy hyperbolic differential equations in the lateral boundary (0, T ) × ∂Ω.In particular, these boundary traces can be determined ahead of determining the solutions.
There is a long-running program of investigating conormal asymptotic expansions of solutions to elliptic partial differential equations as an integral part of the structure, initiated by different people, see e.g.MELROSE [15] or REMPEL AND SCHULZE [19].Recently, there have been attempts to extend this program to include hyperbolic partial differential equations, see e.g.HINTZ AND VASY [6].This paper can also be seen as a contribution in this direction.
1.1.Formulation of the problem and main results.In this paper, we investigate well-posedness of the Cauchy problem for first-order hyperbolic systems with totally characteristic boundary.More specifically, we consider the Cauchy problem for N × N systems where A ∈ C ∞ ([0, T ]; Diff 1 (Ω; C N )), Ω is C ∞ manifold with non-empty boundary ∂Ω, and Ω = Ω \ ∂Ω.Our standing assumptions are that the differential operator (A1) L = ∂ t + A(t, ̟, D ̟ ) is symmetrizable hyperbolic and that the lateral boundary (A2) (0, T ) × ∂Ω is totally characteristic for L .
The latter condition means that σ 1 ψ (A)(t, ̟, ν(̟)) = 0 for (t, ̟) ∈ [0, T ] × ∂Ω, where σ 1 ψ (A) is the principal symbol of A and ν(̟) ∈ T * ̟ Ω is conormal with respect to the boundary ∂Ω (i.e., ν(̟) T̟(∂Ω) = 0).A first observation is that the characteristic curves of L stay inside the lateral boundary when they started out there.In particular, they are tangent to the boundary.Consequently, no boundary conditions are required in order to solve Eq. (1.1).Besides, there is no need for a Lopatinskii condition or an replacement of it in one or the other form.
1.1.1.The result for standard Sobolev spaces.We begin with describing the result for the standard Sobolev spaces H s (Ω), where Ω ⊆ R n is a C ∞ domain and s ≥ 0. This case deserves special attention for two reasons: firstly, the proof here is considerably simpler than in the general case, secondly, it helps to develop some additional intuition for the problems studied later.
In addition, the boundary traces for ℓ ∈ N 0 , ℓ < s + σ − 1/2 (defined by extending ν to a C ∞ vector field in a neighborhood of ∂Ω) are uniquely determined as solutions to certain hyperbolic Cauchy problems in (0, T ) × ∂Ω.
Remark.The latter observation is actually one of the guiding principles in what follows.Start with the standard Sobolev spaces H s (Ω), remove the asymptotic terms arising from a Taylor series expansion at the boundary ∂Ω to arrive at the spaces H s 0 (Ω), then adjust the reference conormal order from 0 to δ (see, e.g., Lemma 3.10) and affix asymptotic terms once again, now possibly of a different asymptotic type P .This yields the function spaces H s,δ P,θ (Ω) in which well-posedness for Eq.(1.1) will be shown to hold as well.
Remark.The regularity of the boundary traces γ ℓ u results from the trace theorems for the standard Sobolev spaces.It is a half an order less than what is obtained in case of a non-characteristic boundary when the uniform Lopatinskii condition hold and the usual compatibility conditions between initial and boundary data are satisfied.is a closed half-space.We state the result in this situation next.
We assume that the coefficients of the differential operator We further set n = 1 + d and write the spatial coordinates as (x, y) ∈ R + × R d .Then the Cauchy problem to be investigated becomes , and A is C ∞ up to x = 0. Further, we assume that the linear differential operator Having concrete applications in mind, apart from asymptotics resulting from a Taylor series expansion at x = 0 as in Theorem 1.1, we consider more general asymptotics of the form where (p, k) ∈ C × N 0 with ℜp → −∞ as |p| → ∞.(The precise conditions are stated in Definition 3.1.)Such asymptotics arise in many applications, both linear and nonlinear.The choice of the exponent −p (in place of p) and the appearance of the factor (−1) k /k! is related to the normalization of the Mellin transform (see Appendix A.1) and simplifies certain formulas later on, especially (1.8), (1.10).We shall denote the uniquely determined coefficients u pk in those asymptotic expansions by γ pk u and regard these coefficients as boundary traces as before.
Incorporating the asymptotic information provided by (1.4) into function spaces H s,δ P,θ (R 1+d + ) (neglecting the dependence on t at this point, see Section 3.2.3), the main result of this paper is as follows: Theorem 1.2.Let s ∈ R, σ ∈ N 0 , P ∈ As δ , and θ 0 ≥ . . .≥ θ σ ≥ 0. Under the assumptions stated above, given u 0 ∈ H s+σ,δ P,θ 0 (R In addition, for all (p, k) where the expression R pk [u] appearing in the right-hand side of (1.5) stands for a term that depends linearly on γ ql u for q − p ∈ N 0 and l > k if q = p.
Notice that Eq. (1.5) is a hyperbolic Cauchy problem for γ pk u in (0, T ) × R d .It follows that the coefficients γ pk u in the asymptotic expansion (1.4) can be successively computed and are uniquely determined by the corresponding coefficients of the initial data u 0 and the right-hand side f .Remark.Theorem 1.1 is a special case of Theorem 1.2 when Ω = R 1+d + .Here, s ≥ 0, δ = 0, P = P 0 , and θ r = s − r + σ for 0 ≤ r ≤ σ (see the example after Definition 3.1).
Remark.Strictly speaking, we only deal with constant discrete asymptotics.It is likely that similar results also hold for continuous asymptotics as well as for variable discrete asymptotics (see, e.g., [5,9,19] for elliptic problems).
1.2.Outline of the argument.In the sequel, we fix a δ ∈ R as reference conormal order.The weighted L 2 space K 0,δ (R 1+d + ) (see Definition 3.4) will be our basic Hilbert space, replacing the space L 2 (R 1+d + ) = K 0,0 (R 1+d + ).Denote by the norm and by , the inner product in K 0,δ (R 1+d + ).One basic problem is to define the asymptotic terms appearing in asymptotic expansions like appropriately (see (1.4)).The asymptotic expansion (1.6) is with respect to an increasing flatness as x → +0, where the term (−1) k /k! x −p log k x v pk (y) has conormal order 1/2 − ℜp − 0. For v ∈ H s,δ P,θ (R 1+d + ), this asymptotic expansion breaks off at conormal order δ + θ so that effectively only finitely many terms in the right-hand side of (1.6) have to be taken into account.Nonetheless, as v ∈ H s,δ P,θ (R 1+d + ) implies that both ℜp < 1/2 − δ and v pk ∈ H s+ℜp+δ−1/2, k (R d ), as a rule we have that (−1) k /k! x −p log k x v pk (y) / ∈ H s,δ P,θ (R 1+d + ) near x = 0.The correct form of the asymptotic term is given by Γ pk v pk , where and ϕ is a cut-off function.(See Section 1.4 for the notation used.)Especially, Γ pk w ∈ C ∞ (R 1+d + ) for w ∈ S ′ (R d ), Γ pk w x=0 = w, and Γ pk w is supported for x 1.The decisive property which makes the approach work, however, is We construct a calculus + ) of cone-degenerate pseudodifferential operators on the half-space R 1+d + , where and the pseudodifferential operators contained exhibit a prearranged behavior as x → +0.The basic idea is taken from Schulze [22,23].In particular, near x = 0, the operators A ∈ Ψ µ c (R 1+d + ) are to the leading order of the form A = ϕ(x) op M (h)ϕ 0 (x), where h(z) = h(y, z, D y ) is an entire family of pseudodifferential operators in Ψ µ cl (R d ) subject to further conditions, op M (h) = M −1 z→x h(z)M with M being the Mellin transform, and ϕ, ϕ 0 are cut-off functions.Compared to the cone calculus of Schulze, where the coefficients v pk would be in S (R d ) in the situation considered here, we now had to show that operators A ∈ Ψ µ c (R 1+d + ) act in an appropriate way on the asymptotic terms given by (1.7).Indeed, it holds that Furthermore, a symbolic symmetrizer for the hyperbolic operator L = ∂ t + A(t, x, y, xD x , D y ) is indeed a symmetrizer b(t, x, y, ξ, η) for the compressed principal symbol σ1 ψ (A)(t, x, y, ξ, η) of A. Using Gårding's inequality in a routine way yields an operator + ; C N )), i.e., ℜσ 1 ψ (BA) = 0. Together with the fact that integration by parts produces no boundary terms, i.e., • BAu, v = u, (BA in a standard manner.Once the calculus of cone-degenerate pseudodifferential operators mentioned above is established, this essentially means that one can treat the Cauchy problem (1.2) in (0, T ) × R 1+d + like a Cauchy problem in free space (0, T ) × R 1+d .From estimate (1.9), one obtains well-posedness of Eq. (1.2) in the basic Hilbert space K 0,δ (R 1+d + ), i.e., the first part of Theorem 1.2 for s = 0, σ = 0, and θ 0 = 0. Well-posedness in the weighted Sobolev spaces K s,δ (R 1+d + ), i.e., the first part of Theorem 1.2 in all other cases with θ 0 = 0, then likewise follows using order reductions that exist in the pseudodifferential calculus considered.
To establish the well-posedness results in the Sobolev spaces H s,δ P,θ (R 1+d + ) with asymptotics is a considerably more involved task.The crucial observation is that the boundary traces γ pk u solve hyperbolic Cauchy problems in the lateral boundary.To see this, one needs to know that, besides the compressed principal symbol σµ and they determine the manner in which asymptotics are mapped by A. More precisely, it holds that where the finite sum in the right-hand side is over those (j, ℓ, r) such that ℜp + j < 1/2 − δ.Thus, applying γ pk to both sides of the equation in (1.1) results in an equation for γ pk u, where the term R pk [u] has a similar meaning as in (1.5).In fact, a compatibility condition between σ 1 ψ (σ 0 c (A(t))) and σ1 ψ (A(t)) ensures that the operator ∂ t + σ 0 c (A(t)) is symmetrizable hyperbolic.Hence, one obtains existence, uniqueness, and higher regularity for the boundary traces γ pk in the correct regularity classes.Subtracting the boundary terms from the prospective solution u, one ends up in weighted Sobolev spaces K s−θ 0 ,δ+θ 0 (R 1+d + ), in which well-posedness has been shown before.Note that at this place it is crucial that the cone-degenerate pseudodifferential operators in Ψ µ c (R 1+d + ) have holomorphic conormal symbols (as opposed to finitely meromorphic ones, one usually sees in a cone pseudodifferential calculus), which implies that the action of is the same for any conormal order γ ∈ R (in the sense that it agrees on K s,γ (R 1+d + ) ∩ K s+µ,γ ′ (R 1+d + ) independently of whether this intersection is seen as a subspace of K s,γ (R 1+d + ) or K s,γ ′ (R 1+d + )).Hence, the argument provided for well-posedness in function spaces with conormal order δ works for any other conormal order just the same.

Comparison with other results and open problems. One of the big open problems in the
field is to provide satisfactory answers concerning well-posedness for hyperbolic boundary problems with a uniformly characteristic boundary.There only exist several partial results in the literature, see e.g.[1,2,14,18,24].
Totally characteristic hyperbolic boundary problems (for higher-order scalar equations) were treated by SAKAMOTO [21].She obtained results comparable to ours by showing (in our own notation) well-posedness in the scales H s,δ P,θ (Ω), where P = T δ P 0 , with P 0 being the type for Taylor asymptotics, and θ = s ≥ 0. Our results are slightly more general in that respect that we now allow general asymptotic types P , general weight intervals given by θ ≥ 0, and also negative Sobolev orders s.In addition, we show that the boundary traces are given as solutions to hyperbolic Cauchy problems in the lateral boundary.This later result appears to be new.
SAKAMOTO [21] used pseudodifferential techniques to establish her results, albeit in a different manner.Our approach might have the advantage that it yields a symmetrizer also in the uniformly characteristic case, upon further developing an adapted pseudodifferential calculus.It seems to be evident that this calculus has to be some sort of an edge calculus, as in SCHULZE [23].On the level handled in this paper the difference between an edge calculus and a cone calculus is rather marginal: In the latter one first performs the Fourier transform F y→η with respect to the y-variables and then the Mellin transform M x→z with respect to the x-variable, while in the former these transforms are performed in the opposite order.As both operations commute, M x→z F y→η = F y→η M x→z , it is possible to recast the cone calculus utilized here in the form of an edge calculus, up to some technical details.Note that the form of the asymptotic terms as given in (1.7) is already typical of edge problems.
Throughout the paper, we shall especially employ the following notation: • δ ∈ R denotes the reference conormal order, which is fixed once and for all.
• For β ∈ R, we introduce the weight line Γ β = {z ∈ C | ℜz = β}.The weight line that corresponds to the reference conormal order δ is for |x| 1.Likewise, ϕ 0 , ϕ 1 are also cutoff functions satisfying, in addition, ϕϕ 0 = ϕ and • γ pk for (p, k) ∈ P , where P is an asymptotic type, is a trace operator.Similarly, Γ pk for (p, k) ∈ P is a potential operator.
is the class of cone-degenerate pseudodifferential operators, of order µ ∈ R, utilized here.Elements of this space are symbolically written as A(x, y, xD x , D y ).
denotes the compressed cotangent bundle over R 1+d + .

WELL-POSEDNESS IN STANDARD SOBOLEV SPACES
We start with proving Theorem 1.1.Notice that it is enough to treat the case σ = 0. Cases with σ ≥ 1 then follow by differentiating the equation σ times with respect to t, as in the proof of Proposition 4.5 below.
The precise form of the term R ℓ [u] will be given in (4.7) below.
Proof.Extend the matrix-valued coefficients A, A j , B in Eq. (1.1) to matrix-valued functions ) so as to obtain a uniformly symmetrizable hyperbolic system where , and F (0,T )×Ω = f .Because this system is on whole space, Eq. (2.2) possesses a unique solution U ∈ C ([0, T ]; H s (R n ; C N )).As characteristics of this system are tangent to the hypersurface (0, T ) × ∂Ω, it follows that whatever starts out in the region in which (0, T ) × (R n \ Ω) stays in that region for all times.Therefore, u = U (0,T )×Ω only depends on u 0 , f , in particular, u is independent of all the extensions chosen.We conclude that u is the unique solution to the original problem (1.2).Differentiating (2.2) a number of times with respect to x and setting x = 0 yields (2.1).

CONE-DEGENERATE PSEUDODIFFERENTIAL OPERATORS
The main technical tool to prove the result in Theorem 1.2 is a calculus for a certain class of cone-degenerate pseudodifferential operators on R 1+d + .Here we briefly introduce this pseudodifferential calculus.Calculi for cone-degenerate pseudodifferential operators have been developed by B.-W. SCHULZE [22,23], see also [4].We closely follow his approach and refer to the said references for details.An equivalent calculus is the b-calculus of R. MELROSE AND G. MENDOZA [15,16].For our purposes, Schulze's cone calculus is preferable as it is more analytic in flavor and, therefore, easier to adapt to our needs.
Compared to [22,23], there are a few differences.First of all, the base of the cone is R d which is a non-compact manifold.This non-compactness introduces no additional difficulties, as we are not interested in the Fredholm property of elliptic operators, but solely in the construction of a symmetrizer.Secondly, the coefficients u pk in the asymptotic expansions (1.4) do not belong to finite-dimensional subspaces of C ∞ (R d ; C N ), but instead can be any functions from the space H s+ℜp+δ−1/2, k (R d ; C N ), see Definition 3.8 and Proposition 3.9 for details.This then requires a special treatment of the asymptotic terms, see Definition 3.7.In fact, the function spaces H s,δ p,θ (R 1+d + ; C N ) employed below are modeled after the edge Sobolev spaces of Schulze (see [11] for a discussion of this point).Lastly, the cone-degenerate pseudodifferential operators we utilize do not produce any further asymptotic information, but instead preserve the given one.This is in the sense that the given asymptotic type, P , which collects the (p, k) appearing in (1.4), is preserved, while certainly the coefficients u pk are, in general, altered when applying an operator A belonging to the calculus to u (see Proposition 3.26 for the way in which this happens).Accordingly, the conormal symbols σ −j c (A) of operators A in the calculus (see Definition 3.20) are holomorphic functions of the Mellin covariable z ∈ C, while for general cone pseudodifferential calculi these conormal symbols are finitely meromorphic functions of z ∈ C. Again, our choice is justified by the fact that we do not have to construct parametrices for elliptic cone-degenerate pseudodifferential operators (with the exception of Proposition 3.28 where we establish the existence of order reductions).
3.1.Asymptotic types.The functional-analytic approach of handling the asymptotic expansions (1.6) starts with collecting the data (p, k) appearing in (1.6) into so-called asymptotic types.Recall that we fix a δ ∈ R as a reference conormal order.Definition 3.1.The set As δ of asymptotic types associated with the conormal order δ ∈ R consists of discrete subsets P ⊂ C × N 0 with the following properties: Remark.Property (iv) is needed to guarantee the coordinate invariance of the constructions.
We set π C P = p ∈ C | (p, k) ∈ P for some k ∈ N 0 and m p = max{k + 1 | (p, k) ∈ P } with the convention that m p = 0 if p / ∈ π C P .An asymptotic type P is then completely determined by the m p .In this sense, P can be thought of as a non-negative divisor (in the sense of complex analysis) having additional properties.Still, regarding P as a discrete subset of C × N 0 comes in handy in the notation employed below.
3.2.Function spaces.Next we introduce suitable weighted Sobolev spaces and Sobolev spaces with asymptotics.These are the function spaces in which we will establish the energy inequalities.

Weighted Sobolev spaces.
Definition 3.2.Let γ ∈ R. For s ∈ N 0 , the weighted Sobolev space H s,γ (R 1+d + ) is defined to consist of all functions u = u(x, y) such that For general s ∈ R, the spaces H s,γ (R 1+d + ) are then introduced by complex interpolation and duality.We can characterize the space H s,γ (R 1+d + ) via the Mellin transform (see Section A.1).
where ũ(z, •) is the Mellin transform of u(x, •) with respect to x.
As said in the introduction, we are mostly interested in the behavior of solutions to Eq. (1.2) near x = 0. Hence, we make a generic choice for their possible behavior near x = ∞ (also compare with Eq. (4.8)).Note that u ∈ H s,γ (R 1+d + ) implies that ϕu ∈ H s,γ (R 1+d + ).Definition 3.4.For s, γ ∈ R, we set In view of H s,γ (R 1+d + ) ⊂ H s loc (R 1+d + ), the space K s,γ (R 1+d + ) is independent of the choice of the cut-off function ϕ.Moreover, K s,γ (R 1+d + ) is a Hilbert space in a natural way.Now fix δ ∈ R. In the sequel, K 0,δ (R 1+d + ) will serve as reference Hilbert space.Write , for the inner product and for the norm in K 0,δ (R 1+d + ).

The asymptotic terms.
The next two lemmas prepare for introducing the terms occurring in the asymptotic expansions (1.6).Recall that the asymptotic expansions (1.6) are formal in the sense that, in general, we do not have enough regularity for the coefficients v pk to write the asymptotic terms as tensor products.
Lemma 3.6.Let (p, k) ∈ C × N 0 .Suppose that w ∈ H s, k (R d ) for some s ∈ R. Then Proof.We set ) and proceed by induction on k.
For k = 0, one has and Then, by Lemma 3.5 and the induction hypothesis, as well as This finishes the proof.
We are now ready to introduce potential operators.
Definition 3.7.For (p, k) ∈ C × N 0 , the potential operator Γ pk acting on functions w = w(y) is given by The role played by the normalizing factor (−1) k /k! becomes apparent from Lemma A.2 in conjunction with (3.7).Based on Lemma 3.6, we have that, for any s ∈ R,

Sobolev spaces with asymptotics.
The definition of the Sobolev spaces with asymptotics postulates an improvement of the conormal order up to δ + θ upon subtracting finitely many asymptotic terms.This works as long as no singular exponent p from the asymptotic type P comes to lay on the weight line Γ 1/2−δ−θ .This leaves out a discrete set of values for θ > 0. The general case is then handled by complex interpolation.
(ii) For general θ ≥ 0, the space H s,δ P,θ (R 1+d + ) is then defined by complex interpolation with respect to the parameter θ.
For s ≥ 0, we also write H s,δ P (R Example.The spaces H s,δ P,θ (R 1+d + ) constitute a natural generalization of the standard Sobolev spaces in view of the following two facts: (i) For s ≥ 0, H s (R 1+d + ) = H s,0 P 0 (R 1+d + ), where P 0 is the type for Taylor asymptotics.(ii) For s ≥ 0, H s 0 (R , where O is the empty asymptotic type. It is not hard to see that the coefficients v pk in (3.4) are uniquely determined.We then introduce, for (p, k) ∈ P with ℜp > 1/2 − δ − θ, the trace operators Essentially by definition, we have the following trace theorem.
Proposition 3.9.Let s ∈ R, P ∈ As δ , θ ≥ 0, and The next result tells us that, for theoretical purposes, one can adjust the reference conormal order δ to be any given real number.Lemma 3.10.Let s ∈ R, P ∈ As δ , θ ≥ 0, and ̺ ∈ R. Then multiplication by ψ ̺ (x) realizes an isomorphism between H s,δ P,θ (R ) .First we let S 0 (R 1+d + ) be the space of all u ∈ S (R 1+d + ) that vanish to infinite order at x = 0. Definition 3.11.For P ∈ As δ , the space S P (R 1+d + ) consists of all v for which there are sequences The space S P (R 1+d + ) is a nuclear Fréchet space in a natural way.Furthermore, upon an appropriate choice of (c p ) depending on (v pk ), the series in the left-hand side of (3.6) converges absolutely in + ) for any s, θ and, for v as in (3.6) and (p, k) ∈ P , we have γ pk v = v pk .Indeed, the short sequence Lemma 3.12.The space S P (R 1+d + ) is dense in H s,δ P,θ (R 1+d + ).
Proof.By complex interpolation, we can assume that . In view of (3.4) and as S (R d ) is dense in H r, l (R d ) for any (r, l) ∈ R × N 0 , it is enough to show that Γ pk w ∈ S P (R 1+d + ) for (p, k) ∈ P and w ∈ S (R d ).The latter, in turn, will follow from the relation which, however, is apparently true.

3.3.
Calculus of cone-degenerate pseudodifferential operators.We now introduce the class + ) of cone-degenerate pseudodifferential operators on the half-space R 1+d + mentioned in the introduction.To make connection to the theory of cone-degenerate pseudodifferential operators, note that the closed half-space R 1+d + is considered as a blowup of the cone (R + × R d )/({0} × R d ).We do not provide proofs for results that can be found in the literature in the form as stated or in a similar form (then with no essential changes in the proofs).Notable exceptions are Propositions 3.25 through 3.29.For other results, we refer to the literature, e.g., HARUTYUNYAN AND SCHULZE [4] or SCHULZE [23].

Note that any
is nowhere vanishing.In the elliptic case, A admits a parametrix, i.e., there exists a . This parametrix B is essentially unique, i.e., it is unique modulo S(R; Ψ −∞ (R d )).Moreover, A(τ ) is invertible for |τ | large and B can be chosen to satisfy B(τ ) = A(τ ) −1 for |τ | large.Note also that Ψ µ cl (R d ; R) equipped with its canonical system of seminorms is a nuclear Fréchet space.
and is parameter-dependent elliptic.

Holomorphic Mellin symbols. Recall that we write
as the set of holomorphic Mellin symbols of order µ.
These Mellin symbols are entire functions of z = β + iτ taking values in the nuclear Fréchet space Ψ µ cl (R d ) and are also smooth functions of β taking values in Ψ µ cl (R d ; R τ ).Note that, as a consequence of the Cauchy-Riemann equations, the principal symbol Proof.This is proven as in WITT [25].
The following result provides a means to control the action of op M (h) Proof.Let m(z) be the Mellin transform of ϕ(x)x −p .Then the Mellin transform of ϕ(x)x −p log k−r x for 0 ≤ r ≤ k equals m (k−r) (z).Doing the computations modulo ǫ>0 K s−ǫ,1/2−ℜp+ǫ (R 1+d + ), we find that (see Lemma 3.6) where we have used that for 0 ≤ r ≤ k.
Notice that any operator A of the form in (3.8) is a pseudodifferential operator on R 1+d + .The point of this definition is to enforce control on the behavior as x → +0 in a specific way (see, e.g., the mapping properties in Proposition 3.25 below).
Remark.Although the weight factor ψ −2δ (x) appears explicitly in the definition of the residual class ) are in fact independent of δ ∈ R.

Symbolic structure.
For the rest of this section, we develop certain elements of the calculus of pseudodifferential operators in ).We start with the symbolic structure.First note that the vector fields a(x, y)x∂ x + d j=1 a j (x, y)∂ j tangent to ∂R 1+d + , where a, a j ∈ C ∞ (R 1+d + ), are the C ∞ sections of a vector bundle over R 1+d + that we denote by T * R 1+d + and call it the compressed cotangent bundle (see, e.g., MELROSE [15]).Indeed, the covariable to can be taken to be ( ξ, η) with ξ = ψ(x)ξ.
Definition 3.18.The compressed principle symbol σµ Consequently, the compressed principal symbol σµ ψ (A) provides control on operators in Ψ µ c (R 1+d + ) up to lower-order perturbations.
Still, control of the asymptotic behavior as x → +0 is achieved with the help of the full sequence (σ −j c (A)) j∈N 0 of conormal symbols.
Definition 3.20.For A ∈ Ψ µ c (R 1+d + ) written as in (3.8), the conormal symbol σ −j c (A) of conormal order −j for j ∈ N 0 is defined as Lemma 3.21.There is a compatibility condition between σµ ψ (A) and σ 0 c (A), namely σµ It is this compatibility condition which later will guarantee that the governing equations for the coefficents γ pk u are symmetrizable hyperbolic.
Remark.In order to provide a heuristic explanation of how control on the asymptotic behavior as x → +0 is achieved by the sequence (σ −j c (A)) j∈N 0 notice that, informally, we have that A ∼ ).Then it is readily checked that (i) σ1 ψ (A)(x, y, ξ, η) = a(x, y) ξ + d j=1 a j (x, y)η j .(ii) σ 0 c (A)(z) = ia(0, y)z + d j=1 a j (0, y)D j + b(0, y).(iii) σ 1 ψ (σ 0 c (A))(y, η, τ ) = − a(0, y)τ + d j=1 a j (0, y)η j .3.3.5.Compositions and adjoints.Now we get into the calculus of cone-degenerate pseudodifferential operators.First consider their formal adjoints.Let A * be formal adjoint operator of ) of cone-degenerate pseudodifferential operators is closed under taking adjoints in the sense that whenever Another result is that cone-degenerate pseudodifferential operators are closed under compositions.Proof.This is a standard result in the theory of cone-degenerate pseudodifferential operators.In particular, it relies on the fact that the conormal symbols are assumed to be holomorphic.Proposition 3.25.For µ ∈ R, where intersection is over all s ∈ R, P ∈ As δ , and θ ≥ 0.
. That this holds for all γ ∈ R is again a consequence of the holomorphy of the conormal symbols.
It is crucial that an integration by parts produces no boundary terms.It is precisely this property which allows us to treat the initial-boundary value problem Eq. (1.1) as a Cauchy problem.

Lemma 3.27. For
+ ; C N ), the result follows.The existence of so-called order reductions is assured next.Proposition 3.28.Let µ ∈ R. Then there exists a selfadjoint, positive definite operator is an isomorphism for all s ∈ R, P ∈ As δ , and θ ≥ 0.
Proof.One way to prove the result is to start with a parameter-dependent version It follows that the operator B = B 0 + C has the desired properties.
Next we derive energy estimates for Eq.(4.1) in the weighted Sobolev spaces K s,δ (R 1+d + ; C N ).We start with the case s = 0.As usual, the proof of the next proposition relies on the following facts (as was already mentioned in the introduction): • B(t)u, u is equivalent to u 2 uniformly in t ∈ [0, T ], • Integration by parts produces no boundary terms (see Lemma 3.27),
As an immediate consequence we have the next result.
+ ), as constructed in Lemma 3.28.Then, Λ s u solves the system As Λ s • is an equivalent norm on K s,δ , we obtain estimate (4.3).
The base case σ = 0 was treated in Proposition 4.4.
4.2.Well-posedness in Sobolev spaces with asymptotics.Here we establish Theorem 1.2 in full generality.In fact, we only derive the fundamental energy inequality.Then the rest of the proof is completely analogous to the proof in the previous section, and it is omitted.

1+d+
, we use coordinates (x, y) with x ≥ 0 and y ∈ R d .• The weighted Sobolev spaces K s,γ (R 1+d + ) are introduced in Definition 3.4 and the Sobolev spaces H s,δ P,θ (R 1+d + ) with asymptotics in Definition 3.8.

3. 3 . 1 .
Parameter-dependent pseudodifferential operators.We start with parameter-dependent pseudodifferential operators.Definition 3.13.For µ ∈ R, the class Ψ µ cl (R d ; R) of classical parameter-dependent pseudodifferential operators on R d , with parameter τ ∈ R, consists of all families A

j≥0x
j op M (σ −j c (A)(z)) as x → +0 (3.10) upon performing a Taylor series expansion of A M at x = 0. (See also Proposition 3.26.)Example.The first-order differential operator A = a(x, y)ψ(x)D x + d j=1 a j (x, y)D j + b(x, y), where a, a j , b ∈ C ∞ b (R 1+d + ), belongs to Ψ 1 c (R 1+d +
1.1.2.The local problem.Most of the effort in this paper is put into the local situation, where Ω = R n + The Schwartz class with asymptotics of type P .At last, we introduce a replacement for the space S (R 1+d + ) and H s,δ+̺ T ̺ P,θ (R 1+d + ).