# On the regularity of the solution for the second initial boundary value problem for hyperbolic systems in domains with conical points

Nguyen Manh Hung1, Nguyen Thanh Anh1* and Phung Kim Chuc2

Author Affiliations

1 Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam

2 Department of Mathematics, Can Tho University, Can Tho City, Vietnam

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Boundary Value Problems 2011, 2011:17  doi:10.1186/1687-2770-2011-17

 Received: 22 January 2011 Accepted: 25 August 2011 Published: 25 August 2011

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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 ℝn, 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 Sn-1. For each t, 0 < t ≤ ∞, denote Qt = Ω × (0, t), Ωt = Ω × {t}. Especially, we set Q = Q, Γ = ∂Ω\{0}, S = Γ × [0, +∞).

For each multi-index p = (p1,..., pn) ∈ ℕn, we use notations |p| = p1 + ... + pn, . For a complex-valued vector function u = (u1 ,..., us) defined on Q, we denote .

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

Hl (Ω) - the usual Sobolev space of vector functions u defined in Ω with the norm

- the space of traces of vector functions from Hl (Ω) on Γ with the norm

Hl,0 (Q, γ) (γ ∈ ℝ)- the weighted Sobolev space of vector functions u defined in Q with the norm

Especially, we set L2(Q, γ) = H0,0(Q, γ).

Hl,1 (Q, γ) (γ ∈ ℝ)- the weighted Sobolev space of vector functions u defined in Q with the norm

- the closure of with respect to the norm

where .

-the weighted Sobolev space of vector functions u defined in Ω with the norm

If l ≥ 1, then denote the spaces consisting of traces of functions from respective spaces on the boundary Γ with the respective norms

- the weighted Sobolev space of vector functions u defined in Q with the norm

From the definitions it follows the continuous imbeddings

(2.1)

and

(2.2)

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

(2.3)

with the norms being equivalent.

Now we introduce the differential operator

where apq = apq (x, t) are the s × s matrices with the bounded complex-valued components in . We assume that for all |p|, |q| ≤ m, where is the transposed conjugate matrix to apq. This means the differential operator L is formally self-adjoint. We assume further that there exists a positive constant μ such that

(2.4)

for all ηp ∈ℂs, |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 Nj = Nj (x, t, D), j = 1, 2,..., m on S such that integration equality

(2.5)

holds for all and for all t ∈ [0, ∞). The order of the operator Nj is 2m - j for j = 1, 2,..., m.

In this paper, we consider the following problem:

(2.6)

(2.7)

(2.8)

A complex vector-valued function u Hm,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

(2.9)

holds for all η(x, t) ∈ Hm,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

(3.1)

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

(3.2)

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

(3.3)

holds for all u Hm(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 apq together with their derivatives with respect to t are bounded on . Then there exists a positive real number γ0 such that for each γ > γ0, if f L2(Q, σ) for some nonnegative real number σ, the problem (2.6)-(2.8) has a unique generalized solution u in the space Hm,1(Q, γ + σ) and

(3.4)

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 is an orthogonal basis of Hm(Ω) which is orthonormal in L2(Ω). Put

where are the solution of the system of the following ordinary differential equations of second order:

(3.5)

with the initial conditions

(3.6)

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

(3.7)

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

(3.8)

With noting that, for some positive real number ρ,

we can rewrite (3.8) as follows

(3.9)

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 ε1, we have

where C1 = C1(ε1) is a nonnegative constant independent of uN, f and τ. Now using again the Cauchy and interpolation inequalities, for an arbitrary positive number ε2 with ε2 < μ, it holds that

where C2 = C2(ε2) is a nonnegative constant independent of uN, f and τ. For the terms III and IV, by the Cauchy inequality, we have

and

where ε3 > 0, arbitrary. Combining the above estimations we get from (3.9) that

(3.10)

Now fix ε1, ε2 and consider the function

We have

We see that the function g has a unique minimum at

We put

(3.11)

Now we take real numbers γ, γ1 arbitrarily satisfying γ0 < γ1 < γ. Then there are positive real numbers ε1, ε2, (ε2 < μ), ρ (ρ > C2(ε1, ε2)) and ε3 such that

(3.12)

From now to the end of the present proof, we fix such constants ε1, ε2, ε3 and ρ. Let stand for the left-hand side of (3.10). It follows from (3.10) and (3.12) that

(3.13)

where . By the Gronwall-Bellman inequality (3.2), we receive from (3.13) that

(3.14)

We see that

Hence, it follows from (3.14) that

(3.15)

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

(3.16)

It is clear that |||.|||Q,γ+σ is a norm in Hm,1(Q, γ + σ) which is equivalent to the norm . Thus, it follows from (3.16) that

(3.17)

From this inequality, by standard weakly convergent arguments (see, e.g., [[10], Ch. 7]), we can conclude that the sequence possesses a subsequence convergent to a vector function u Hm,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 apq together with their derivatives with respect to t up to the order h are bounded on . Let γ0 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 > γ0 the generalized solution u in the space Hm,1(Q, γ + σ) of the problem (3.6)- (3.7) has derivatives with respect to t up to the order h with for k = 0, 1,..., h and

(3.18)

where C is a constant independent of u and f.

Proof. From the assumptions on the regularities of the coefficients apq and of the function f it follows that the solution of the system (3.5), (3.6) has generalized derivatives with respect to t up to the order h + 2. Now take an arbitrary real number γ1 satisfying γ0 < γ1 < γ. We will prove by induction that

(3.19)

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 . Assuming by induction that (3.19) holds for 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

(3.20)

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

(3.21)

Now multiplying both sides of (3.20) by , then taking sum with respect to l from 1 to N, we get

(3.22)

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

(3.23)

This equality has the form (3.8) with uN replaced by and the last term of the righthand side of (3.8) replaced by the following expression

Since the coefficients apq 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

(3.24)

From this inequality, by again standard weakly convergent arguments, we can conclude that the sequence possesses a subsequence convergent to a vector function u(k) Hm,1(Q, (k +1)γ +σ), moreover, u(k) is the kth 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), Nj (x, t, D) in the form

Let L0(x, t, D), N0j (x, t, D), be the principal homogeneous parts of L(x, t, D), Nj (x, t, D). It can be directly verified that the derivative Dα can be written in the form

where Pα, p (ω, ω) are differential operators of order ≤ |α| - p with smooth coefficients on , r = |x|, ω is an arbitrary local coordinate system on Sn-1, , . Thus we can write L0(0, t, D) and N0j (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 continuously maps

For some fixed t ∈ (0, ∞), a complex number λ0 is called an eigenvalue of if there exists φ0 H2m(G) such that φ0 ≠ 0 and . It is well known that the spectrum of the operator for each 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

(4.1)

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

(4.2)

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, t0 be a fixed number in [0, ∞), and let be a solution of the following elliptic boundary value problem

(4.3)

(4.4)

where . Then and the following estimate

(4.5)

holds with the constant C independent of u, f, gj and t0.

Proof. Without generality we assume that the domain Ω coincides with the cone K in the unit ball. Set Ω0 = {x ∈ Ω: |x| ≥ 2-1},

and Γk = ∂Ω ∩ ∂Ωk, 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, gj and t0. By making change of variable x = 2-k x' for a positive integer k, we get from (4.3), (4.4) that

(4.6)

(4.7)

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

(4.8)

with the constant C independent of u, f, gj, t0 and k. Let be arbitrary extensions of gj to Ω, j = 1,..., m. Then we have from (4.8) that

(4.9)

Returning to variable x with noting that, in Ωk+2, 2-k-2 r ≤ 2-k-1, from (4.9) we have

(4.10)

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

(4.11)

Here it is noted that can be estimated by the right-hand side of (4.11). It follows from (4.11) that

with the constant C independent of u, f, gj and t0.   □

Lemma 4.3. Let t0 be a fixed number in [0, ∞), be a generalized solution of the following elliptic boundary value problem

(4.12)

(4.13)

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

(4.14)

holds with the constant C independent of u, and f, gj, and t0.

Proof. Firstly, since for j = 1,..., m, by (2.3), it holds that , and therefore, . Moreover, it is obvious that . Hence, and for .

If , then by (2.3) and (2.2). Thus the assertion of the lemma follows from Lemma 4.2 with noting that the space is continuously imbedded in according to (2.1).

Now consider the case . Let be the greatest integer not exceeding . Then we have . According to [[2], Th. 7.1.1], the function u which belongs to has the representation

where , and

with the following estimates

(4.15)

(4.16)

Here , and C is a constant independent of u and t0. Put . From (4.16) we see easily that and

where C is a constant independent of u and t0. From this we have

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

(4.17)

(4.18)

Now we can apply Lemma 4.2 to conclude from (4.17) and (4.18) that . Therefore, with the estimate (4.14). The lemma is completely proved.

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

(4.19)

According to Theorem 3.4 it holds that , k h. In particular, utt L2(Q, 2γ + σ). Thus, from the equality (2.9) it follows that

(4.20)

for all η Hm(Ω) and a.e. t ∈ (0, ∞). Since f (·, t) - utt(·, t) ∈ L2(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 for a.e. 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

(4.21)

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

(4.22)

for all η Hm(Ω), a.e. t ∈ (0, ∞). Since , by the induction assumption, we have from (4.21) that

(4.23)

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

(4.24)

for all and a.e. t ∈ (0, ∞), where

Similarly as above, it follows from (4.24) that for a.e. 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

(4.25)

(4.26)

Since for a.e. t ∈ [0, ∞), by Lemma 4.3, it follows from (4.25) and (4.26) that for a.e. t ∈ (0, ∞) and

where C is a constant independent of u, f1 and t. Since the trip

does not contain any eigenvalue of for all t ∈ [0, +∞), and by the definition of the number 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 and

(4.27)

where C is a constant independent of u, f1 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

(4.28)

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

(4.29)

(4.30)

By the induction assumption, it holds that

Moreover,

by the assumption of the theorem and

by Theorem 3.4. Thus, for a.e. t ∈ (0, ∞), we have , and

where C is the constant independent of u, f and t. Now we can repeat the arguments above to conclude that with the estimate (4.2) for k = h -1. The proof is completed.

### 5 An example

In this section we apply the previous results to the Cauchy-Neumann problem for the classical wave equation. We consider the following problem:

(5.1)

(5.2)

(5.3)

where Δ is the Laplace operator.

For problem (5.1)-(5.3) it can be directly verified that the constants μ, μ1 and γ0 are now defined by

The operator pencil associated with the problem (5.1)-(5.3) is now defined by (see [[13], Sec. 2.3])

(5.4)

where δ is the Laplace-Beltrami operator on the unit sphere Sn-1. It is well known that (see also [[13], Sec. 2.3]) the trip

(5.5)

does not contains any eigenvalue of the operator pencil . We see that if 0 ≤ α ≤ 1 and n > 4 - 2α, then the trip (4.1) with m = 1 is contained in the trip (5.5), since ε can be chosen as zero or an arbitrary small positive real number. Thus, we can apply Theorem 4.1 to receive the following result.

Theorem 5.1. Let h be a nonnegative integer and α be a real number, 0 ≤ α ≤ 1. Assume that the vector function f satisfy the following conditions for some nonnegative real number σ

(i)

(ii)

Assume further that n > 4 - 2α. Then for an arbitrary positive real number γ the problem (5.1)-(5.3) has a unique generalized solution u in the space H1,1(Q, γ + σ) which has derivatives with respect to t up to the order h with for k = 0, 1,..., h - 1, and

where C is a constant independent of u and f.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors typed, read and approved the final manuscript.

### Acknowledgements

This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam, under project no. 101.01.58.09.

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