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In this paper, we study the Cauchy problem in a strip for a two-dimensional hyperbolic equation containing the sum of a differential operator and a shift operator acting on a spatial variable that varies over the real axis. An operating scheme is used to construct the solutions of the equation. The solution of the problem is obtained in the form of a convolution of the function found using the operating scheme and the function from the initial conditions of the problem. It is proved that classical solutions of the considered initial problem exist if the real part of the symbol of the differential-difference operator in the equation is positive.
In recent years, functional-differential equations, or, their special case, differential equations with a deviating argument, have become widespread in applications of mathematics. The systematic study of equations with a deviating argument began in the 1940s in connection with applications to automatic control theory and it was associated with the research by Pinney , Bellman and Cooke , Hale , and other authors.
Interest in problems for differential-difference equations is due to their numerous applications: in the mechanics of a deformable solid body; in relativistic electrodynamics; when studying the processes of vortex formation and the formation of complex coherent spots; when solving some problems related to plasma; in simulation of vibrations of the crystal lattice; in problems of nonlinear optics; in the study of neural networks; when studying models of population dynamics in mathematical biology; in the study of environmental and economic processes; in a wide range of tasks in the theory of automatic control; when solving problems of optimizing the treatment of oncological diseases (see, for example, works [4,5,6]); etc.
Differential-difference equations form a special class of functional-differential equations for which the theory of boundary value problems is currently developed. Problems for elliptic differential-difference equations in bounded domains have been studied quite comprehensively by now; the theory for such equations was created and developed by Skubachevskii [7,8].
Problems for parabolic and hyperbolic differential-difference equations have been studied to a much lesser extent [9,10,11].
As far as the authors know, at present, there are few papers dealing with hyperbolic differential-difference equations containing shifts with respect to the spatial variable. In [12,13,14], the families of classical solutions are constructed for two-dimensional hyperbolic equations with shifts in the only space variable x ranging over the real line; the shifts occur either in the potentials or in the highest derivative. Some similar problems for elliptic equations were studied in [15,16].
In this paper, we study the solvability of the Cauchy problem in a strip for a two-dimensional hyperbolic equation with a nonlocal potential.
Let be the coordinate plane area , where is the given real number, . Let us consider in the domain D the hyperbolic differential-difference equation, which contains the sum of the differential operator and the shift operator with respect of the spatial variable x:
where a, , are given real numbers.
Suppose that for all the inequality
Inequality (2) means that the real part of the symbol of the differential-difference operator in Equation (1) is positive.
Consider the function , . The derivative of this function is
Since at , and at , then the derivative is non-negative on the interval if
In this case, the function at is non-decreasing and its smallest value is equal to ; hence,
for all .
Since the function is even, this value b is the smallest for all real . Thus, the condition (2) is satisfied if the coefficients a, b and the shift h of the equality (1) satisfy the inequalities (3).
Formulation of the problem. Find a function that satisfies the conditions
where the initial function satisfies the conditions , and .
A function is called a classical solution of the problem (5)–(7), if
It is continuous and continuously differentiable with respect to the variables x and t in the set ;
It has continuous derivatives and in the domain D;
For each point , the limits of the functions and at exists and are equal to zero.
This paper studies the initial problem (5)–(7) for two-dimensional hyperbolic equation with a nonlocal term. The solution of the problem is obtained in the form of a convolution of the function found by using the operating scheme and the initial conditions (7).
2. Construction of Solutions of the Equation
The fundamental solution of a linear differential operator L with constant coefficients is a generalized function , that satisfies the equation
where is the Dirac -function.
We formally apply the Fourier transform with respect to the variable x to Equation (1), and passes to the dual variable . For the function , we obtain the equation
where is the Heaviside step function and the function satisfies the equation
with the initial conditions
The characteristic equation for Equation (11) has the roots
where the functions and are denoted by
Note that, whenever the condition (2) is satisfied, Functions (13) and (14) are defined correctly.
The general solution of Equation (11) is determined by the formula
where and are arbitrary constants depending on the parameter . To find these constants, substitute we substitute the last expression to the conditions (12):
As a result, the solution to the problem (11), (12) has the form
Taking into account Equation (10), the solution of Equation (9) is determined by the formula
Applying the inverse Fourier transform to the last expression, we obtain
Transform this expression using the equalities and :
Define the functions
Since and , we can write in the form
We will use the resulting integral to construct a solution to the system (5)–(7).
We introduce a weight function (according to ), that is continuous, non-negative for each , and satisfies the conditions:
(1) For any arbitrarily small number :
(2) Improper integrals
converge for each ;
(3) Improper integrals
converge for each .
As an example of such weight function , which is continuous and non-negative for any value of , and satisfies conditions (16)–(18), one can take any function where and are any real constants.
Indeed, Function (13) is represented in the following form:
that is, for , the function is equivalent to the function , where is any arbitrarily small number.
From Formula (14), it follows that , which means that . Thus, for Function (15) for we have the estimate
Since the inequalities
hold for any arbitrarily small number and , we obtain the conditions
Using the inequalities
and Lopital’s rule, one can show that
Thus, for the function , the first condition from (16) is satisfied.
Similarly, taking into account the second inequality from (19), we can check that the second condition from (16) for the function is also satisfied.
To prove the convergence of the integrals (17) and (18), we use the criterion for the convergence of improper integrals: if there is a finite limit at , then the integral converges. The existence of finite limits
implies the convergence of the integrals (18) for the function for any fixed values , and .
Using the inequalities (19) and Lopital’s rule, one can check that all four limits
are equal to zero for any arbitrarily small number . This means that the integrals (17) converge for the function (, ) and each .
satisfies the equality (1) in the classical sense. Here, is a definite and non-negative for any value function, which satisfies the conditions (16)–(18); functions and are determined by Equation (15).
As noted above, if the condition (2) is satisfied, the functions and are defined correctly for all values of the parameters a, b, h, and . Moreover, the function is non-zero. That is, the integrand in (20) is continuous at every point as a composition of continuous functions.
Let us first investigate the convergence of the integral:
It follows from Formulas (4) and (13) that for the function satisfies the estimate
Consider the expression
where is any arbitrarily small number. Taking into account (22), we have
Similarly, it can be shown that when the second condition in (16) is satisfied, the integral
Let us now check that Function (21) satisfies Equation (1). To do this, we formally differentiate Function (21) with respect to the variables x and t up to the second order under the integral sign.
Taking into account Equation (15), we obtain . Since is determined by Equation (14), the inequality is satisfied, and therefore, . Therefore,
Taking into account the inequality (2) and Equation (13), we deduce that , whence, .
If the inequality and the condition (2) are satisfied, we can calculate
Using the obtained expressions for and , from equality (27) we obtain
Substitute the obtained expressions of the derivatives and to Equation (1):
Thus, Function (21) satisfies Equation (1) in the classical sense.
Now let us prove the uniform convergence of the integrals (24) and (25) with respect to the variable x on any segment , and the uniform convergence of the integrals (26) and (28) with respect to the variable t on any segment . Note that the integrands of all these integrals have no singularities at the point .
Let us investigate the integral (24) for the uniform convergence, taking into account the estimate (22) and using the Weierstrass criterion:
According to the convergence of the integrals (17), the integral on the right-hand side of the latter inequality converges and the integrand does not depend on the variable x; hence, the integral (24) converges uniformly with respect to the variable x on any finite segment .
Let us investigate the uniform convergence of the integral (25) with respect to the variable x on any finite segment . Using the inequality (22), we calculate
By virtue of the convergence of the integrals (17), the integral on the right-hand side of the latter inequality converges, and the integrand does not depend on the variable x. Thus, the integral (25) converges uniformly, which means that differentiation under the integral sign of Function (21) with respect to the variable x up to the second order including was legal.
It remains to check the uniform convergence of the integral (26) with respect to the variable t on any finite segment , and the integral (28) with respect to the variable t on any finite segment . Using the definition (15), we can write Function (26) as
Since the integral (18) converges, the integrals on the right side of the latter inequality converge and not depend on the variable t. Therefore, the integral (26) converges uniformly for any value of .
Since the integrand in the latter integral does not depend on t, then, according to the Weierstrass test, the integral (28) converges uniformly in the variable t on any finite segment . This means that differentiation under the integral sign in (21) with respect to the variable t up to the second order including was legal if the integral
converges for any value . And this is true, since the integrals (17) and (18) converge.
It can be shown similarly that the improper integrals obtained after the formal differentiation under the sign of the integral over the variables x and t up to the second order including of the integrand in (23) converge uniformly if the integrals
converge. Their convergence was considered above, see (17) and (18).
Thus, we have proved that the function , defined by Equation (20), exists at every point of the area D and satisfies the equality (1) in the classical sense. Hence, the lemma is proved.
Function (29) exists in the region D if the condition
Since is an integrable function for all , we are to check the fulfillment of condition for any .
Let us find the majorant of Function (20). From the equality (13), we obtain the estimate
The equality (14) implies that , hence; . Then, from (15) and (30), we have
Taking into account the inequalities (22), (31) and , we can estimate the function
Replace the integration variable in the latter integral according to the formula for :
The integrals on the right-hand side of (32), due to conditions (17) and (3), converge for any and any . Thus, we have shown that the function is majorized by the function , where , , where is an absolute constant. This implies the convergence of the integral , that is for any . This means that Function (20) exists in the domain D and, by virtue of the proved Lemma, it is a classical solution of Equation (1).
Note also that, by virtue of the same Lemma, Function (29) belongs to the class (the integrand in (29) is continuous, the integrals and converge uniformly in the variable x on any finite segment , the integrals and converge uniformly in the variable t on any finite segment , and the integral converges at the border ). Thus, the theorem is proved.
3. Fulfillment of the Initial Conditions of the Problem
Let . In the equality (29), make the change of variable , and consider the difference
Using the inequality (2) and the equality , we write the functions (15) as follows:
Obviously, the resulting radical expressions in (35) and (36) are always non-negative.
and the function can be written as
After the substitution , from the latter equality we obtain
Denote the functions
Using this notation, we can write
2. Let us now prove that the limit relation
is satisfied uniformly with respect to . To do this, it suffices to show that for any arbitrarily small number there is a number such that for any and holds the inequality
Represent the function in the form and consider the difference
We use the formulas for the sine of the sum and difference in the third and fourth integrals in the last expression. Taking into account their squaring and the formula for the sine of a double angle, we obtain
In the resulting expression, we expand the sine of the sum in the integrand in the first integral, which we then group with the third integral, and write the sine of the difference in the second integral and group it with the fourth integral as follows:
3. Consider first the integral . Using the definition (37) and inequality (4), transform the expression
Taking into account that and the inequality (4), from the last expression, we obtain
Using the inequalities (22) and (42), we estimate the integral :
Since the asymptotic representation of the function is valid for , the latter integral for can be bounded as
After the reverse change of the variable according to the formula , we finally obtain
Since the resulting integral on the right-hand side is a convergent integral from the series (18), the last inequality for implies the estimate
4. Similarly, we estimate the integrals and for .
From the last two expressions for , we can deduce
for any arbitrarily small fixed number .
5. Let us now estimate the integral . Taking into account the inequalities (22) and (42), we obtain
Since the resulting integral converges as the partial case of (18), the latter chain of inequalities for implies the estimate
for any arbitrarily small number .
6. Consider now the integral . Let the function be such that for the integrand in the integral is majorized (in absolute value) by the function , where is a real constant. Therefore, you can choose such a large enough number such that the inequality
will be satisfied.
Fix the number and evaluate for the expression
Let at the value of the function
be equal to one for any z. Let us show that the function defined in this way tends to unity at uniformly with respect to .
Suppose the contrary, that is, there is such a number that for any positive number there are and such that the inequality
Consider the sequence , . There exist the sequences and for which the inequality
holds for any . Without loss of generality, we can assume that the sequence of elements converges, and we denote its limit as . Obviously, and ; hence, , which contradicts the definition of the function on the axis . Thus, we have proved the inequality
So, we have proved the estimate
7. Similarly, it can be shown that if, for , the value of the function
is equal to one for any z, then the following inequality holds:
Estimates (43)–(45), (47), and (49) prove the fulfillment of inequality (41), which implies the validity of relation (40).
8. Let us now estimate the expression (34). From the inequality (32), it follows that for and the inequality
is satisfied, where is a constant. Hence, for and , the estimate
holds. This means that the integrand in the integral is majorized by the function , that is, for any , the inequality
is satisfied. Choose the number sufficiently large so that we obtain the estimate
for any arbitrarily small number .
The integral is estimated in a similar way:
Write the integrand in the integral in the following form:
Let us write down . Given Equation (40), there is such that the inequality
is satisfied for any and , and for a chosen sufficiently large number . Let , then for any we obtain the inequalities
that is, .
Since the function is continuous over the real line (together with its first derivative), there exists , such that the inequality
is satisfied for any and . Thus, it is proved that for the following estimate holds:
The resulting estimates , and , by virtue of an arbitrary choice of and , prove that the first relation in (33) holds.
9. Let us now prove the second relation in (33). Using the definition (15), we first calculate
Let be an arbitrary value. Consider the function
In the last expression, make the change of variable and obtain
Since the condition
is satisfied, the function exists in the region .
Since, according to the condition of the theorem, the function is integrable over the real line, it is enough to check the condition . To do this, taking into account inequalities (30), we estimate the function
The integrals on the right-hand side in the latter expression converge as the partial case of the integrals (18) for any . Thus, we have shown that for any , which means that the function exists in the area .
Thus, for , the estimate
is satisfied for any arbitrarily small number . This implies the fulfillment of the second relation in (33). Thus, the theorem is proved.
Conceptualization, V.V. and N.Z.; methodology, V.V. and N.Z.; validation, V.V. and N.Z.; formal analysis, V.V. and N.Z.; investigation, V.V. and N.Z.; data curation, V.V. and N.Z.; writing and original draft preparation, N.Z.; writing and review and editing, V.V. and N.Z. All authors have read and agreed to the published version of the manuscript.
The second author was financially supported by the Ministry of of Science and Higher Education of the Russian Federation within the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No 075-15-2022-284.
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
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