### Abstract

In this paper, we deal with the second initial boundary value problem for higher order hyperbolic systems in domains with conical points. We establish several results on the well-posedness and the regularity of solutions.

### 1 Introduction

Boundary value problems in nonsmooth domains have been studied in differential aspects. Up to now, elliptic boundary value problems in domains with point singularities have been thoroughly investigated (see, e.g, [1,2] and the extensive bibliography in this book). We are concerned with initial boundary value problems for hyperbolic equations and systems in domains with conical points. These problems with the Dirichlet boundary conditions were investigated in [3-5] in which the unique existence, the regularity and the asymptotic behaviour near the conical points of the solutions are established. The Neumann boundary problem for general second-order hyperbolic systems with the coefficients independent of time in domains with conical points was studied in [6]. In the present paper we consider the Cauchy-Neumann (the second initial) boundary value problem for higher-order strongly hyperbolic systems in domains with conical points.

Our paper is organized as follows. Section 2 is devoted to some notations and the formulation of the problem. In Section 3 we present the results on the unique existence and the regularity in time of the generalized solution. The global regularity of the solution is dealt with in Section 4.

### 2 Notations and the formulation of the problem

Let Ω be a bounded domain in ℝ* ^{n}*,

*n*≥ 2, with the boundary ∂Ω. We suppose that ∂Ω is an infinitely differentiable surface everywhere except the origin, in a neighborhood of which Ω coincides with the cone

*K*= {

*x*:

*x*/|

*x*| ∈

*G*}, where

*G*is a smooth domain on the unit sphere

*S*

^{n-1}. For each

*t*, 0 <

*t*≤ ∞, denote

*Q*= Ω × (0, t), Ω

_{t }*= Ω × {t}. Especially, we set*

_{t }*Q*=

*Q*

_{∞}, Γ = ∂Ω\{0},

*S*= Γ × [0, +∞).

For each multi-index *p *= (*p*_{1},..., *p _{n}*) ∈ ℕ

*, we use notations |*

^{n}*p*| =

*p*

_{1 }+ ... +

*p*,

_{n}*u*= (

*u*

_{1 },...,

*u*) defined on

_{s}*Q*, we denote

Let us introduce the following functional spaces used in this paper. Let *l *denote a nonnegative integer.

*H ^{l }*(Ω) - the usual Sobolev space of vector functions

*u*defined in Ω with the norm

*H ^{l }*(Ω) on Γ with the norm

*H*^{l,0 }(*Q*, *γ*) (*γ *∈ ℝ)- the weighted Sobolev space of vector functions *u *defined in *Q *with the norm

Especially, we set *L*_{2}(*Q*, *γ*) = *H*^{0,0}(*Q*, *γ*).

*H*^{l,1 }(*Q*, *γ*) (*γ *∈ ℝ)- the weighted Sobolev space of vector functions *u *defined in *Q *with the norm

where

*u *defined in Ω with the norm

If *l *≥ 1, then

*u *defined in *Q *with the norm

From the definitions it follows the continuous imbeddings

and

for arbitrary nonnegative integers *l*, *k *and real number *α*. It is also well known (see [[2], Th. 7.1.1]) that if

with the norms being equivalent.

Now we introduce the differential operator

where *a _{pq }*=

*a*(

_{pq }*x*,

*t*) are the

*s*×

*s*matrices with the bounded complex-valued components in

*p*|, |

*q*| ≤

*m*, where

*a*. This means the differential operator

_{pq}*L*is formally self-adjoint. We assume further that there exists a positive constant

*μ*such that

for all *η _{p }*∈ℂ

*, |*

^{s}*p*| =

*m*, and all

Let *v *be the unit exterior normal to *S*. It is well known that (see, e.g., [[7], Th. 9.47]) there are boundary operators *N _{j }*=

*N*(

_{j }*x*,

*t*,

*D*),

*j*= 1, 2,...,

*m*on

*S*such that integration equality

holds for all
*t *∈ [0, ∞). The order of the operator *N _{j }*is 2

*m*-

*j*for

*j*= 1, 2,...,

*m*.

In this paper, we consider the following problem:

A complex vector-valued function *u *∈ *H*^{m,1}(*Q*, *γ*) is called a generalized solution of problem (2.6)-(2.8) if and only if *u*|_{t = 0 }= 0 and the equality

holds for all *η*(*x*, *t*) ∈ *H*^{m,1}(*Q*) satisfying *η*(*x*, *t*) = 0 for all *t *≥ *T *for some positive real number *T*.

### 3 The unique solvability and the regularity in time

First, we introduce some notations which will be used in the proof of Theorems 3.3
and 3.4. For each vector function *u*,*v *defined in Ω and each nonnegative integer *k*,

For vector functions *u *and *v *defined in *Q *and *τ *> 0, we set

Especially, we set

From the formally self-adjointness of the operator *L*, we see that

Next, we introduce the following Gronwall-Bellman and interpolation inequalities as two fundamental tools to establish the theorems on the unique existence and the regularity in time.

**Lemma 3.1 **([8], Lemma 3.1) *Assume u*, *α*, *β are real-valued continuous on an interval *[*a*, *b*], *β is nonnegative and integrable on *[*a*, *b*], *α is nondecreasing satisfying*

*Then*

From [[9], Th. 4.14], we have the following lemma.

**Lemma 3.2**. *For each positive real number ε and each integer j*, 0 < *j *< *m, there exists a positive real number C *= *C *(Ω, *m*, *ε*) *which is dependent on only *Ω, *m and ε such that the inequality*

*holds for all u *∈ *H ^{m}*(Ω).

Now we state and prove the main theorems of this section.

**Theorem 3.3**. *Let h be a nonnegative integer. Assume that all the coefficients a _{pq }together with their derivatives with respect to t are bounded on
*

_{0 }

*such that for each γ*>

*γ*

_{0},

*if f*∈

*L*

_{2}(

*Q*,

*σ*)

*for some nonnegative real number σ, the problem*(2.6)-(2.8)

*has a unique generalized solution u in the space H*

^{m,1}(

*Q*,

*γ*+

*σ*)

*and*

*where C is a constant independent of u and f*.

*Proof*. The uniqueness is proved by similar way as in [[4], Th. 3.2]. We omit the detail here. Now we prove the existence by Galerkin approximating
method. Suppose
*H ^{m}*(Ω) which is orthonormal in

*L*

_{2}(Ω). Put

where

with the initial conditions

Let us multiply (3.5) by
*l *from 1 to *N*, and integrate the obtained equality with respect to *t *from 0 to *τ *(0 < *τ *< ∞) to receive

Now adding this equality to its complex conjugate, then using (3.1) and the integration by parts, we obtain

With noting that, for some positive real number *ρ*,

we can rewrite (3.8) as follows

By (2.4), the left-hand side of (3.9) is greater than

We denote by *I*, *II*, *III*, *IV *the terms from the first, second, third, and forth, respectively, of the right-hand
sides of (3.9). We will give estimations for these terms. Firstly, we separate *I *into two terms

Put

Then, by the Cauchy inequality, we have

By the Cauchy inequality and the interpolation inequality (3.3), for an arbitrary
positive number *ε*_{1}, we have

where *C*_{1 }= *C*_{1}(*ε*_{1}) is a nonnegative constant independent of *u ^{N}*,

*f*and

*τ*. Now using again the Cauchy and interpolation inequalities, for an arbitrary positive number

*ε*

_{2 }with

*ε*

_{2 }<

*μ*, it holds that

where *C*_{2 }= *C*_{2}(*ε*_{2}) is a nonnegative constant independent of *u ^{N}*,

*f*and

*τ*. For the terms

*III*and

*IV*, by the Cauchy inequality, we have

and

where *ε*_{3 }> 0, arbitrary. Combining the above estimations we get from (3.9) that

Now fix *ε*_{1}, *ε*_{2 }and consider the function

We have

We see that the function *g *has a unique minimum at

We put

Now we take real numbers *γ*, *γ*_{1 }arbitrarily satisfying *γ*_{0 }< *γ*_{1 }< *γ*. Then there are positive real numbers *ε*_{1}, *ε*_{2}, (*ε*_{2 }< *μ*), *ρ *(*ρ *> *C*_{2}(*ε*_{1}, *ε*_{2})) and *ε*_{3 }such that

From now to the end of the present proof, we fix such constants *ε*_{1}, *ε*_{2}, *ε*_{3 }and *ρ*. Let

where

We see that

Hence, it follows from (3.14) that

Now multiplying both sides of this inequality by *e*^{-2(γ+σ)τ}, then integrating them with respect to *τ *from 0 to ∞, we arrive at

It is clear that |||.|||_{Q,γ+σ }is a norm in *H*^{m,1}(*Q*, *γ *+ *σ*) which is equivalent to the norm

From this inequality, by standard weakly convergent arguments (see, e.g., [[10], Ch. 7]), we can conclude that the sequence
*u *∈ *H*^{m,1}(*Q*, *γ *+ *σ*) which is a generalized solution of problem (2.6)-(2.8). Moreover, it follows from
(3.17) that the inequality (3.4) holds. □

**Theorem 3.4**. *Let h be a nonnegative integer. Assume that all the coefficients a _{pq }together with their derivatives with respect to t up to the order h are bounded on
*

_{0 }

*be the number as in Theorem 3.3 which was defined by formula*(3.11).

*Let the vector function f satisfy the following conditions for some nonnegative real number σ*

(i)

(ii)

*Then for an arbitrary real number γ satisfying γ *> *γ*_{0 }*the generalized solution u in the space H*^{m,1}(*Q*, *γ *+ *σ*) *of the problem *(3.6)- (3.7) *has derivatives with respect to t up to the order h with *
*k *= 0, 1,..., *h and*

*where C is a constant independent of u and f*.

*Proof*. From the assumptions on the regularities of the coefficients *a _{pq }*and of the function

*f*it follows that the solution

*t*up to the order

*h*+ 2. Now take an arbitrary real number

*γ*

_{1 }satisfying

*γ*

_{0 }<

*γ*

_{1 }<

*γ*. We will prove by induction that

and for *k *= 0,..., *h*, where the constant *C *is independent of *N*, *f *and *τ*. From (3.15) it follows that (3.19) holds for *k *= 0 since the norm |||·||| is equivalent to the norm
*k *= *h *- 1, we will show it to be true for *k *= *h*. To this end we differentiate *h *times both sides of (3.5) with respect to *t *to receive the following equality

From these equalities together with the initial (3.6) and the assumption *(ii)*, we can show by induction on *h *that

Now multiplying both sides of (3.20) by
*l *from 1 to *N*, we get

Adding the equality (3.22) to its complex conjugate, we have

Integrating both sides of this equality with respect to *t *from 0 to a positive real *τ *with using the integration by parts and (3.21), we arrive at

This equality has the form (3.8) with *u ^{N }*replaced by

Since the coefficients *a _{pq }*together with their derivatives with respect to

*t*up to the order

*h*are bounded, by the Cauchy and interpolation inequalities and the induction assumption, we see that

and

Thus, repeating the arguments which were used to get (3.15) from (3.8), we can obtain
(3.19) for *k *= *h *from (3.23).

Now we multiply both sides of (3.19) by *e*^{-2((k+1)γ+σ)τ}, then integrate them with respect to *τ *from 0 to ∞ to get

From this inequality, by again standard weakly convergent arguments, we can conclude
that the sequence
*u*^{(k) }∈ *H*^{m,1}(*Q*, (*k *+1)*γ *+*σ*), moreover, *u*^{(k) }is the *k*th generalized derivative in *t *of the generalized solution *u *of problem (2.6)-(2.8). The estimation (3.18) follows from (3.24) by passing the weak
convergences. □

### 4 The global regularity

First, we introduce the operator pencil associated with the problem. See [11] for more detail. For convenience we rewrite the operators *L*(*x*, *t*, *D*), *N _{j }*(

*x*,

*t*,

*D*) in the form

Let *L*_{0}(*x*, *t*, *D*), *N*_{0j }(*x*, *t*, *D*), be the principal homogeneous parts of *L*(*x*, *t*, *D*), *N _{j }*(

*x*,

*t*,

*D*). It can be directly verified that the derivative

*D*can be written in the form

^{α }

where *P _{α, p }*(

*ω*,

*∂*) are differential operators of order ≤ |

_{ω}*α*| -

*p*with smooth coefficients on

*r*= |

*x*|,

*ω*is an arbitrary local coordinate system on

*S*

^{n-1},

*L*

_{0}(0,

*t*,

*D*) and

*N*

_{0j }(0,

*t*,

*D*) in the form

The operator pencil associated with the problem is defined by

For every fixed *λ *∈ ℂ and *t *∈ (0, ∞), the operator

For some fixed *t *∈ (0, ∞), a complex number *λ*_{0 }is called an eigenvalue of
*φ*_{0 }∈ *H*^{2m}(*G*) such that *φ*_{0 }≠ 0 and
*t *∈ (0, ∞), is an enumerable set of eigenvalues (see [[2], Th. 5.2.1]).

Now let us give the main theorem of this section:

**Theorem 4.1**. *Suppose that all the assumptions of Theorem 3.4 hold for a given positive integer
h. Assume further that the strip*

*does not contain any eigenvalue of *
* for all t *∈ (0, +∞) *and for some real numbers ε and α satisfying *0 ≤ *α *≤ *m *+ *ε*,
* if *
*and n *is *even, otherwise ε *= 0. *Then *
*for k *= 0, 1,..., *h *- 1 *and*

*where C is a constant independent of u and f*.

To prove Theorem 4.1 we need to establish some following lemmas.

**Lemma 4.2**. *Let l be a nonnegative integer, t*_{0 }*be a fixed number in *[0, ∞), *and let
*

*where *
*Then *
*and the following estimate*

*holds with the constant C independent of u*, *f*, *g _{j }and t*

_{0}.

*Proof*. Without generality we assume that the domain Ω coincides with the cone *K *in the unit ball. Set Ω^{0 }= {*x *∈ Ω: |*x*| ≥ 2^{-1}},

and Γ* ^{k }*= ∂Ω ∩ ∂Ω

*,*

^{k}*k*= 0, 1 .... According to well known results on the regularity of solutions of elliptic boundary problems in smooth domains (see, e.g., [12]), we have

with the constant *C *independent of *u*, *f*, *g _{j }*and

*t*

_{0}. By making change of variable

*x*= 2

^{-k }

*x*' for a positive integer

*k*, we get from (4.3), (4.4) that

Similarly as above, from (4.6), (4.7), we have

with the constant *C *independent of *u*, *f*, *g _{j}*,

*t*

_{0 }and

*k*. Let

*g*to Ω,

_{j }*j*= 1,...,

*m*. Then we have from (4.8) that

Returning to variable *x *with noting that, in Ω^{k+2}, 2^{-k-2 }≤ *r *≤ 2^{-k-1}, from (4.9) we have

Taking sum both sides of these inequalities with respect to *k *from 1 to ∞, we have

Here it is noted that

with the constant *C *independent of *u*, *f*, *g _{j }*and

*t*

_{0}. □

**Lemma 4.3**. *Let t*_{0 }*be a fixed number in *[0, ∞), *
*

*where *
*j *= 1,..., *m*, *ε is defined as in Theorem 4.1. Then *
*Moreover, the following inequalities*

*holds with the constant C independent of u, and f, g _{j}, and t*

_{0}.

*Proof*. Firstly, since
*j *= 1,..., *m*, by (2.3), it holds that

If

Now consider the case
*u *which belongs to

where

with the following estimates

Here
*C *is a constant independent of *u *and *t*_{0}. Put

where *C *is a constant independent of *u *and *t*_{0}. From this we have

Hence, it follows from (4.12) and (4.13) that

Now we can apply Lemma 4.2 to conclude from (4.17) and (4.18) that

**Proof of Theorem 4.1**: First, we show by induction on *h *that

According to Theorem 3.4 it holds that
*k *≤ *h*. In particular, *u _{tt }*∈

*L*

_{2}(

*Q*, 2

*γ*+

*σ*). Thus, from the equality (2.9) it follows that

for all *η *∈ *H ^{m}*(Ω) and a.e.

*t*∈ (0, ∞). Since

*f*(·,

*t*) -

*u*(·,

_{tt}*t*) ∈

*L*

_{2}(Ω) for a.e.

*t*∈ (0, ∞), according to results for elliptic boundary value problem in domains with smooth boundaries, it follows from (4.20) that

*t*∈ (0, ∞), moreover, the function

*u*satisfies the following equalities:

for a.e. (*x*, *t*) ∈ *Q *and

in the trace sense. Thus, the assertion (4.19) holds for *h *= 1, and by (2.5) we also have

for all
*t *∈ (0, ∞). Assume now (4.19) holds for *h *- 2. It follows from (4.20) that

for all *η *∈ *H ^{m}*(Ω), a.e.

*t*∈ (0, ∞). Since

for all
*t *∈ (0, ∞). Combining (4.22) and (4.23) we obtain

for all
*t *∈ (0, ∞), where

Similarly as above, it follows from (4.24) that
*t *∈ (0, ∞), and therefore, (4.19) holds for *h *- 1.

Now we prove the assertion of the theorem by induction on *h*. Let us consider first the case *h *= 1. We rewrite (2.6), (2.7) in the form

Since
*t *∈ [0, ∞), by Lemma 4.3, it follows from (4.25) and (4.26) that
*t *∈ (0, ∞) and

where *C *is a constant independent of *u*, *f*_{1 }and *t*. Since the trip

does not contain any eigenvalue of
*t *∈ [0, +∞), and
*ε*, we can apply Theorem 7.2.4 and the note below Theorem 7.3.5 of [2] to conclude from (4.25), (4.26) that

where *C *is a constant independent of *u*, *f*_{1 }and *t*. Now multiplying both sides of (4.27) with *e*^{-2(2γ+σ)t}, then integrating with respect to *t *from 0 to ∞ and using estimates from Theorem 3.4, we obtain

where *C *is a constant independent of *u *and *f*. Hence, the theorem is valid for *h *= 1.

Assume that the theorem is true for some nonnegative *h *- 2. We will prove it for *h *- 1. Differentiating (*h *- 1) times both sides of (4.25), (4.26) with respect to *t*, we have