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 wellposedness 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 [35] 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 secondorder hyperbolic systems with the coefficients independent of time in domains with conical points was studied in [6]. In the present paper we consider the CauchyNeumann (the second initial) boundary value problem for higherorder 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^{n1}. For each t, 0 < t ≤ ∞, denote Q_{t }= Ω × (0, t), Ω_{t }= Ω × {t}. Especially, we set Q = Q_{∞}, Γ = ∂Ω\{0}, S = Γ × [0, +∞).
For each multiindex p = (p_{1},..., p_{n}) ∈ ℕ^{n}, we use notations p = p_{1 }+ ... + p_{n}, . For a complexvalued vector function u = (u_{1 },..., u_{s}) defined on 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
 the space of traces of vector functions from 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
 the closure of with respect to the norm
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
and
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
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 complexvalued components in . We assume that for all p, q ≤ m, where is the transposed conjugate matrix to a_{pq}. This means the differential operator L is formally selfadjoint. 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 and for all t ∈ [0, ∞). The order of the operator N_{j }is 2m  j for j = 1, 2,..., m.
In this paper, we consider the following problem:
A complex vectorvalued 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 selfadjointness of the operator L, we see that
Next, we introduce the following GronwallBellman 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 realvalued 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 . Then there exists a positive real number γ_{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 is an orthogonal basis of H^{m}(Ω) which is orthonormal in L_{2}(Ω). Put
where are the solution of the system of the following ordinary differential equations of second order:
with the initial conditions
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
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 lefthand 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 righthand 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 stand for the lefthand side of (3.10). It follows from (3.10) and (3.12) that
where . By the GronwallBellman inequality (3.2), we receive from (3.13) that
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 . Thus, it follows from (3.16) that
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 ∈ 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 . Let γ_{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
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 for 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 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 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 . 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
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 , then taking sum with respect to 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 and the last term of the righthand side of (3.8) replaced by the following expression
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 possesses a subsequence convergent to a vector function u^{(k) }∈ H^{m,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), 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^{α }can be written in the form
where P_{α, p }(ω, ∂_{ω}) are differential operators of order ≤ α  p with smooth coefficients on , r = x, ω is an arbitrary local coordinate system on S^{n1}, , . Thus we can write 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 continuously maps
For some fixed t ∈ (0, ∞), a complex number λ_{0 }is called an eigenvalue of if there exists φ_{0 }∈ H^{2m}(G) such that φ_{0 }≠ 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
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 be a solution of the following elliptic boundary value problem
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 be arbitrary extensions of g_{j }to Ω, j = 1,..., m. Then we have from (4.8) that
Returning to variable x with noting that, in Ω^{k+2}, 2^{k2 }≤ r ≤ 2^{k1}, 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 can be estimated by the righthand side of (4.11). It follows from (4.11) 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, ∞), be a generalized solution of the following elliptic boundary value problem
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 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
with the following estimates
Here , and C is a constant independent of u and t_{0}. Put . From (4.16) we see easily that and
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 . Therefore, with the estimate (4.14). The lemma is completely proved.
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, 2γ + 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 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
for all and a.e. 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 , by the induction assumption, we have from (4.21) that
for all and a.e. t ∈ (0, ∞). Combining (4.22) and (4.23) we obtain
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
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, f_{1 }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
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
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 CauchyNeumann problem for the classical wave equation. We consider the following problem:
where Δ is the Laplace operator.
For problem (5.1)(5.3) it can be directly verified that the constants μ, μ_{1 }and γ_{0 }are now defined by
The operator pencil associated with the problem (5.1)(5.3) is now defined by (see [[13], Sec. 2.3])
where δ is the LaplaceBeltrami operator on the unit sphere S^{n1}. It is well known that (see also [[13], Sec. 2.3]) the trip
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
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 H^{1,1}(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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