The goal of this paper is to establish the regularity of the solution of the first initialboundary value problem for general higherorder hyperbolic equations in cylinders with the bases containing cuspidal points.
1. Introduction
Initial boundaryvalue 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
where is a smooth domain in . Then the mapping
takes the set onto the halfcylinder . Moreover, it follows that
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
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:
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
and by , the spaces consisting of all functions such that generalized derivatives exist and belong to , (see [9]), with norms
For shortness, we set
Finally, we define the weighted Sobolev space as a set of all functions defined in such that
To simplify notation, we continue to write instead of .
3. Formulation of the Problem
Let us consider the partial differential operator of order
where are functions with complex values, ( denotes the transposed conjugate of ) and are infinitely differentiable in . Moreover, we assume that the functions
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:
which is valid for all and a.e. , where
We also suppose that the form is elliptic uniformly with respect to , that is, the inequality
is valid for all and all , where is the positive constant independent of and . In this paper, we consider the following problem:
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
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
Then problem (3.6), (3.8) has the unique generalized solution , and the following estimate holds
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:
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
where .
Proof.
Let be an orthogonal basis of the space . Setting , where , and substituting the function into (3.9), we conclude that
We will denote by
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
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 parameterdepending boundary value problem
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
where the constant is independent of .
Proof.
Setting
by the Friederichs inequality, we have
therefore,
for all . Hence,
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,
where , we obtain (cf. [7, Lemma ]). This implies . Using the fact that
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
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
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
where the constant is independent of .
Proof.
Let us first prove that belong to the for and satisfy
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
where
By the supposition of the theorem and the inductive assumption, the righthand side of (4.13) belongs to . By the arguments analogous to the proof of Lemma 4.2, we get and
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
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
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
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
Differentiating both sides of (3.6) again with respect to times, we obtain
By the supposition of the theorem and the inductive assumption, the righthand 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
where . By the inductive assumption with respect to , belongs to for a.e. . Thus, the righthand 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
Therefore,
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
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
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
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
The proof is complete.
5. Examples
In this section, we apply the results of the previous section to the CauchyDirichlet problem for the wave equation. The assumptions can be described as follows: is a bounded domain in is smooth,
where , , , as and , and , .
We consider the CauchyDirichlet problem for the wave equation in :
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
Moreover, if , and for , then and satisfies
For the twodimensional case , and letting , we consider problem (5.2) in the cylinder , where is a bounded domain in is smooth, and
Thus, the change of variables
transforms
With notations , we have
Hence, the differential operator , which arises from the differential operator via the coordinate change , turns out to be
Clearly, coefficients of differential operator stabilize for , and the limit differential operator of (denoted by for convenience) is
We denote also by the operator corresponding to the parameterdepending boundary value problem
Eigenvalues of are roots of the Bessel function
has only real roots (see [10, Theorem , page 94]). Therefore, they are
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
Moreover, if , and for , then and satisfies
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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