## Abstract

For the parabolic Shilov-type systems with a negative genus, a method of studying the properties of a fundamental solution of the Cauchy problem is proposed. This method allows to improve the known estimates of Zhitomirskii fundamental solution for systems with dissipative parabolicity and describe the features of this solution more accurately. It opens wide possibilities for constructing a classical theory of the Cauchy problem for parabolic systems with negative genus and variable coefficients.

### Keywords

- parabolic Shilov systems
- negative genus
- fundamental solution
- Cauchy problem
- matriciant
- dissipative parabolicity

## 1. Introduction

The theory of parabolic equations dates back to the time of the classical equation of thermal conductivity [1]. However, it acquired its most distinct features from the fundamental work by I.G. Petrovskii [2] published in 1938. There he describes and investigates a fairly wide class of systems of linear equations with partial derivatives, the fundamental solution of which has typical properties of the fundamental solution of the thermal conductivity equation:

(here

In 1955, G.Ye. Shilov formulates a new definition of parabolicity, which generalizes the concept of “

The presence of a gap between

Another anomalous phenomenon of the systems with “dissipative parabolicity” is their parabolic instability with respect to changes in the coefficients, even of those found at zero derivative. This fact was first pointed out by U Hou-Sin in 1960, who gave the example of a parabolically unstable system [17]. In this regard, the question of the study of parabolic Shilov-type systems with variable coefficients is problematic and still remains open.

Zhitomirskii’s estimates (2) show that the fundamental solution of

The answer to this question is given in this paper. A method for studying the function

The main content of the work is as follows. Section 2 contains the necessary information on the concept of parabolicity by Shilov. One class of systems with dissipative parabolicity and variable coefficients is described in Section 3. The study of the properties of the fundamental solution of the Cauchy problem for parabolic Shilov-type systems with a negative genus is carried out in Section 4. The final Section 5 is the conclusions.

## 2. Preliminary information

Let

Let us fix

in which

matrix differential expression with coefficients

Let us denote by

The Shilov-type parabolicity of the system (3) depending on the constancy or variability of its coefficients, is defined differently.

In the case when the coefficients

the system (3) on the set *Shilov-type parabolic* system with the parabolicity index

where

If the coefficients of the system (3) depend on

For the system (3) we shall write the corresponding dual by Fourier system

The *matriciant of the system* (8) is such a matrix solution of the system

(here

Under the condition of continuity of the coefficients of the system (3), the matriciant

The system (3) with continuous coefficients on *Shilov-type parabolic* system on the set

with some positive constants

It should be noted that for Shilov-type parabolic systems with constant coefficients, the condition (11) is a direct consequence of the corresponding condition of parabolicity (7) [15]. For parabolic systems (3) with

The Eq. (10) allows us to extend the matriciant

we find that

(here, a

The smoothness of the matriciant

from

The *genus*

Similarly to *-parabolicity*.

It should be noted that the fundamental solution of the Cauchy problem for

The following section gives an example of a

## 3. One class of parabolically resistant systems

Due to the difficulty of establishing the fundamental condition (11), for the system (3) with variable coefficients, the definition of parabolability according to Shilov is somewhat specific. It is known [4] that the corresponding condition (11) is satisfied for

Let us consider a system of Eq. (3), in which the differential expression

where

Let us assume that the corresponding system

is

(A):

**Example** of system (3) with condition (A). Let

is the system of kind (3) with condition (A). Indeed, putting

and solving the appropriate equation

we obtain that

**Theorem 1** *Let (3) be a system with continuous coefficients, for which the conditions formulated in this clause are satisfied. Then it is an**-parabolic system with variable coefficients.*

**Proof.** According to the definition of

On condition of continuity of the coefficients, the matriciant

Thus, the correct equality

in which

Having solved the Cauchy problem (26), (25), we obtain the image

It should be noted that

(here the positive constant

from which we come to the ratio

Using now the classic Gr

This inequality, in combination with condition (A), ensures the existence of positive constants

The theorem is proved.

**Remark 1** *The proof of Theorem 1 is based on the classical idea of establishing an estimate (11) for**-parabolic systems with the coefficients continuously depending on**. Therefore, analyzing this proof, especially its last part, we can understand why, in contrast to the**-parabolicity, in the case of**the difficulties in establishing the*condition (11).

The study of the properties of the matriciant

## 4. Properties of fundamental solution

Let us move on to the search for an answer to the question posed in Section 1 concerning the accuracy of Zhitomirskii’s estimates (2) in the case of a system (3) of genus

**Theorem 2** *Let the system (3)**be parabolic with the negative genus**, and let**and**be such arbitrarily fixed numbers that**and**. Then*

*where*

**Proof.** To simplify the calculations, we put

Let us consider the functional matrix

for which, according to the definition of the genus

the estimate is performed

with positive values

To estimate the derivatives

in which

Let us put

where

that

that is

at some

First of all it should be noted that

Since

then

Now let us estimate the value

Let us start with the simpler case when

We assume that

If

where

Therefore, for each

We show that the statement (48) is also true in the case of

We shall fix arbitrarily

Now let

we obtain:

Hence we arrive at performing (48) at

According to the estimates (45), (48) and equality

we find:

Together with (39), these estimates ensure the existence of such positive constants

in which

Next, we shall use the image

Identity

in which

at

Hence, after integrating by parts

from which we obtain that

for all

Having considered the estimate (54), for

Then

(here positive values

Thus, for all

in which the values

The theorem is proved.

**Remark 2** *Zhitomirskii’s estimates (2) are obtained from (33) for**and*

Given that

**Corollary 1** *For**-parabolic system (3) with genus**there are such positive constants**and**that for all**and**the next estimate is performed*

Therefore, according to the corollary 1, the fundamental solution

**Corollary 2** *Let (3)**be a parabolic system with negative genus**, then for all**and**estimate is performed*

in which the positive values

**Proof.** Estimates (65) are obtained directly from (33) at

The established estimates (65) provide exponential decrease when changing

## 5. Conclusions

The class of systems with dissipative parabolicity and variable coefficients defined in Section 3 proves that the class of parabolic Shilov-type systems with coefficients

Analyzing the obtained estimates (33) of the fundamental solution of the systems (3) with dissipative parabolicity, we conclude that in the case of the negative genus