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Regularity of complexified hyperbolic equations with integral conditions

This paper considers hyperbolic wave equations with non-local in time conditions involving integrals with respect to time. It is shown that regularity of the solution can be achieved for complexified problem with integral conditions involving harmonic complex exponential weights. The paper establishes existence, uniqueness, and a regularity of the solutions.


Introduction
The most common type of boundary conditions for evolution partial differential equations are Cauchy initial conditions. It is know that these conditions can be replaced, in some cases, by nonlocal conditions, for example, including integrals over time intervals. There is a significant number of works devoted to these boundary values problems.
For parabolic equations with non-local in time boundary conditions, some results and references can be found, e.g., in [9,16]. For Schrödinger equations, some results and references can be found in, e.g., [3,6,10,18]. In [3,6,18], the non-local in time condition was dominated by the initial value, and the approach was based on the contraction mapping theorem. In [3,18], the nonlocal conditions connected solutions in a finite set of times. In [6], the conditions were quite general and allowed to include integrals with respect to time. In [10], integral conditions without dominating initial value have been considered.
For hyperbolic equations with mixed highest order derivative, regularity results were obtained in [2]. In [1], systems of hyperbolic equations with integral conditions have been considered.
In [12,13,17], hyperbolic wave equations under integral conditions with respect to time have been considered. In [12,13], the eigenfunction expansion method has been used, and the regularity result has been affected by the so-called "small denominators" ("small divisors") problem that often causes instability of solutions for hyperbolic wave equations with non-local in time conditions. The solvability was obtained in [12,13] for the case where the spectrum for the inputs and solutions does not contain resonance points. In [17], a regularity condition without these restrictions on the spectrum have been obtained for the hyperbolic wave equation with a Laplacian. This condition imposed certain restrictions on the kernel in the integral condition which has vanish with a certain rate at the end of the time interval.
The paper readdresses the problem of regularity for solutions of boundary value problems for hyperbolic wave equations with non-local in time conditions. The paper suggests a complexification of boundary value problems that ensures regularity of the solutions. This complexification requires to consider integral conditions T 0 e iωt u(t)dt = g, for a real nonzero ω that can be arbitrarily small, where u(t) is the solution of the hyperbolic wave equation with real coefficients that take values in an appropriate Hilbert space. This allows to bypass, for this particular setting, the "small denominators" (or "small divisors") problem. We establish existence, uniqueness, and a regularity of the solutions for the complexified equation. The proofs are based the spectral expansion, similarly to the setting from [9,11,14,10]. The eigenfunction expansion for the solution is presented explicitly. This allows to derive a numerical solution.
The rest of the paper is organized as follows. In Section 2, we introduce a boundary value problem with averaging over time, and we present the main result (Theorem 1). In Section 3, we present the proofs. Section 4 gives a numerical example of the impact of the presence of small ω = 0 on the appearance of small denominators.. Section 5 presents conclusions and discusses future research.

Some definitions
For a Banach space X, we denote the norm by · X . For a Hilbert space X, we denote the inner product by (·, ·) X . We denote the Lebesgue measure and the σ-algebra of Lebesgue sets in R n bȳ ℓ n andB n , respectively.
Let D ⊂ R n be a domain, and let H = L 2 (D,B n ,l 1 ; C) be the space of complex-valued functions.
Let A : H → H be a self-adjoint operator defined on an everywhere closed subset D(A) of H such that if u is a real valued function then Au is also a real valued function.
Let H BC be a set functions from H such that H BC is everywhere dense in H, that the set D(A) ∩ H is everywhere dense in H, and that (u, Av) H is finite and defined for u, v ∈ H BC as a continuous extension from D(A) × D(A).

Consider an eigenvalue problem
We assume that this equation is satisfied and that v k are eigenfunctions for (1), i.e., for some λ k ∈ (0, +∞) such that λ k → +∞ as k → +∞.
These assumptions imply that the operator A is self-adjoint with respect to the boundary conditions defined by the choice of H BC .
For q = −1, 0, 1, 2, let H q be the Hilbert spaces obtained as the closure of the set H BC ∩ D(A) in the norms respectively. According to this definition, H 0 = H. For q = −1, 0, 1, 2, and r ∈ [1, +∞], introduce the spaces and the spaces considered as Banach spaces with the norms, respectively,

Problem setting and the main result
Let T > 0, ω ∈ R, ω = 0, a ∈ H 1 , and g ∈ H 2 , be given. We consider the boundary value problem T 0 e iωt u(t)dt = g.
For u ∈ W 1 1 , we accept that equation (3) is satisfied as an equality that holds in H −1 for all s, t such that 0 ≤ s ≤ t ≤ T , and that conditions (4) and (4) are satisfied as equalities in H.

Theorem 1
Assume that e 2iωT = 1. In this case, for any a ∈ H 1 and g ∈ H 2 , there exists a unique solution u ∈ W 1 ∞ . Moreover, there exists c > 0 such that Here c > 0 depends only on H BC , A, T , and ω.
By Theorem 1, problem (3)-(5) is well-posed in the sense of Hadamard for a ∈ H 1 and g ∈ H 2 .
The proof of this theorem is given below; it is based on explicit representation of the solution u for given g and a via eigenfunction expansion. It can be noted that, since ω = 0 and ω ∈ R, the solution u of problem (3)- (5) is not real valued even if both a and g are real valued.
Remark 1 It is known that real valued problem (3)-(5) with ω = 0 is solvable for some real valued α and g but it does not feature stable solutions due to "small denominator" problem arising for certain frequencies; see, e.g. examples in [17], p. 42. For small ω → 0, the part v of the solution (v, w) from Remark 2 can be used as a stable approximation of the real valued solution of problem (5) with ω = 0. In this case, sin(ωt)w(t) can be considered as some small stabilizing term.
Remark 2 One can reformulate the setting with complex valued solutions as a setting with two where v(t) = Re u(t) and w(t) = Im u(t).
Then problem (3)-(5) can be rewritten as For a ∈ H 1 and b ∈ H, consider a boundary problem with the Cauchy condition For u ∈ W 1 1 , we accept again that equation (8) is satisfied as an equality (6) that holds in H −1 for all s, t such that 0 ≤ s ≤ t ≤ T , and that conditions (9) and (10) are satisfied as equalities in Proposition 1 For any a ∈ H 1 and b ∈ H, there exists a unique solution u ∈ W 1 1 of problem (6)- (10). This solution u is such that u ∈ W 1 ∞ . Moreover, there exists c > 0 such that Here c > 0 depends only on H BC , A, T , and ω.
The statement of Proposition 1 represents a minor modification of well known results adapted to our choice of spaces; however, we provided its proof in Section 3 below for the sake of completeness.

Proofs
Proof of Proposition 1. Let a and b be expanded as Here the coefficients α k are such that +∞ k=1 |α k | 2 = a H < +∞. We look for the solution u expanded as where y k (t) are solutions of equations In this case, It can be seen that for some C k , D k ∈ C.
The coefficients C k and D k are defined from the system This gives For the case of real α k and β k , we have that C k =D k .
For p = 0, 1 and q = 0, 1 such that p + q = 1, we have that This means that estimate (11) holds.
Clearly, equations (6)-(10) hold for the case where a, b, and u, are replaced by their truncated We have that u N ∈ W 2 ∩ W 1 . Estimate (18) and completeness of the Banach space W 1 ensures that u N → u in W 1 as N → +∞. This u is the solution (6)- (10), and that energy estimate (18) holds. This completes the proof of Proposition 1.
To proceed to the proof of Theorem 1, we need to adjust the approach used to the case of the integral boundary conditions.
Let a and g be expanded as Here the coefficients α k are such that +∞ k=1 λ k |α k | 2 = a H 1 < +∞. We look for the solution u expanded as where y k (t) are the solutions of equations (14) defined by (15), where C k and D k are defined from the system Lemma 1 Solution (C k , D k ) of system (22) exists and is uniquely defined for any k. Moreover, there exists c > 0 that depends on T and ω only and such that for all k.
Proof of Lemma 1. Let In this case, Suppose that we can prove that In this case, d k = 0 for all k, and there exists c p > 0 such that, for k ∈ Λ + 1 , This would imply the proof of the lemma.
Let us prove that (24) holds. Let Clearly, these sets are disjoint, and the set Λ 0 is either an empty set or a singleton.
The assumption of Theorem 1 that e 2iωT = 1 excluded the case where e iθ k T = e iωT and e iθ k T = e −iωT simultaneously, since this would imply that e iωT = e −iωT , or e 2iωT = 1. Hence The case where k ∈ Λ 0 Let us consider first the case where Λ 0 = {k} is a singleton, i.e., θ k = ω or θ k = −ω.
The case where k ∈ Λ 1 Let us consider the case where k ∈ Λ 1 . By the definitions, it follows that ω = ±θ k . In this case, and Let us consider first the case where e iθ k T = e iωT . In this case, we have that By the assumptions on ω and T , e 2iωT = 1. Hence Using a similar approach where e iθ k T = e −iωT , we obtain that inf k∈Λ 1 Hence the statement of Lemma 1 holds for k ∈ Λ 1 .

The case where k ∈ Λ 2
Let us consider the most typical case where k ∈ Λ 2 , i.e. e iθ k T = e iωT , e iθ k T = e −iωT . (28) In particular, we have in this case that ω = θ k and ω = −θ k for all k in this case.
Let us show first that the values d k (iω + iθ k ) are separated from zero for large k. We have that Hence Clearly, we have that ζ k → 0 and z k → 1 as k → +∞. Let a k = Re ζ k and b k = Im ζ k . We have that ζ k = a k + ib k and lim k→+∞ (a k cos(ωT ) − b k sin(ωT ) − z k ) = −1.
This would imply that cos(xT ) = cos(ωT ) and Re e ixT = Re e iωT , and this in turn would imply that either e ixT = e iωT or e ixT = e −iωT . However, this case is excluded for k ∈ Λ 2 . Therefore, Hence (24) holds. This completes the proof of Lemma 1.
We are now in the position to prove Theorem 1.
Proof of Theorem 1. We have that By Lemma 1, it follows that D 2 k ≤ c 0 (α 2 k + λ k γ 2 k ) for some c 0 > 0 that depends on H BC , A, T , and ω. Hence for some c 1 > 0, c 2 > 0, and c 2 > 0, that depend on H BC , A, T , and ω. By Lemma 1, estimate (7) holds. This completes the proof of Theorem 1.

A numerical example of impact of the presence of ω = 0
Consider a toy example for the problem where g ∈ H 2 . This is a special case of problem (3)-(5) with n = 1, D = (0, π), A = d 2 /dx 2 . It is known that λ k = k 2 and v k (x) = sin(kx), k = 1, 2, ...,, are the corresponding eigenvalues and eigenfunctions Respectively, θ k = k.
For simplicity, we assume that ω / ∈ {1, 2, 3, ...}. In this case, θ k ∈ Λ 2 , in the notations of the proof of Theorem 1. In addition, we assume that e 2iωT = 1. The solution constructed in the proof of Theorem 1 is defined by (26) and (33), i.e., and where C k and D k are defined from the system (26), (27) such that where d k defined by (27), i.e.
Let z(m) ∆ = inf k≤m |d k |(1 + θ k ). As can be seen from the proof of Lemma 1, this values should be separated from zero to ensure regularity of solutions claimed in Theorem 1.
This illustrates that that the impact of including even a small enough ω = can prevent appearance of "small denominators" (small divisors).

Conclusions and discussion
The paper establishes solvability and regularity of a complexified boundary value problem for linear hyperbolic wave equations where a Cauchy condition is replaced by a integral condition T 0 e iωt u(t)dt = g for the solution. It is shown that this new problem is well-posed in a wide class of solutions given the presence of the weight function e iωt , where ω ∈ R \ {0} can be arbitrarily small; the only condition is that e 2iT ω = 1. This leads to complex valued solutions of the boundary value problem with this integral condition. This boundary value problem would be equivalent to a boundary value problem for a system of two real valued hyperbolic equations, for the real and imaginary parts of the complex valued solution respectively, with an integral condition connecting solutions. In this case, the real part of the solution can be considered as an approximation as ω → 0 of the solution of the real valued solution with ω = 0. The setting considered in the paper allows many modifications and extensions. Most likely, the results can be extended on the case where the eigenvalues for A can be non-positive, and where the weight e iωt in (5) is replaced by e rt+iωt for r ∈ R. We leave it for the future research.
So far, we have considered the case where the solutions can be expanded via the basis from the eigenfunctions. It would be interestingly to extend the result on the more general case, as it was done in [11] for wave equations with two point conditions. We leave it for the future research as well.