The Cauchy Problem for Parabolic Equations with Degeneration

One of the important directions of development of the modern theory of partial differential equations is the study of nonlocal boundary value problems for different types of differential equations and partial differential equation systems and establishing the conditions for their correct solvability. Such problems arise when modeling different phenomena and processes in modern science, quantum mechanics, technology, economics, etc. Multipoint problems for partial differential equations were studied in the papers by Ptashnyk and his disciples. In particular, paper [1] is dedicated to multipoint problems for partial differential equations, not resolved relative to the highest time derivative. The question of the existence and qualitative properties of solutions for equations with limited order of degeneration has been studied in papers [2–4]. The Dirichlet problem with impulse action for a parabolic equation with power singularities of arbitrary order on time and spatial variables is investigated in paper [5]. In paper [6], a multipoint one-sided boundary value problem is studied. In paper [7], presented is the result of research of optimal control of the system described by the problem with an oblique derivative and an integral condition on time variable for parabolic equations with power singularities of arbitrary order. For a second-order parabolic equation, a multipoint (in time) problem with oblique derivative is considered in paper [8]. Conditions for the existence and uniqueness of solution of the posed problem in Hölder spaces with power weight are established. In this paper, we investigate a multipoint time-varying Cauchy problem for a parabolic second-order equation with power singularities and arbitrary order degenerations in the coefficients of spatial and time variables at some set of points. We also find conditions for the existence and uniqueness of the solution of formulated problem in Hölder spaces with power weight.


Introduction
One of the important directions of development of the modern theory of partial differential equations is the study of nonlocal boundary value problems for different types of differential equations and partial differential equation systems and establishing the conditions for their correct solvability. Such problems arise when modeling different phenomena and processes in modern science, quantum mechanics, technology, economics, etc.
Multipoint problems for partial differential equations were studied in the papers by Ptashnyk and his disciples. In particular, paper [1] is dedicated to multipoint problems for partial differential equations, not resolved relative to the highest time derivative. The question of the existence and qualitative properties of solutions for equations with limited order of degeneration has been studied in papers [2][3][4]. The Dirichlet problem with impulse action for a parabolic equation with power singularities of arbitrary order on time and spatial variables is investigated in paper [5]. In paper [6], a multipoint one-sided boundary value problem is studied. In paper [7], presented is the result of research of optimal control of the system described by the problem with an oblique derivative and an integral condition on time variable for parabolic equations with power singularities of arbitrary order. For a second-order parabolic equation, a multipoint (in time) problem with oblique derivative is considered in paper [8]. Conditions for the existence and uniqueness of solution of the posed problem in Hölder spaces with power weight are established.
In this paper, we investigate a multipoint time-varying Cauchy problem for a parabolic second-order equation with power singularities and arbitrary order degenerations in the coefficients of spatial and time variables at some set of points. We also find conditions for the existence and uniqueness of the solution of formulated problem in Hölder spaces with power weight.
Let D be an arbitrary closed domain in R n , Q ðkÞ = ½t k , We define the functional space in which we study problem (1) and (2).
We assume that the initial problems (1) and (2) satisfy the following conditions: (a) For the arbitrary vector ξ = ðξ 1 , ⋯, ξ n Þ, ∀ðt, xÞ ∈ Π \ Q ð0Þ , the following inequality is true, where c 1 , c 2 are fixed positive constants and s 1 ðμ Advances in Mathematical Physics Let us formulate the main result of the paper.

Theorem 1.
Let conditions (a) and (b) be satisfied for problem (1) and (2). Then, there exists a unique solution of problem (1) and (2) in the space H 2+α ðγ ; β ; 0 ; ΠÞ, and the following estimate is correct: To study problem (1) and (2), we construct a sequence of solutions of problems with a smooth coefficient limit value of which is the solution of problem (1) and (2).

Evaluation of Solutions of Problems with Smooth Coefficients
Let Π ðkÞ m = Π ðkÞ ∩ fðt, xÞ ∈ Π ðkÞ | s 1 ð1, tÞ ≥ m −1 1 , s 2 ð1, xÞ ≥ m −1 2 g, m = ðm 1 , m 2 Þ, m 1 > 1, m 2 > 1 be a sequence of domains that for m 1 → ∞, m 2 → ∞ converges to Π ðkÞ . In the domain Π, we consider the problem of finding the functions u m ðt, xÞ that satisfy the equations and the time variable t condition Here, the coefficients a ij , a i , a 0 , and functions f m , φ ðkÞ m , into the domains Π ðkÞ m coincide with A ij , A i , A 0 , f , φ k , respectively, and in the domains Π \ Π ðkÞ m are continuous prolongations of coefficients A ij , A i , A 0 and functions f , φ k from domains Π ðkÞ m into domains ΠΠ ðkÞ m with preservation of their smoothness and norm ( [9], p. 82).
To solve problem (7) and (8), we have a correct theorem.

Theorem 2.
Let u m ðt, xÞ be the classical solutions of problem (7) and (8) in the domain Π and let conditions (a) and (b) be satisfied. Then, for u m ðt, xÞ, the following estimate is true.
Proof. Let max Q ðkÞ u m ðt, xÞ = u m ðM 1 Þ. If M 1 ∈ Q ðkÞ , then at the point M 1 , the following correlations are true and equation (7) is satisfied. Considering (10) and equation (7) at the point M 1 , the inequality are true and equation (7) is satisfied. Considering correlation (12) and equation (7) at the point M 2 , we have In the case of M 1 ∈ Q ðkÞ ∩ ðt = t k Þ or M 2 ∈ Q ðkÞ ∩ ðt = t k Þ from condition (8), we obtain Considering inequality (11), (13), and (14), we obtain The theorem is proved. Now, we find estimates of the derivatives of solutions u m ðt, xÞ. In the space C l ðΠÞ, we introduce a norm ku m ; γ ; β ; q ; Πk l which is equivalent, at fixed m 1 , m 2 , to the Hölder norm, which is defined by the same way as the 3 Advances in Mathematical Physics norm ku ; γ ; β ; q ; Πk l ; only, instead of functions s 1 ðq ð1Þ , tÞ and s 2 ðq ð2Þ , xÞ, we take d 1 ðq ð1Þ , tÞ and d 2 ðq ð2Þ , xÞ, respectively, Theorem 3. Let conditions (a) and (b) be satisfied. Then, for the solution of problem (7) and (8), the estimate is true.
We denote and choose a thrice differentiable function ψðt, yÞ, which satisfies the conditions We denote the function Z m ðt, yÞ = v m ðt, yÞψðt, yÞ which is a solution of the Cauchy problem On the basis of Theorem 5.3 from ([2], p. 364), for the solution of problem (31) and (32) and arbitrary points ðM 1 , The definition of the space H l ðγ ; β ; q ; QÞ implies the satisfaction of inequalities