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.
- parabolic Shilov systems
- negative genus
- fundamental solution
- Cauchy problem
- dissipative parabolicity
The theory of parabolic equations dates back to the time of the classical equation of thermal conductivity . However, it acquired its most distinct features from the fundamental work by I.G. Petrovskii  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 – is the coefficient of thermal conductivity, and – is the Euclidean norm in ). These systems were later called “parabolic by Petrovskii” or “-parabolic” systems. Due to the efforts of many researchers, the theory of -parabolic systems developed rapidly throughout the second half of the 20th century. At that, there were considered the systems with both fixed and variable coefficients having different properties. Comprehensive results were obtained on the structure and properties of solutions, as well as on the correct solvability of boundary value problems, in particular, the Cauchy problem, in different functional spaces [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].
In 1955, G.Ye. Shilov formulates a new definition of parabolicity, which generalizes the concept of “-parabolicity” and significantly expands the class of Petrovskii’s systems with constant coefficients by those systems, in which the order is no longer necessarily even, and may not coincide with the parabolicity index . The parabolic Shilov-type systems, mostly with constant coefficients, were studied in [15, 16, 17, 18, 19, 20, 21, 22, 23, 24].
The presence of a gap between and in such systems produces a peculiar “dissipation” effect, the measure of which may be a special characteristic of the system – its genus : . The parabolic systems, in which , − the classical equation of thermal conductivity, in particular, as well as all -parabolic systems, − have the genus , while for the systems with , generally speaking, the genus is . Besides, the more the parabolicity index deviates from the order of the system , the more its genus , decreasing, gets further away from 1. In systems with such a dissipation, even with constant coefficients, deviations from the standards set by the classical thermal equation are observed. First of all, for their fundamental solution , the analytic properties in the complex space  are getting worse, and the order of exponential behavior on the real hyperplane changes :
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 . 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 parabolic systems with the positive genus on the set shows the behavior typical for : it decreases exponentially and has a peculiarity at only one point . This fact allowed us to successfully develop the classical theory of the Cauchy problem for parabolic systems with variable coefficients and non-negative genus in [25, 26, 27, 28]. However, according to these estimates, in the case of the function may have a peculiarity on the entire hyperplane , . This point significantly complicates the substantiation of the convergence of the process of successive approximations, in particular, while making the fundamental solution of the Cauchy problem for systems with variable coefficients using the Levy method. In this regard, a natural question arises: How accurate are the estimates (2) for systems of the genus ?
The answer to this question is given in this paper. A method for studying the function for parabolic Shilov-type systems of genus , which allows us to more accurately describe the behavior of this function in the vicinity of the point is also suggested in this research paper. In addition, one class of systems with dissipative parabolicity is also defined here. These systems are parabolically stable to changes in their lower coefficients.
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 – be the set of all natural numbers; ; and – real and complex space of dimension respectively; – the set of all -dimensional multi-indices; , ; – imaginary unit; – scalar product in the space ; , if ; , , , , if , , ; – is the partial derivative with the variable .
Let us fix , arbitrarily and consider the system of partial differential equations of order
in which , – is an unknown vector-function and
matrix differential expression with coefficients .
Let us denote by the matrix symbol of the differential expression :
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 are constant, i.e., when
the system (3) on the set is referred to as Shilov-type parabolic system with the parabolicity index , , if 
where - characteristic numbers of the matrix symbol , .
If the coefficients of the system (3) depend on (continuously), then the Shilov-type parabolicity of this system is defined somewhat differently, using the concept of the matriciant of the linear differential equations system.
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 that
(here – a single matrix of order).
Under the condition of continuity of the coefficients of the system (3), the matriciant has the structure 
The system (3) with continuous coefficients on is called a Shilov-type parabolic system on the set with parabolicity index , , if for the matriciant of the corresponding dual by Fourier system (8) the following estimation is performed 
with some positive constants and . Here
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) . For parabolic systems (3) with -dependent coefficients at , this fact generally cannot be confirmed by classical means of the theory of parabolic systems due to the parabolic instability of such systems to changing their coefficients.
The Eq. (10) allows us to extend the matriciant into the complex space to the complete analytical function. Taking into account the estimation
we find that
(here, a and are positive constants independent of , and ).
The smoothness of the matriciant together with the estimates (11), (14), according to the statement of the theorem of the Phragmén-Lindelöf type [30, p. 247], ensure the existence of the area
from with , in which the following estimate is performed
The genusof the Shilov-type parabolic system (3) is the exact upper boundary of the indices , with which in the domain for the matriciant the estimate (16) is performed 
Similarly to -parabolicity, it is convenient to call the Shilov-type parabolicity a -parabolicity.
It should be noted that the fundamental solution of the Cauchy problem for -parabolic system (3) is represented by the function 
The following section gives an example of a -parabolic system and defines a class of systems with dissipative parabolicity, each of which is a -parabolic system with variable coefficients.
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  that the corresponding condition (11) is satisfied for -parabolic systems (3) with continuous coefficients. However, it is impossible to confirm the fulfillment of this condition in a similar way for systems (3) with based on the condition (7). Therefore, it is important to be aware of the richness of the class of the Shilov-type systems with variable coefficients, in particular, of the examples of such systems that are not parabolic by Petrovskii.
Let us consider a system of Eq. (3), in which the differential expression allows an image
Let us assume that the corresponding system
is -parabolic on the set , and the coefficients of the differential expression are continuous complex-valued functions defined on , while the values , and satisfy the condition
Example of system (3) with condition (A). Let , , and , , are some continuous on complex-valued functions. Then the system
is the system of kind (3) with condition (A). Indeed, putting
and solving the appropriate equation
we obtain that , , and . For these values and , obviously the condition (A) holds.
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 -parabolicity for the system (3) with variable coefficients, it is enough to show that for the matrix of the corresponding dual by Fourier system (8) on the set , , the estimate (11) is performed.
On condition of continuity of the coefficients, the matriciant is the only solution of the Cauchy problem for the system (8) with the initial condition
Thus, the correct equality
Having solved the Cauchy problem (26), (25), we obtain the image
It should be noted that is the matriciant of the dual by Fourier system to -parabolic system (20), therefore, the estimate (11) is performed for it. Hence, considering the inequality
(here the positive constant in independent of and ), the next estimate is obtained
from which we come to the ratio
Using now the classic Grnwall’s lemma , we get
This inequality, in combination with condition (A), ensures the existence of positive constants and , with which for all the estimate (11) is performed.
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 ofthe difficulties in establishing thecondition (11).
The study of the properties of the matriciant for systems with a negative genus will be continued in the next section.
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 letandbe such arbitrarily fixed numbers thatand. Then
Proof. To simplify the calculations, we put . The general case is realized similarly.
Let us consider the functional matrix
for which, according to the definition of the genus of the system (3), on the set
the estimate is performed
with positive values and , independent of , and .
To estimate the derivatives we use the Cauchy integral formula
in which – is a circle with the center in the point of the radius
Let us put and fix a fairly small positive so that (the existence of such is substantiated in (, p. 287) when proving the theorem 4 of the Phragmén-Lindelöf type in the case of independent variables). Then, according to the estimate (36), we have
where – fixed points with such coordinates
at some .
First of all it should be noted that
Now let us estimate the value .
Let us start with the simpler case when .
We assume that , then
If , then
Therefore, for each there are such positive constants and that for all and the estimate is performed
We show that the statement (48) is also true in the case of .
We shall fix arbitrarily and further consider that . Then for , we have:
Now let , and be such that the condition: is satisfied. Taking into consideration that
Hence we arrive at performing (48) at .
According to the estimates (45), (48) and equality
Together with (39), these estimates ensure the existence of such positive constants , and that for all , and the following inequality is true
in which .
Next, we shall use the image
at allows to write the previous equality in the form
Hence, after integrating by parts times, we arrive at the relation
from which we obtain that
for all and .
Having considered the estimate (54), for and we find:
(here positive values , and do not depend on , , , and ).
Thus, for all , , and the correct estimates are
in which the values , , and do not depend on , , and .
The theorem is proved.
Remark 2 Zhitomirskii’s estimates (2) are obtained from (33) for,and.
Given that , and , from the theorem 2 we arrive at the following statement.
Corollary 1 For-parabolic system (3) with genusthere are such positive constants,andthat for all,,andthe next estimate is performed
Therefore, according to the corollary 1, the fundamental solution in the case of negative genus also has a singularity only at the point .
Corollary 2 Let (3)be a parabolic system with negative genus, then for all,,andestimate is performed
in which the positive values , and do not depend on , , and ; and are integer and fractional parts of the number respectively.
Proof. Estimates (65) are obtained directly from (33) at , and .
The established estimates (65) provide exponential decrease when changing on the set derivatives of the function in case . Similarly to the case considered in [25, 26, 27, 28], this will allow us to successfully study the Cauchy problem for wide classes of -parabolic systems (3) with negative genus and variable coefficients . Moreover, this will also allow us to describe in a similar way the sets of classical solutions of such systems with generalized limit values on the initial hyperplane and to study the local behavior of these solutions when changing on that part of where the functional has good properties etc.
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 is quite broad and cannot be confined to the class of -parabolic systems (3) with continuous coefficients only.
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 the function on the set has only one singular point . Similarly to the case , these estimates allow to perform the expansion of the Shilov class -parabolic systems by supplementing it with the systems with negative genus and coefficients depending on space variable, and to successfully develop the theory of the Cauchy problem for it using the classical means. Moreover, the estimates (33) open wide possibilities for studying the properties of solutions of parabolic systems of the genus at the approximation of the initial hyperplane.