On the Cauchy Problem for Pseudohyperbolic Equations with Lower Order Terms

We consider a class of strictly pseudohyperbolic equations with lower order terms. The solvability of the Cauchy problem in Sobolev spaces with special weights is established. The uniqueness of the solution is proven and estimates are obtained.


Introduction
Pseudohyperbolic equations are equations that are unsolvable with respect to the highest derivative and have the form where L 0 (D x ) 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 W r 2,γ with the exponential weight e −γt .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 W l,r 2,γ , 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 W l,r 2,γ 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 W l,r 2,γ,κ q ,σ .This class contains the spaces W l,r 2,γ , i.e., W l,r 2,γ ⊂ W l,r 2,γ,κ q ,σ .Functions from this class and their generalized derivatives belong to Lebesgue spaces with the exponential weight e −γt and special power weights to x ∈ R n .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].

The Main Results
Recall the definition of pseudohyperbolic operators without lower order terms [1]: We assume that the operators satisfy the following conditions.
Condition 3. The equation has only real roots η 1 (ξ), . . ., η l (ξ).Consider strictly pseudohyperbolic operators with lower order terms of the form i.e., L(D t , D x ) is representable as where the principal part L(D t , D x ) is a strictly pseudohyperbolic operator, and Here, the symbols L l−k (iξ) of the operators L l−k (D x ), k = 0, . . ., l, satisfy the inequalities where We study the class of the operators (4) for which the equation has distinct real roots η 1 (ξ), . . ., η l (ξ), and the function for all (η, ξ) ∈ R n+1 , γ ≥ 0 satisfies the estimates where a 1 , a 2 > 0 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 G ⊆ R n+1 .We use the symbol to denote the Sobolev space with the weight e −γt .The norm in W l,r 2,γ (G) is defined as follows: u(t, x), W l,r 2,γ (G) = e −γt u(t, x), W l,r 2 (G) .The Cauchy problem (10) was studied in [18] in the case of The unique solvability of (10) in W l,r 2,γ (R n+1 + ) was proven under the assumption that D k t u(t, x) ∈ W 0,(1−kα 0 )r 2,γ (R n+1 + ), k = 0, . . ., l.In this article, we investigate the solvability of the Cauchy problem (10) in a wider scale of weighted Sobolev spaces A locally integrable function u(t, x) belongs to the space W l,r 2,γ,κ q ,σ (G), if u(t, x) has the generalized derivatives D and The norm is defined as follows: Denote the Fourier transform of u γ (t, x) = e −γt u(t, x) ∈ L 2 (R n+1 ) by u γ (η, ξ), its partial Fourier transform in x by u γ (t, ξ), and its partial Fourier transform in t by u γ (η, x).
We seek a solution to (10) in W l,r 2,γ,κ q ,1 (R n+1 + ) and assume that the following generalized derivatives exist in R n+1 + : We prove the following theorems.
Theorem 1.For every function u(t, x) ∈ W l,r 2,γ,κ q ,1 (R n+1 ), such that we have the estimate where the constant c > 0 does not depend on u(t, x).
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 W l,r 2,γ,κ q ,1 (R n+1 + ).An estimate of such type is called the energy inequality for strictly hyperbolic operators [20,21].
Remark 2. Note that in the case of κ q = 0, the theorem of the unique solvability of (10) was proven in [18].

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 W l,r 2,γ (R n+1 + ), γ > 0, was proven in [1,17,18].Let us show that one can prove the uniqueness of the solution to the Cauchy problem in W l,r 2,γ,κ q ,1 (R n+1 + ), |α|/2 > κ q , in a similar way.
In the next three sections, we prove that, under the conditions of Theorem 2, the Cauchy problem has a solution in W l,r 2,γ,κ q ,1 (R n+1 + ).

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 ξ, which is obtained by formally applying the Fourier transform in x to the problem ( 10) Since κ q > 0, the coefficient L 0 (iξ) + L 0 (iξ) at the highest derivative has a singularity at ξ = 0. We study the problem (18) for ξ ∈ R n \{0}.
The solution to this problem can be represented as where Γ(ξ) is a contour in the complex plane surrounding all the roots of Equation (7).Note that the integral J(t, ξ) is a solution to the following Cauchy problem: Since the roots η k (ξ) of ( 7) are real and distinct, then the following lemmas hold (see [1,18]).

Lemma 2. The representation holds
Proof.The proof of the lemma follows directly from (20) since Taking into account that the roots η 1 (ξ), . . ., η l (ξ) are different and using the residue theorem, we obtain the required representation.
Lemma 3. The estimate holds Proof.Let η 1 (ξ), . . ., η l (ξ) be the roots of (7) for ξ ∈ R n \{0}.We introduce the functions It is easy to verify that ψ l (t, ξ) is a solution to the Cauchy problem Comparing it with (21), due to the uniqueness the solution, we obtain the identity to prove the lemma, it suffices to estimate |ψ l (t, ξ)| for t ≥ 0. Taking into account the recurrence relations (22) and the realness of the roots η k (ξ), we have Consequently, the required inequality follows from (23).

Lemma 4. The identities are valid
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 ξ to v(t, ξ) 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 |ξ| → 0 and v(t, ξ) can have a nonintegrable singularity at ξ = 0. 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 {v m (t, ξ)}, where By (19) and (20), v m (t, ξ) have no singularities at ξ = 0. Since f (t, x) ∈ W 0,s 2,γ (R n+1 + ), then the inverse Fourier transform operator F −1 in ξ is applicable to v m (t, ξ) and we can define the sequence of the functions {u m (t, x)}, where Further, we show that the functions u m (t, x) m 1 can be considered as approximate solutions to the problem (10).

Estimates of Approximate Solutions to the Cauchy Problem
In this section, we estimate {u m (t, x)} in the norm of W l,r 2,γ,κ q ,1 (R n+1 + ) and prove that this sequence is fundamental.
with a constant c > 0 independent of m and f (t, x); moreover, for every k ≥ 1, we have Proof.By Parseval's equality, we have We extend the function f (τ, ξ) by zero for t < 0, keeping the notation.Then, taking into account (24), we obtain By (14), from the equality, it follows that Since κ q ≤ βα ≤ 1, we obtain the required estimate.Convergence is proven in the same way.
The lemma is proven.
Convergence can be proven in the same way.The lemma is proven.
with a constant c > 0 independent of m and f (t, x); moreover, for every k ≥ 1, we have as m → ∞.
The lemma is proven.
Proof.For the function v(t, ξ) defined in (19), by (21), the following equality is valid: Using Parseval's equality and the Heaviside function θ(t), we conclude that By the property of the Fourier transform of convolution, we have Using the representation (25), we obtain The identity , (6), and Lemma 1 ensure the inequality (31).Similar arguments imply (32).The lemma is proven.
In the next section, relying on Lemmas 5-11, we prove that the sequence {u m (t, x)} is convergent in W l,r 2,γ,κ q ,1 (R n+1 + ), and the limit function u(t, x) is a solution to the Cauchy problem (10) and satisfies (12).

Solvability of the Cauchy Problem
As noted above, the uniqueness of the solution to the Cauchy problem (10) in the space W l,r 2,γ,κ q ,1 (R n+1 + ) follows from the energy inequality when |α|/2 > κ q .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 W l,r 2,γ,κ q ,1 (R n+1 + ) 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.
Theorem 2 is proven.

Conclusions
The Cauchy problem was studied for a class of strictly pseudohyperbolic equations with lower order terms.A new class of weighted Sobolev spaces W l,r 2,γ,κ q ,σ was introduced.By definition, functions from this class and their generalized derivatives belong to Lebesgue spaces with the exponential weight e −γt and special power weights in x ∈ R n .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.