The goal of this paper is to establish the regularity of the solution of the first initial-boundary value problem for general higher-order hyperbolic equations in cylinders with the bases containing cuspidal points.
1. Introduction
Initial boundary-value problems for hyperbolic and parabolic type equations in a cylinder with the base containing conical points have been developed sufficiently by us [1–4], the main results of which are about the unique existence of the solution and asymptotic expansions of the solution near a neighborhood of a conical point. However, those problems mentioned above in cylinder with base containing cuspidal point, also interesting for applied sciences, have not been studied yet.
In the present paper, we are concerned with the first initial boundary value problems for higher hyperbolic equation in a cylinder, whose base containing cuspidal points.
In [5, 6] we showed the existence of a sequence of smooth domains 
such that 
and 
. Furthermore, we proved the existence, the uniqueness, and the smoothness with respect
to time variable of the generalized solution by approximating boundary method, which
can be applied for nonlinear equations. With the help of the results in [5, 6] as well as the results for elliptic boundary value problems in [7, 8], we can deal with the regularity with respect to both time variables and spatial
ones of the solution.
Our paper is organized as follows: in Section 2, we introduce exterior cusp domain and weight Sobolev spaces. In Section 3, we will state the formulation of the problem. The main results, Theorems 4.3, 4.6, and 4.7, are stated in Section 4, and examples are given in Section 5.
2. Cusp Domain and Weighted Sobolev Spaces
Let 
be an infinitely differentiable positive function on the interval 
satisfying the following conditions:
(i)
for 
(ii)
.
These conditions are satisfied; the function 
if 
is an example. Obviously, conditions (i) and (ii) imply 
. We assume that 
is a bounded domain in 
is smooth, and
(21)where 
is a smooth domain in 
. Then the mapping
(22)takes the set 
onto the half-cylinder 
. Moreover, it follows that
(23)We extend the functions 
to an infinitely differentiable positive function on the interval 
The space 
can be defined as the closure of the set 
with respect to the norm
(24)It is known that 
, then 
(see [7, Lemma 
]).
We also denote by 
the Sobolev space of functions 
and 
that have generalized derivatives 
, 
. The norm in this space is defined as follows:
(25)The space 
is the completion of 
in norm of the space 
.
Set 
; we proceed to introduce some functional spaces. Let 
be Banach spaces, we denote by 
the spaces consisting of all measurable functions 
with norm
(26)and by 
, 
the spaces consisting of all functions 
such that generalized derivatives 
exist and belong to 
, (see [9]), with norms
(27)For shortness, we set
(28)Finally, we define the weighted Sobolev space 
as a set of all functions defined in 
such that
(29)To simplify notation, we continue to write 
instead of 
.
3. Formulation of the Problem
Let us consider the partial differential operator of order 
(31)where 
are functions with complex values, 
(
denotes the transposed conjugate of 
) and 
are infinitely differentiable in 
. Moreover, we assume that the functions
(32)satisfy the condition of stabilization for 
for a.e. 
in 
(see [7, Section 
]). Then the coefficients of the operators 
, which arise from operators 
via the coordinate change 
, stabilize for 
. If we replace the coefficients of the differential operator 
by their limits for 
, we get differential operator which has coefficients depending only on 
and 
(for the convenience in use, we denote also by 
).
In the paper, we usually use the following Green's formula:
(33)which is valid for all 
and a.e. 
, where
(34)We also suppose that the form 
is 
-elliptic uniformly with respect to 
, that is, the inequality
(35)is valid for all 
and all 
, where 
is the positive constant independent of 
and 
. In this paper, we consider the following problem:
(36)
(37)
(38)where 
and 
are derivatives with respect to the outer unit normal of 
.
Definition 3.1.
A function 
is called a generalized solution of problem (3.6)–(3.8) if and only if 
belongs to 
and the equality
(39)holds for all 
.
The existence, the uniqueness and the smoothness with respect to the time variable
for the generalized solution of problem (3.6)–(3.8) in the Sobolev space 
were established in [5, 6] according to following theorems:
Theorem 3.2.
Assume that 
and there exists a positive number 
such that
(310)Then problem (3.6), (3.8) has the unique generalized solution 
, and the following estimate holds
(311)where 
is a constant independent of 
and 
.
Theorem 3.3.
Suppose that the following hypotheses are satisfied:
(i)
(ii)
(iii)
.
Then the generalized solution 
of problem (3.6), (3.8) has generalized derivatives with respect to 
up to order h in 
and satisfies the following estimate:
(312)where 
is a constant independent of 
and 
.
Owing to the support of the following proposition, we can apply the results of the Dirichlet problem for elliptic equation in domains with exterior cusps.
Proposition 3.4.
Suppose that 
is a generalized solution of problem (3.6)–(3.8) and 
. Then for a.e. 
, 
is a generalized solution in 
of the Dirichlet problem for elliptic equation
(313)where 
.
Proof.
Let 
be an orthogonal basis of the space 
. Setting 
, where 
, and substituting the function 
into (3.9), we conclude that
(314)We will denote by
(315)that 
Noting that 
and using Fubini's theorem, we obtain from (3.14) that 
in 
, where 
is a set of measure zero. Since 
are dense in 
, the following equality
(316)holds for all 
for all 
. It follows that 
is a generalized solution in 
of the Dirichlet problem for elliptic equation (3.13), for a.e. 
.
4. The Main Results
In this section, we would like to present the main results of the study which is based
on our previous results (cf. [5, 6]) and the results of elliptic equations in cusp domains (cf. [7]). For the start of this section, we denote by 
the operator corresponding to the parameter-depending boundary value problem
(41)For each 
we have the operator pencil 
to be Fredholm, and its spectrum consists of a countable number of isolated eigenvalues.
Similarly to Theorem 
in [7], we have the following lemma.
Lemma 4.1.
Assume that 
, where 
are real numbers. Additionally, the authors suppose that no eigenvalues of 
line in strips 
and 
, where 
and 
are eigenvalues of 
, and 
. Then the generalized solution 
of the Dirichlet problem for elliptic equation (3.13), such that 
if 
, belongs to the 
and satisfies the inequality
(42)where the constant 
is independent of 
.
Proof.
Setting
(43)by the Friederichs inequality, we have
(44)therefore,
(45)for all 
. Hence,
(46)Let 
be the function which arises from 
via the coordinate change 
. We set 
; then from the properties of the mapping (2.2) and from inequality (4.6), it follows
that 
. Since 
is the solution of an elliptic equation in 
with coefficients which stabilize for 
, that is,
(47)where 
, we obtain 
(cf. [7, Lemma 
]). This implies 
. Using the fact that
(48)as 
if 
, we conclude that 
From Corollary 
in [7] it follows that 
. Furthermore, (4.2) holds.
Lemma 4.2.
Suppose that 
, and the strip 
does not contain eigenvalues of 
. Then the generalized solution 
of problem (3.6)–(3.8), such that 
if 
, belongs to the 
and satisfies the inequality
(49)where the constant 
is independent of 
.
Proof.
Using the smoothness of the generalized solution of problem (3.6)–(3.8) with respect
to 
in Theorem 3.3 and Proposition 3.4, we can see that for a.e. 
is the generalized solution of Dirichlet problem for (3.13) with 
. From Lemma 4.1, it implies that 
for a.e. 
and satisfies the inequality
(410)By integrating the inequality above with respect to 
from 
to 
, and using the estimates for derivatives of 
with respect to 
again, we obtain 
, which satisfies inequality (4.9).
Theorem 4.3.
Let the assumptions of Lemma 4.2 be satisfied, and 
, 
, 
for 
. Then the generalized solution 
of problem (3.6)–(3.8), such that 
if 
, belongs to the 
and satisfies the inequality
(411)where the constant 
is independent of 
.
Proof.
Let us first prove that 
belong to the 
for 
and satisfy
(412)The proof is an induction on 
. According to Lemma 4.2, it is valid for 
. Now let this assertion be true for 
; we will prove that this also holds for 
. Due to Lemma 4.2, 
satisfies (3.6). By differentiating both sides of (3.6) with respect to 
times, we obtain
(413)where
(414)By the supposition of the theorem and the inductive assumption, the right-hand side
of (4.13) belongs to 
. By the arguments analogous to the proof of Lemma 4.2, we get 
and
(415)where 
is a constant independent of 
, and 
.
By using (4.15) and estimates for derivatives of 
with respect to 
in Theorem 3.3, we have
(416)Remark 4.4.
Let 
be a sufficiently small positive number. Suppose that 
and the strip 
contains no eigenvalues of 
, 
; then the generalized solution 
of problem (3.6)–(3.8), such that 
if 
, belongs to the 
. In fact, setting 
, we obtain the first initial boundary value problem which differs little from (3.6)–(3.8).
Therefore, 
, and then 
. Using the remark above and Lemma 4.1, we obtain the following theorem.
Theorem 4.5.
Let the assumptions of Lemma 4.1 be satisfied. Furthermore, the authors assume that
, 
and 
for 
. Then the generalized solution 
of problem (3.6)–(3.8), such that 
if 
, belongs to the 
and satisfies the inequality
(417)where the constant 
is independent of 
.
This theorem is proved by arguments analogous to those proofs of Lemma 4.2 and Theorem 4.3. Next, we will prove the well regularity of the generalized solution of problem (3.6)–(3.8).
Theorem 4.6.
Let the assumptions of Lemma 4.1 be satisfied. Furthermore, the authors assume that
, 
and 
for 
. Then the generalized solution 
of problem (3.6)–(3.8), such that 
if 
, belongs to the 
and satisfies the inequality
(418)where the constant 
is independent of 
and 
.
Proof.
The theorem is proved by induction on 
. Thanks to Theorem 4.5, this theorem is obviously valid for 
. Assume that the theorem is true for 
, we will prove that it also holds for 
. It is only needed to show that
(419)Differentiating both sides of (3.6) again with respect to 
times, we obtain
(420)By the supposition of the theorem and the inductive assumption, the right-hand side
of (4.20) belongs to 
for a.e. 
. Using Lemma 4.1, we conclude that 
. It implies that (4.19) holds for 
. Suppose that (4.19) is true for 
and set 
, we obtain
(421)where 
. By the inductive assumption with respect to 
, 
belongs to 
for a.e. 
. Thus, the right-hand side of (4.21) belongs to 
. Applying Lemma 4.1 again for 
, we get that 
for a.e. 
. It means that 
belongs to 
. Furthermore, we have
(422)Therefore,
(423)It implies that (4.19) holds for 
. The proof is complete.
Now we will prove the global regularity of the solution.
Theorem 4.7.
Let the hypotheses of Lemma 4.1 be satisfied. Furthermore, suppose 
, 
and 
,for 
. Then the generalized solution 
of problem (3.6)–(3.8) belongs to the 
and satisfies the inequality
(424)where the constant 
is independent of 
and 
.
Proof.
We denote by 
the unit ball, and suppose that 
and 
in the neighborhood of the origin 
. It is easy to get that
(425)where 
is a differential operator, whose coefficients have compact support in a neighborhood
of the origin. By arguments analogous to the proof of Theorem 4.6, we obtain
(426)Set 
, then 
in a neighborhood of the origin and 
, and using the smoothness of the solution of this problem in domain with smooth boundary,
we get
(427)The proof is complete.
5. Examples
In this section, we apply the results of the previous section to the Cauchy-Dirichlet
problem for the wave equation. The assumptions can be described as follows: 
is a bounded domain in 
is smooth,
(51)where 
, 
, 
, 
as 
and 
, and 
, 
.
We consider the Cauchy-Dirichlet problem for the wave equation in 
:
(52)where 
. It follows the results of Section 4 that if 
, where 
is the least positive root of the Bessel function 
, then problem (5.2) has a unique solution 
in 
and we have the estimate
(53)Moreover, if 
, and 
for 
, then 
and satisfies
(54)For the two-dimensional case 
, and letting 
, we consider problem (5.2) in the cylinder 
, where 
is a bounded domain in 
is smooth, and
(55)Thus, the change of variables
(56)transforms
(57)With notations 
, we have
(58)
(59)Hence, the differential operator 
, which arises from the differential operator 
via the coordinate change 
, turns out to be
(510)Clearly, coefficients of differential operator 
stabilize for 
, and the limit differential operator of 
(denoted by 
for convenience) is
(511)We denote also by 
the operator corresponding to the parameter-depending boundary value problem
(512)Eigenvalues of 
are roots of the Bessel function
(513)
has only real roots (see [10, Theorem 
, page 94]). Therefore, they are
(514)It is easy to see that 
is the least positive root of the Bessel function 
. From arguments above in combination with Theorems 4.6 and 4.7, we obtain the following
results:
Theorem 5.1.
Suppose that 
, 
, 
, 
is a real number and 
for 
. Then problem (5.2) has a unique solution 
in 
and we have the estimate
(515)Moreover, if 
, and 
for 
, then 
and satisfies
(516)In case that boundary of 
has some cuspidal points, then by arguments analogous to Section 4, we consequently
obtain the similar results.
Acknowledgment
This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam.
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