About the Nature of the Spectrum of the Periodic Problem for the Heat Equation with a Deviating Argument

Abstract

We study the nature of the spectrum of the periodic problem for the heat equation with a lower-order term and with a deviating argument. A significant influence of the lower-order term on the correct solvability of this problem is found. We obtain a criterion for the strong solvability of the above-mentioned problem.

1. Introduction

In paper [1] we constructed a spectral theory of a model differential equation of first order with a deviating argument. The main idea of the article [1] has been further developed in [2, 3].

The equation that we studied belongs to a class of functional-differential equations. These equations have only recently become a subject of research of individual authors; particularly, such equations are studied in [4–6]. Functional-differential equations have been actively studied recently by some authors in papers [7, 8].

In this paper using the methods of paper [1] we have found a solution of the mixed problem for the heat equation with a deviating argument. As a result we got that the classical solvability of the boundary value problem requires a certain smoothness of the right-hand side of the equation (see (11) of Theorem 4) and the strong solvability of the problem is provided by the properties of the coefficient of the lower-order term of the equation (Theorem 11).

Let Ω⊂𝑅² be a rectangle bounded by the following segments: 𝐴𝐵∶0≤𝑡≤𝑇,𝑥=0,𝐵𝐶∶0≤𝑥≤𝑙,𝑡=𝑇,𝐶𝐷∶0≤𝑡≤𝑇,𝑥=𝑙,𝐷𝐴∶0≤𝑥≤𝑙,𝑡=0. (1)

We denote by 𝐶^2,1(Ω) the set of functions 𝑢(𝑥,𝑡) that are twice continuously differentiable with respect to the variable 𝑥 and once continuously differentiable with respect to the variable 𝑡. The boundary of the area Ω is a set of segments 𝐵=𝐴𝐵∪𝐴𝐷∪𝐶𝐷.

Periodic Problem. Find a solution of the equation 𝐿𝑢=𝑢_𝑡(𝑥,𝑇−𝑡)+𝑢_𝑥𝑥(𝑥,𝑡)+𝑎𝑢_𝑥=𝑓(𝑥,𝑡) (2) satisfying the conditions 𝑢|_𝑡=0𝑢|=0,_𝑥=0−𝑢|_𝑥=𝑙=0,𝑢_𝑥𝑥=0−𝑢_{𝑥𝑥=𝑙}=0, (3)(4) where 𝑓∈𝐿²(Ω) and 𝑎 is constant.

Further the coefficient 𝑎 will be called the coefficient of influence.

Definition 1. The function 𝑢(𝑥,𝑡)∈𝐿²(Ω) is called a strong solution of (2)–(4) if there exists a sequence of functions {𝑢_𝑛}∈𝐶^2,1(Ω)∩𝐶^1,0(Ω), 𝑛=1,2,3,…, that satisfies the boundary conditions of the problem and such that {𝑢_𝑛} and {𝐿𝑢_𝑛}, 𝑛=1,2,3,…, converge, respectively, to 𝑢 and 𝑓.

Definition 2. The boundary value problem (2)–(4) is called strongly solvable if a strong solution of the problem exists for any right-hand side 𝑓(𝑥,𝑡)∈𝐿²(Ω) and this solution is unique.

The aim of this study is to investigate the nature of the spectrum and the influence of the lower-order term on the strong solvability of the problem (2)–(4) in the space 𝐿²(Ω).

Using Fourier method we obtained the conditions of the existence of a strong solution of the boundary value problem (2)–(4) in the space 𝐿²(Ω) in the form of Theorem 4. With the help of the spectral theory of linear operators we have established criteria of strong solvability of this problem, presented in the form of Theorem 11, which is the main result of this paper.

2. Results

Problem about Spectrum. Examine the nature of the spectrum of the functional-differential operator 𝐿𝑢=𝑢_𝑡(𝑥,𝑇−𝑡)+𝑢_𝑥𝑥(𝑥,𝑡)+𝑎𝑢_𝑥,𝐷(𝐿)=𝑢∈𝐶^2,1(Ω)∩𝐶Ω,𝑢|_𝑡=0=𝑢|_𝑥=0−𝑢|_𝑥=𝑙=𝑢_𝑥𝑥=0−𝑢_{𝑥𝑥=𝑙}=0. (5)

Theorem 3. The spectral problem 𝐿𝑢=𝑢_𝑡(𝑥,𝑇−𝑡)+𝑢_𝑥𝑥(𝑥,𝑡)+𝑎𝑢_𝑥()(),𝑢|𝑥,𝑡=𝜆𝑢𝑥,𝑡_𝑡=0=𝑢|_𝑥=0−𝑢|_𝑥=𝑙=𝑢_𝑥𝑥=0−𝑢_{𝑥𝑥=𝑙}=0 (6) has an infinite number of eigenvalues​​: 𝜆_𝑚𝑛=(−1)^𝑛1𝑛+2𝜋𝑇−2𝑚𝜋𝑙²+2𝑚𝜋𝑖𝑙𝑎,𝑚=0,±1,±2,…,𝑛=0,1,2,…, (7) and the corresponding eigenfunctions 𝑢_𝑚𝑛()=2𝑥,𝑡√𝑇𝑙exp2𝑚𝜋𝑖𝑙𝑥1⋅sin𝑛+2𝜋𝑡𝑇,𝑚=0,±1,±2,…,𝑛=0,1,2,…, (8) which form an orthonormal basis of the space 𝐿²(Ω), where Ω=[0,𝑙]×[0,𝑇].

Proof. By method of separation of variables we get two spectral problems. The first problem is the Sturm-Liouville problem with Dirichlet condition. The second problem is the Cauchy problem for the first-order equation with deviating arguments, which is studied in detail in [1]. The rest is elementary.

Theorem 4. For the existence and uniqueness of a strong solution of the boundary problem (2)–(4), it is necessary and sufficient to fulfill the condition (−1)^𝑛1𝑛+2⋅𝜋𝑇−2𝑚𝜋𝑙²−2𝑚𝜋𝑙⋅Im𝑎²+2𝑚𝜋𝑙⋅Re𝑎²≠0,∀𝑚=0,±1,±2,…,𝑛=0,1,2,…. (9)
When this condition is fulfilled a strong solution of the problem exists and has the form 𝑢(𝑥,𝑡)=^∞∞_{𝑚=−∞𝑛=0}𝑓,𝑢_𝑚𝑛𝜆_𝑚𝑛⋅𝑢_𝑚𝑛(𝑥,𝑡) (10) for all 𝑓(𝑥,𝑡)∈𝐿²(Ω) satisfying ^∞∞_{𝑚=−∞𝑛=0}𝑓,𝑢_𝑚𝑛𝜆_𝑚𝑛²<+∞, (11) where the eigenvalues 𝜆_𝑚𝑛 and the eigenfunctions 𝑢_𝑚𝑛(𝑥,𝑡) are given by (7) and (8).

The proof is omitted since this theorem is a simple consequence of the preceding theorem.

Theorem 5. If 𝑎+𝑎=0, then the differential operator (5) is essentially self-adjoint in the space 𝐻=𝐿²(Ω), where Ω=[0,𝑙]×[0,𝑇] is a rectangle, lying in the upper half-plane (𝑥,𝑡)∈𝑅².

Proof. It easily follows from the symmetry of the operator 𝐿 and the completeness of its eigenvectors.

From Theorems 4 and 5 there follows the Theorem 6.

Theorem 6. If (a)Re𝑎=0,(b)(−1)^𝑛(𝑛+1/2)⋅(𝜋/𝑇)−(2𝑚𝜋/𝑙)²−(2𝑚𝜋/𝑙)⋅Im𝑎≠0, then the inverse operator (𝐿)⁻¹ exists, which is also self-adjoint.

Proof. From Theorem 4 there follows the existence of the operator (𝐿)⁻¹; the rest follows from the series of equalities (𝐿⁻¹)^∗=(𝐿^∗)⁻¹=(𝐿)⁻¹, in which Theorem 5 and the known equality (𝐴⁻¹)^∗=(𝐴^∗)⁻¹ were used.

Definition 7. A linear operator 𝐴 (not necessarily bounded) in space 𝐻 is called normal if it is densely defined and closed and satisfies the condition 𝐴^∗𝐴=𝐴𝐴^∗.

Lemma 8 (see [9]). Let 𝐴 be a densely defined operator in a Hilbert space 𝐻. Then(a)𝐴^∗ is closed;(b)𝐴 allows a closure if and only if 𝐷(𝐴^∗) is dense, and in this case 𝐴=𝐴^∗∗;(c)if 𝐴 allows a closure, then (𝐴)^∗=𝐴^∗.

Lemma 9 (see [10]). Let 𝐴 be a normal operator in the space 𝐻. Then(a)𝐷(𝐴)=𝐷(𝐴^∗);(b)‖𝐴𝑥‖=‖𝐴^∗𝑥‖ for ∀𝑥∈𝐷(𝐴);(c)𝐴 is the maximum normal operator.

Using these lemmas we obtain the next theorem.

Theorem 10. If 𝑎+𝑎≠0, then the closure of the operator 𝐿 is a normal operator; that is, the equality 𝐿^∗𝐿=𝐿𝐿^∗ is satisfied.

Proof. This formula can be verified directly.

3. About the Nature of the Spectrum of the Operator 𝐿

Theorem 11. (a) If Re𝑎≠0, then there exists an inverse operator 𝐿⁻¹, which is normal and compact. We have the estimate 𝐿⁻¹≤𝐾⁻¹𝜋,𝐾=max,2𝑇2𝜋𝑙|Re𝑎|. (12)
The spectrum of the operator 𝐿 is discrete, that is, has no limit points in the finite part of the plane.
(b) If Re𝑎=0, Im𝑎≠((−1)^𝑛(𝑛+1/2)/2𝑚)⋅(𝑙/𝑇)−2𝑚𝜋/𝑙, 𝑛=0,1,2,…,𝑚=0,±1,±2,…, and both values of 2𝜋𝑇/𝑙² and 𝑇⋅Im𝑎/𝑙 are rational, then the inverse operator 𝐿⁻¹ exists and is bounded but not compact. The operator 𝐿 is self-adjoint; its spectrum consists of an infinite number of eigenvalues, among which there are an infinite number of infinite multiple eigenvalues.
(c) If Re𝑎=0, Im𝑎≠((−1)^𝑛(𝑛+1/2)/2𝑚)⋅(𝑙/𝑇)−2𝑚𝜋/𝑙, 𝑛=0,1,2,…, 𝑚=0,±1,±2,… and at least one of the values of ​​2𝜋𝑇/𝑙² and 𝑇⋅Im𝑎/𝑙 is irrational, then the inverse operator 𝐿⁻¹ exists, but is not bounded. The operator 𝐿 is self-adjoint and its spectrum consists of an infinite number of eigenvalues ​​and continuous spectrum filling the entire real axis (−∞,+∞). The points of the continuous spectrum are the limit points of eigenvalues.
(d) If Re𝑎=0 and Im𝑎=(((−1)^𝑛(𝑛+1/2))/2𝑚)⋅(𝑙/𝑇)−2𝑚𝜋/𝑙 for some values 𝑛=0,1,2,…; 𝑚=0,±1,±2,…, the inverse operator 𝐿⁻¹ does not exist. The operator 𝐿 is self-adjoint. If both values of ​​2𝜋𝑇/𝑙² and 𝑇⋅Im𝑎/𝑙 are rational, the spectrum consists of an infinite number of eigenvalues, among which there are an infinite number of infinite multiple eigenvalues. If at least one of the values of 2𝜋𝑇/𝑙²,𝑇⋅Im𝑎/𝑙 is irrational, then the spectrum of the operator 𝐿 fills the entire real line (−∞,+∞).

Proof. In the work [11] Weyl introduced the concept of a sequence uniformly distributed modulo 1 and also proved a criterion of uniform distribution. In the same paper he gave examples of sequences distributed uniformly modulo 1. The simplest of these sequences is the sequence 1𝜉,2𝜉,3𝜉,… with some irrational number 𝜉.
Weyl’s First Theorem. If 𝜑(𝑧) is a polynomial with a constant term 𝛼₀ and not all coefficients of 𝜑(𝑧)−𝛼₀ are rational, then the sequence of numbers 𝜑(1),𝜑(2),𝜑(3),… is distributed uniformly dense everywhere.
In particular:
Weyl’s Second Theorem. If 𝜉 is some irrational number, the sequence of points 1𝜉,4𝜉,9𝜉,16𝜉,25𝜉,… when winding of a real axis around a circle of length 1 covers it evenly dense everywhere. The same will hold if the squares of numbers are replaced by their cubes or fourth degrees, and so forth.
Next, we show that the set of eigenvalues {𝜆_𝑚𝑛} is compacted with an increase in the indices 𝑛 and 𝑚.
The eigenvalues that we found are of the form 𝜆_𝑚𝑛=(−1)^𝑛1𝑛+2𝜋𝑇−2𝑚𝜋𝑙²+2𝑚𝜋𝑖𝑙𝑎,𝑎≠0,𝑚=0,±1,±2,…,𝑛=0,1,2,…. (13)
Let us consider the neighborhood of the origin. If it was a limit point of the set of eigenvalues, then the inverse operator (𝐿)⁻¹ would be unbounded.
If 𝜆_𝑚𝑛→0 for some subsequence, then 𝜆_𝑚𝑛²=(−1)^𝑛1𝑛+2𝜋𝑇−2𝑚𝜋𝑙²−2𝑚𝜋𝑙⋅Im𝑎²+2𝑚𝜋𝑙⋅Re𝑎²⟶0. (14)
This is impossible when Re𝑎≠0. If Re𝑎≠0, then 𝜆_𝑚𝑛²≥2𝜋𝑙Re𝑎²⟹𝜆,∀𝑚=1,2,…,_𝑚𝑛≥2𝜋𝑙𝜆Re𝑎,∀𝑚=1,2,…,_0𝑛2=1𝑛+2𝜋𝑇²≥𝜋2𝑇²,𝜆_0𝑛≥𝜋2𝑇,∀𝑛=0,1,2,…. (15)
Hence, 𝜆_𝑚𝑛𝜋≥max,2𝑇2𝜋𝑙|Re𝑎|,∀𝑚=0,±1,±2,…;𝑛=0,1,2,… (16)
Thus, when Re𝑎≠0 the inverse operator 𝐿⁻¹ exists and is bounded. If a subsequence {𝜆_𝑘𝑙} of the sequence {𝜆_𝑚𝑛} converges to a point 𝜆₀ in the complex plane, then the sequence {|𝜆_𝑘𝑙|} is bounded; therefore, the second index 𝑙 takes only a finite number of values. Then the first index takes a finite number of values too. We have a contradiction, since assumption on {𝜆_𝑘𝑙} is an infinite convergent sequence. Therefore, the sequence {𝜆_𝑚𝑛} has no limit points in the finite part of the complex plane 𝜆, which means that the spectrum of the operator 𝐿 is discrete.
Now we will investigate whether the operator 𝐿⁻¹ is compact. Any subsequence of the sequence {𝜆_𝑚𝑛},𝑚=0,±1,±2,…,𝑛=0,1,2…, tends to infinity. In fact, let {𝜆_𝑖𝑗} be an arbitrary infinite subsequence of the sequence {𝜆_𝑚𝑛}. Then two situations are possible:(1)either the first index takes an infinite number of values, then 𝜆_𝑖𝑗²≥2𝑖𝜋𝑙Re𝑎²,⟹𝜆_𝑖𝑗⟶+∞, (17)(2)or the first index takes a finite number of values, while the second index takes an infinite number of values; therefore 𝜆_𝑖𝑗²=(−1)^𝑗1𝑗+2𝜋𝑇−2𝑖𝜋𝑙²−2𝑖𝜋𝑙⋅Im𝑎²+2𝑖𝜋𝑙⋅Re𝑎²≥1𝑗+2𝜋𝑇+(−1)^𝑗+12𝑖𝜋𝑙²+(−1)^𝑗+12𝑖𝜋𝑙⋅Im𝑎²⟶∞. (18)
Due to the fact that 𝑗→+∞, the second and third terms are the bounded quantities.

Lemma 12 (see [12]). For the complete continuity of the operator of the normal type it is necessary and sufficient to fulfill the condition lim_𝑛→+∞𝜆_𝑛=0. (19)

On the basis of Lemma 12 and (17) and (18) it follows that the inverse operator 𝐿⁻¹ is completely continuous. Therefore its spectrum is discrete.

Now consider the case Re𝑎=0.

In this case, the eigenvalues have the form 𝜆_𝑚𝑛=(−1)^𝑛1𝑛+2𝜋𝑇−2𝑚𝜋𝑙²−2𝑚𝜋𝑙Im𝑎,Im𝑎≠0. (20)

Suppose that 𝜆_𝑚𝑛≠0, that is, ()Im𝑎≠−1^𝑛(𝑛+1/2)⋅𝑙2𝑚𝑇−2𝑚𝜋𝑙,𝑛=0,1,2,…,𝑚=0,±1,±2,…. (21)

If 𝑛=2𝑘+1 and 𝑘=0,1,2,…, then 𝜆_𝑚,2𝑘+13=−2𝑘+2𝜋𝑇−2𝑚𝜋𝑙²−2𝑚𝜋𝑙Im𝑎⟶+∞ (22) at 𝑚,𝑘→∞; therefore this subsequence has no limit points.

If 𝑛=2𝑘, 𝑘=0,1,2,…, then 𝜆_𝑚,2𝑘=12𝑘+2𝜋𝑇−2𝑚𝜋𝑙²−2𝑚𝜋𝑙Im𝑎,𝑘=0,1,2,…,𝑚=0,±1,±2,…. (23)

Transform this expression to a form convenient for us 𝜆_𝑚,2𝑘=2𝜋𝑇1𝑘+4−2𝑚²𝜋𝑇𝑙²−𝑚𝑇𝑙=Im𝑎2𝜋𝑇1𝑘+4−𝑚²2𝜋𝑇𝑙²+𝑚𝑇⋅Im𝑎𝑙. (24)

For convenience we introduce the following notation: 𝜑(𝑚)=𝑚²2𝜋𝑇𝑙²+𝑚𝑇⋅Im𝑎𝑙, (25) where [𝑥] is the integer part and (𝑥) is the fractional part. Suppose that 𝑘=[𝜑(𝑚)], then 𝜆_𝑚,2𝑘=2𝜋𝑇14+𝜑(𝑚(𝑚=)−𝜑)2𝜋𝑇14−𝜑(𝑚)−𝜑(𝑚)=2𝜋𝑇14−𝜑(𝑚). (26)

Now we use the Weyl’s theorem [11], for this we assume that at least one of the values of 2𝜋𝑇𝑙²,𝑇⋅Im𝑎𝑙 (27) is irrational. Then by Weyl's theorem the fractional part 𝜑(𝑚), that is, (𝜑(𝑚)), fills the interval [0,1] uniformly dense when 𝑚=0,1,2,…. Then the subsequence {𝜆_𝑚,2𝑘},𝑚=0,1,2,…,𝑘=[𝜑(𝑚)], is dense everywhere in the interval [−3𝜋/2𝑇,𝜋/2𝑇].

Assuming 𝑘=[𝜑(𝑚)]+1and𝑘=[𝜑(𝑚)]+2,…, then 𝑘=[𝜑(𝑚)]−1, 𝑘=[𝜑(𝑚)]−2,…, and so on; we obtain that the sequence {𝜆_𝑚𝑛} is uniformly dense everywhere; that is, the continuous spectrum of the operator 𝐿 fills the entire real axis from −∞ to +∞. Let now both values of 2𝜋𝑇𝑙²,𝑇⋅Im𝑎𝑙 (28) be rational; then 𝜑(𝑚) is always rational. To be specific let 2𝜋𝑇𝑙²=𝑝𝑞,𝑇⋅Im𝑎𝑙=𝑟𝑘. (29)

Then 𝜑(𝑚)=𝑚²⋅𝑝𝑞𝑟+𝑚⋅𝑘=𝑚²𝑝+𝑚𝑟=𝜑(𝑚+𝜑(𝑚).𝑞⋅𝑘) (30)

The fractional part 𝜑(𝑚) takes only a finite number of values; they are the remainders of the division 𝑚²𝑝+𝑚𝑟 by 𝑞⋅𝑘; that is 10,,2𝑞⋅𝑘𝑞⋅𝑘,…,𝑞⋅𝑘−1.𝑞⋅𝑘 (31)

When 𝑚 is changing from −∞ to +∞, these values will repeat infinitely many times, at least one or all of them. For us it is important that they do not coincide with 1/4.

Assuming 𝑘=[𝜑(𝑚)],𝑚=0,1,2,…, we see that 𝜆_𝑚,2𝑘=2𝜋𝑇14−𝜑(𝑚),𝑚=0,1,2,…. (32)

This infinite sequence is contained in a segment [−3𝜋/2𝑇,𝜋/2𝑇] and consists of a finite number of fractions of the form 2𝜋𝑇⋅14,2𝜋𝑇⋅14−1,𝑞⋅𝑘2𝜋𝑇⋅14−2𝑞⋅𝑘,…,2𝜋𝑇⋅14−𝑞𝑘−1,𝑞⋅𝑘 (33) so at least one of them, or all, or some of them are repeated infinitely many times. This suggests that some numbers in the segment [−3𝜋/2𝑇,𝜋/2𝑇] are the infinitely multiple eigenvalues​​.

Continuing this reasoning as in the irrational case, we see that the spectrum of the operator 𝐿 consists of an infinite set of eigenvalues and among the eigenvalues there are the infinite set of the infinitely multiple eigenvalues. By our assumption ()Im𝑎≠−1^𝑛(𝑛+1/2)⋅𝑙2𝑚𝑇−2𝑚𝜋𝑙,𝑛=0,1,2,…,𝑚=0,±1,±2,…; (34) therefore the inverse operator exists and is bounded but is not compact in view of the existence of eigenvalues of infinite multiplicity, as the spectrum of the compact operator has a finite multiplicity. If Re𝑎=0 and ()Im𝑎=−1^𝑛(𝑛+1/2)⋅𝑙2𝑚𝑇−2𝑚𝜋𝑙 (35) for some values 𝑛=0,1,2,…,𝑚=0,±1,±2,…, the inverse operator 𝐿⁻¹ does not exist and zero is an eigenvalue, perhaps, of the infinite multiplicity. In this case, if at least one of the values of 2𝜋𝑇𝑙²,𝑇⋅Im𝑎𝑙 (36) is irrational, then the spectrum of the operator 𝐿 fills the entire real axis. If both of these values are rational, then the spectrum of the operator 𝐿 consists of an infinite number of eigenvalues, among which there are an infinite number of infinite multiple eigenvalues.

Finally, we note that the boundary value problem (2)–(4) is strongly solvable in the cases of (a) and (b) of Theorem 11 but in cases (c) and (d) is not.

4. Conclusions

By means of Fourier method, a criterion for strong solvability of the mixed Cauchy problem for the heat equation with a deviating argument was found. The nature of the spectrum of the periodic problem for the heat equation with a deviating argument was studied in detail, and the norm of the inverse operator was estimated through the influence coefficient. The dependence between the coefficient of influence and the nature of the spectrum of the heat operator with a deviating argument was found.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

The authors are grateful to the referees for their critical comments which helped in improving the content of this paper. The work is carried out under the auspices of the Ministry of Education and Science of Kazakhstan (1100/SF, 0754/SF).