1. Introduction
Pseudohyperbolic equations are equations that are unsolvable with respect to the highest derivative and have the form
where
is a quasielliptic operator. The class of pseudohyperbolic equations was introduced in [
1]. Examples of such equations are equations arising in hydrodynamics (for example, the generalized Boussinesq equation [
2,
3,
4,
5,
6,
7]), in elasticity theory (for example, the Vlasov equation [
8,
9]), and in waveguide modeling (see, for example [
10,
11,
12]).
The theory of partial differential equations of the form (1) began to develop after the publication of S. L. Sobolev’s works on the dynamics of a rotating fluid (see his works in [
13]). These works were the first in-depth studies of differential equations unsolvable with respect to the highest derivative. Therefore, equations of the form (1) are often called
Sobolev-type equations. Currently, there is a large number of publications devoted to the study of various problems for such equations. There are also more than dozen monographs on this theme (see, for example [
1,
14,
15,
16]). However, there are still few works on the theory of boundary value problems for pseudohyperbolic equations. The theory of the Cauchy problem is the most developed for such equations (see, for example [
1,
12,
14,
17,
18,
19]). Note that in these papers, the solvability of the Cauchy problem was studied in well-known Sobolev spaces
with the exponential weight
. In the literature, such spaces are often used to prove the solvability of boundary value problems for parabolic and hyperbolic equations. However, when studying the solvability of the Cauchy problem for equations unsolvable with respect to the highest derivative in the spaces
, essential differences from the classical equations may arise (see [
1]). In particular, in [
1,
17,
18,
19], the unique solvability of the Cauchy problem for strictly pseudohyperbolic equations in
was proven under the condition that the right-hand sides of the equations have certain smoothness and are orthogonal to some monomials. In these papers, it was established that the number of the orthogonality conditions is finite and essentially depends on the lower order terms of the equations. Note that the requirements of the orthogonality of the right-hand sides to some monomials for the solvability of the Cauchy problem for equations of the form (1) differ significantly from the solvability conditions for the Cauchy problem for equations of the hyperbolic and parabolic type.
In this paper, we continue the study of the Cauchy problem for strictly pseudohyperbolic equations with constant coefficients and lower order terms. Here, we define a new class of weighted Sobolev spaces
. This class contains the spaces
, i.e.,
. Functions from this class and their generalized derivatives belong to Lebesgue spaces with the exponential weight
and special power weights to
. We prove the existence and the uniqueness of the solution to the Cauchy problem in these spaces under minimal requirements on the right-hand sides. The obtained results strengthen well-known theorems from [
1,
17,
18,
19].
2. The Main Results
Recall the definition of pseudohyperbolic operators without lower order terms [
1]:
We assume that the operators satisfy the following conditions.
Condition 1. The symbol of the operator (2) is homogeneous with respect to some vector , , , , i.e., Condition 2. The operator is quasielliptic, i.e., , , if and only if .
Condition 3. The equationhas only real roots .
Definition 1. The differential operator in (2) is called pseudohyperbolic if Conditions 1–3 hold. If the roots of (3) are distinct real numbers, then the operator is called strictly pseudohyperbolic.
Consider strictly pseudohyperbolic operators with lower order terms of the form
i.e.,
is representable as
where the principal part
is a strictly pseudohyperbolic operator, and
Here, the symbols
of the operators
,
, satisfy the inequalities
where
We study the class of the operators (4) for which the equation
has distinct real roots
, and the function
for all
,
satisfies the estimates
where
are some constants.
We consider the Cauchy problem for strictly pseudohyperbolic equations with lower order terms and zero initial conditions
To study the problem, we follow the scheme of [
1,
18].
Let
. We use the symbol
to denote the Sobolev space with the weight
. The norm in
is defined as follows:
The Cauchy problem (10) was studied in [
18] in the case of
,
. The unique solvability of (10) in
was proven under the assumption that
,
.
In this article, we investigate the solvability of the Cauchy problem (10) in a wider scale of weighted Sobolev spaces
A locally integrable function
belongs to the space
, if
has the generalized derivatives
,
, in
G; moreover,
and
The norm is defined as follows:
Denote the Fourier transform of by , its partial Fourier transform in x by , and its partial Fourier transform in t by .
We seek a solution to (10) in
and assume that the following generalized derivatives exist in
:
We prove the following theorems.
Theorem 1. For every function , such thatwe have the estimatewhere the constant does not depend on . Theorem 2. Let andThen, the Cauchy problem (10) is uniquely solvable in and the solution satisfies the estimatewhere the constant does not depend on . Remark 1. The estimate (11) is the core in the proof of the uniqueness of the solution to the Cauchy problem under consideration in the spaces . An estimate of such type is called the energy inequality for strictly hyperbolic operators [20,21]. Remark 2. Note that in the case of , the theorem of the unique solvability of (10) was proven in [18]. Example 1. We consider the Cauchy problem for the pseudohyperbolic equationwhere , . It follows from [1,17] that the Cauchy problem (13) with is uniquely solvable in , , for every It follows from [18] that the Cauchy problem (13) with , is uniquely solvable in for arbitrary n and under less restrictions on the right-hand side :However, in the case of , , there exists a unique solution to the Cauchy problem (13) in , , for every Theorem 2 gives us a result on the unique solvability of the Cauchy problem (13) with , in , , , , under less restrictions on the right-hand side . Namely, for solvability, it suffices to require that 3. Uniqueness of the Solution to the Cauchy Problem
As is known, the uniqueness of the solution to the Cauchy problem for hyperbolic equations follows from energy estimates [
20,
21]. Using an analog of such energy estimates, the uniqueness of the solution to the Cauchy problem for strictly pseudohyperbolic equations in
,
, was proven in [
1,
17,
18]. Let us show that one can prove the uniqueness of the solution to the Cauchy problem in
,
, in a similar way.
Taking into account (5), (8), (9), the following lemma was proven in [
18].
Lemma 1. There is a constant such thatfor all and . Using Lemma 1, we prove Theorem 1. Due to Parseval’s equality, we have
Hence, (11) directly follows from (14). Theorem 1 is proven.
Note that, using Theorem 1, we can establish the uniqueness of the solution to the Cauchy problem (10) in
,
, for
. Indeed, if
is a solution to the Cauchy problem with
, then, extending it by zero on
, we obtain the function
satisfying the conditions of Theorem 1. Therefore, (11) yields
Then,
Taking into account (5), by Parseval’s equality, we obtain
By Condition 1, the symbol
is homogeneous with respect to the vector
. Since
with constants
independent of
, then, from (5), we have
By the conditions of Theorem 2,
, then
. Consequently, by Theorem 2 from [
22], the quasielliptic equation with lower order terms
,
, has only the zero solution from the Sobolev space with the power weight
, the norm in which is defined as follows
where
Since
with constants
independent of
, then the belonging of the solution to the Cauchy problem (10) to
,
, is equivalent to its belonging to
,
,
. Hence,
for
and (15) has only the solution
,
,
. Thus, if the solution
to (10) exists, then it is uniquely determined.
In the next three sections, we prove that, under the conditions of Theorem 2, the Cauchy problem has a solution in .
4. Construction of Approximate Solutions to the Cauchy Problem
In this section, following [
1,
18], we give formulas for approximate solutions to the Cauchy problem (10). Let
We consider the Cauchy problem for an ordinary differential equation with a real parameter
, which is obtained by formally applying the Fourier transform in
x to the problem (10)
Since
, the coefficient
at the highest derivative has a singularity at
. We study the problem (18) for
.
The solution to this problem can be represented as
where
is a contour in the complex plane surrounding all the roots of Equation (
7).
Note that the integral
is a solution to the following Cauchy problem:
Since the roots
of (7) are real and distinct, then the following lemmas hold (see [
1,
18]).
Lemma 2. The representation holdswhere for , Proof. The proof of the lemma follows directly from (20) since
Taking into account that the roots
are different and using the residue theorem, we obtain the required representation. □
Proof. Let
be the roots of (7) for
. We introduce the functions
It is easy to verify that
is a solution to the Cauchy problem
Comparing it with (21), due to the uniqueness of the solution, we obtain the identity
Since
to prove the lemma, it suffices to estimate
for
.
Taking into account the recurrence relations (22) and the realness of the roots
, we have
Consequently, the required inequality follows from (23). □
Lemma 4. The identities are validfor , . Proof. The proof of the lemma follows from (20), (21).
At first, construct a solution to the Cauchy problem (10). Applying the inverse Fourier operator in
to
in (19), we can obtain a formal solution to the problem (10). However, Lemma 2 implies that the function defined by (20) increases unboundedly as
and
can have a nonintegrable singularity at
. Hence, to obtain a formula for a solution to (10), it is necessary to apply some regularization of the inverse Fourier operator. To this end, we consider the sequence of the functions
, where
By (19) and (20),
have no singularities at
. Since
, then the inverse Fourier transform operator
in
is applicable to
and we can define the sequence of the functions
, where
Further, we show that the functions for can be considered as approximate solutions to the problem (10). □
5. Estimates of Approximate Solutions to the Cauchy Problem
In this section, we estimate in the norm of and prove that this sequence is fundamental.
Lemma 5. Let . Then,with a constant independent of m and ; moreover, for every , we haveas . Proof. By Parseval’s equality, we have
We extend the function
by zero for
, keeping the notation. Then, taking into account (24), we obtain
By (14), from the equality, it follows that
Since
, we obtain the required estimate. Convergence is proven in the same way.
The lemma is proven. □
Lemma 6. Let . Then,with a constant independent of m and ; moreover, for every , we haveas . Proof. We assume that is extended by zero on .
Using Parseval’s equality and taking into account the properties of the function
from (20), and also that
, we have
Using the formula of the Fourier transform of convolution and Lemma 4, we obtain
By Lemma 1, the inequality holds
where
. Taking into account Parseval’s equality, we have
for
and
for
. By (26), we obtain similar estimates for
.
Convergence can be proven in the same way.
The lemma is proven. □
Lemma 7. Let . Then,with a constant independent of m and ; moreover, for every , we haveas . Proof. Note that by Condition 1, the quasielliptic operator
is homogeneous with respect to the vector
, and the symbol of the differential operator
satisfies the inequality (16). By the lemma condition,
, i.e.,
. It follows from [
22] that the quasielliptic operator
establishes an isomorphism.
Therefore, for every
, the estimate holds
with a constant
independent of
. Taking into account the definition of the norm of the weighted Sobolev space, we have
By (17), since
,
, we obtain
We take
for arbitrary fixed
. Substituting
defined by (30) to (29), we have
This implies the required inequality (27). The convergence (28) can be proven in a similar way.
The lemma is proven. □
Lemma 8. Let . Then,with a constant independent of m and ; moreover, for every , we haveas . The proof of the lemma is carried out according to the scheme of the proof of the previous lemma.
Lemma 9. Let . Then,with a constant independent of m and ; moreover, for every , we haveas . Proof. For the function
defined in (19), by (21), the following equality is valid:
Using Parseval’s equality and the Heaviside function
, we conclude that
By the property of the Fourier transform of convolution, we have
Using the representation (25), we obtain
The identity
estimates (5), (6), and Lemma 1 ensure the inequality (31).
Similar arguments imply (32).
The lemma is proven. □
Lemma 10. For every functionsuch thatwe have Proof. Since
, then
By the condition of the lemma, we also have
Using Young’s inequality,
when
,
, we have
and
Taking into account (33), (35), and the notation
, from (38), it follows that
Using (34), (36), from (39), we have
Let us show that
Rewriting
in the form
and using Young’s inequality (37) with
,
, we have
Taking into account (33)–(36), from (38)–(40), we obtain the required result. □
Lemma 11. For every functionsuch that , ,we have The proof follows directly from Lemma 10.
In the next section, relying on Lemmas 5–11, we prove that the sequence is convergent in , and the limit function is a solution to the Cauchy problem (10) and satisfies (12).
6. Solvability of the Cauchy Problem
As noted above, the uniqueness of the solution to the Cauchy problem (10) in the space follows from the energy inequality when . Let us show the existence of a solution under the conditions specified in Theorem 2.
The proof of the solvability of the Cauchy problem in
is carried out in accordance with the scheme described in [
1,
17]. Note that, in contrast to [
1,
17], we study the solvability of the problem in wider weighted spaces and for equations containing lower order terms. Therefore, we use stronger estimates for approximate solutions established in the previous sections.
From Lemmas 5–8, it follows that the sequence of the functions
is fundamental in the weighted Sobolev space
and
where
is a constant independent of
m and
. Since the space
is complete, there exists a limit function
such that
and a similar estimate holds.
By Lemma 9 and the properties of generalized differentiation, the generalized derivatives
exist; moreover,
According to Lemma 11, if
is such that
then we have
for
. Consequently, taking into account the construction of the functions
, we obtain
and
Hence,
is the solution to the Cauchy problem (10) and (12) holds.
Theorem 2 is proven.
7. Conclusions
The Cauchy problem was studied for a class of strictly pseudohyperbolic equations with lower order terms. A new class of weighted Sobolev spaces
was introduced. By definition, functions from this class and their generalized derivatives belong to Lebesgue spaces with the exponential weight
and special power weights in
. In these spaces, we established new results on the unique solvability of the considered Cauchy problem under minimal conditions on the right-hand sides of the equations. Theorem 2 strengthens the well-known results of the works [
1,
17,
18,
19] in which theorems on the solvability of the Cauchy problem were established in known Sobolev spaces with an exponential weight in
t. Theorem 2 can be considered as an analog of theorems on the solvability of the Cauchy problem for strictly hyperbolic equations.
Author Contributions
Conceptualization, G.V.D.; methodology, G.V.D.; validation, L.N.B. and G.V.D.; investigation, L.N.B. and G.V.D.; writing—original draft preparation, L.N.B. and G.V.D.; writing—review and editing, L.N.B. and G.V.D. All authors have read and agreed to the published version of the manuscript.
Funding
The work is supported by the Mathematical Center in Akademgorodok under agreement No. 075-15-2022-282 with the Ministry of Science and Higher Education of the Russian Federation.
Data Availability Statement
Not applicable.
Acknowledgments
The authors thank the anonymous reviewers for their helpful remarks.
Conflicts of Interest
The authors declare no conflict of interest.
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