Peculiarities of the Fundamental Solution of Parabolic Systems with a Negative Genus

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.

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:

$$G_{\circ }\left(t-\tau ;x\right)={\left(\sqrt{4\mathit{\pi a}\left(t-\tau \right)}\right)}^{-n}{e}^{-\frac{\parallel x{\parallel }^2}{4a\left(t-\tau \right)}},t>\tau \ge 0,x\in {\mathbb{R}}^n$$E1

(here $a$ – is the coefficient of thermal conductivity, and $\parallel \cdot \parallel$ – is the Euclidean norm in ${\mathbb{R}}^n$). These systems were later called “parabolic by Petrovskii” or “$2b$-parabolic” systems. Due to the efforts of many researchers, the theory of $2b$-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 “$2b$ -parabolicity” and significantly expands the class of Petrovskii’s systems with constant coefficients by those systems, in which the order $p$ is no longer necessarily even, and may not coincide with the parabolicity index $h$ [14]. 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 $p$ and $h$ in such systems produces a peculiar “dissipation” effect, the measure of which may be a special characteristic of the system – its genus $\mu$: $1-\left(p-h\right)\le \mu \le 1$. The parabolic systems, in which $p=h$, − the classical equation of thermal conductivity, in particular, as well as all $2b$-parabolic systems, − have the genus $\mu =1$, while for the systems with $p\ne h$, generally speaking, the genus is $\mu <1$. Besides, the more the parabolicity index $h$ deviates from the order of the system $p$, the more its genus $\mu$, 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 $G\left(t,\tau ;\cdot \right)$, the analytic properties in the complex space ${\mathbb{C}}^n$ [15] are getting worse, and the order of exponential behavior on the real hyperplane ${\mathbb{R}}^n$ changes [16]:

$$\mid {\partial }_x^{k}G\left(t,\tau ;x\right)\mid \le A_k{\left(t-\tau \right)}^{-\frac{n+\gamma +{\left|k\right|}_{+}}{h}}\left\{\begin{array}{ll}{e}^{-{\delta }_0{\left(\frac{\parallel x\parallel }{{\left(t-\tau \right)}^{\mu /p}}\right)}^{\frac{p}{p-\mu }}},& 0<\mu \le 1,\\ e^{-{\delta }_0{\left(\frac{\parallel x\parallel }{{\left(t-\tau \right)}^{\mu /h}}\right)}^{\frac{h}{h-\mu }}},& \mu \le 0,\end{array}\right.\gamma \ge 0.$$E2

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 $G\left(t,\tau ;x\right)$ parabolic systems with the positive genus $\mu$ on the set $\left(\tau ;+\infty \right)\times {\mathbb{R}}^n$ shows the behavior typical for $G_{\circ }\left(t-\tau ;x\right)$: it decreases exponentially and has a peculiarity at only one point $\left(t;x\right)=\left(\tau ;0\right)$. This fact allowed us to successfully develop the classical theory of the Cauchy problem for parabolic systems with variable coefficients and non-negative genus $\mu$ in [25, 26, 27, 28]. However, according to these estimates, in the case of $\mu <0$ the function $G\left(t,\tau ;x\right)$ may have a peculiarity on the entire hyperplane $t=\tau$, $x\in {\mathbb{R}}^n$. 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 $\mu <0$?

The answer to this question is given in this paper. A method for studying the function $G\left(t,\tau ;x\right)$ for parabolic Shilov-type systems of genus $\mu <0$, which allows us to more accurately describe the behavior of this function in the vicinity of the point $\left(t;x\right)=\left(\tau ;0\right)$ 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 $\mathbb{N}$ – be the set of all natural numbers; ${\mathbb{N}}_m=\left\{1,\dots ;m\right\}$; ${\mathbb{R}}^n$ and ${\mathbb{C}}^n$ – real and complex space of $n\ge 1$ dimension respectively; ${\mathbb{Z}}_{+}^n$ – the set of all $n$-dimensional multi-indices; $\mathbb{R}≔{\mathbb{R}}^1,\mathbb{C}≔{\mathbb{C}}^1$, ${\mathbb{Z}}_{+}≔{\mathbb{Z}}_{+}^1$; $i$ – imaginary unit; $\left(\cdot ,\cdot \right)$ – scalar product in the space ${\mathbb{R}}^n$; $\mid x+\mathit{iy}\mid ≔{\left(x^2+y^2\right)}^{\frac{1}{2}}$, if $\left\{x,y\right\}\subset \mathbb{R}$; $z^l≔z_1^{l_1}\dots z_n^{l_n}$, ${\left|z\right|}^l≔{\left|z_1\right|}^{l_1}\dots {\left|z_n\right|}^{l_n}$, ${\left|z\right|}_{+}^h≔{\left|z_1\right|}^h+\dots +{\left|z_n\right|}^h$, ${\left|z\right|}_{+}≔{\left|z\right|}_{+}^1$, if $z≔\left(z_1;\dots ;z_n\right)\in {\mathbb{C}}^n$, $l≔\left(l_1;\dots ;l_n\right)\in {\mathbb{Z}}_{+}^n$, $h\in \mathbb{R}$; ${\partial }_{\xi }\cdot$ – is the partial derivative with the variable $\xi$.

Let us fix $\left\{m,p\right\}\subset \mathbb{N}$, $T\in \left(0;+\infty \right)$ arbitrarily and consider the system of partial differential equations of $p$ order

$${\partial }_{t}u\left(t;x\right)=A\left(t;i{\partial }_x\right)u\left(t;x\right),\left(t;x\right)\in {\mathrm{\Pi }}_{\left(0;T\right]},$$E3

in which ${\mathrm{\Pi }}_{\left(0;T\right]}≔\left(0;T\right]\times {\mathbb{R}}^n$, $u\left(t;x\right)≔\mathrm{col}\left(u_1\left(t;x\right);\dots ;u_m(t;x)\right)$ – is an unknown vector-function and

$$A\left(t;i{\partial }_x\right)≔{\left(\sum _{{\left|k\right|}_{+}\le p}{a}_k^{\mathit{jl}}\left(t\right)i^{{\left|k\right|}_{+}}{\partial }_x^k\right)}_{j,l=1}^m$$E4

matrix differential expression with coefficients $a_k^{\mathit{jl}}\left(\cdot \right)$.

Let us denote by $\mathcal{A}$ the matrix symbol of the differential expression $A\left(t;i{\partial }_x\right)$:

$$\mathcal{A}\left(t;s\right)={\left(\sum _{{\left|k\right|}_{+}\le p}{a}_k^{\mathit{jl}}\left(t\right)s^k\right)}_{j,l=1}^m,t\in \left(0;T\right],s\in {\mathbb{C}}^n.$$E5

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 $a_k^{\mathit{jl}}$ are constant, i.e., when

$$A\left(t;i{\partial }_x\right)\equiv A\left(i{\partial }_x\right),\mathcal{A}\left(t;\cdot \right)\equiv \mathcal{A}\left(\cdot \right),$$E6

the system (3) on the set ${\mathrm{\Pi }}_{\left[0;T\right]}$ is referred to as Shilov-type parabolic system with the parabolicity index $h$, $0<h\le p$, if [15]

$$\exists {\delta }_0>0\exists \delta \ge 0\forall \xi \in {\mathbb{R}}^n:\underset{j\in {\mathbb{N}}_m}{max}Re{\lambda }_j\left(\xi \right)\le -{\delta }_0\parallel \xi {\parallel }^h+\delta ,$$E7

where ${\lambda }_j\left(s\right)$ - characteristic numbers of the matrix symbol $\mathcal{A}\left(s\right)$, $s\in {\mathbb{C}}^n$.

If the coefficients of the system (3) depend on $t$ (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

$${\partial }_{t}v\left(t;\xi \right)=\mathcal{A}\left(t;\xi \right)v\left(t;\xi \right),0\le \tau <t\le T,\xi \in {\mathbb{R}}^n.$$E8

The matriciant of the system (8) is such a matrix solution of the system ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right),0\le \tau <t\le T,$ that

$${\left.{\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)\right|}_{t=\tau }=E\left(\forall \tau \in \left[0;T\right]\right)$$E9

(here $E$ – a single matrix of $m$ order).

Under the condition of continuity of the coefficients of the system (3), the matriciant ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)$ has the structure [29]

$${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)=E+\sum _{r=1}^{\infty }\underset{\tau }{\overset{t}{\int }}\underset{\tau }{\overset{t_1}{\int }}\dots \underset{\tau }{\overset{t_{r-1}}{\int }}\left(\prod _{j=1}^r\mathcal{A}\left(t_j;\cdot \right)\right){\mathit{dt}}_r\dots {\mathit{dt}}_2{\mathit{dt}}_1.$$E10

The system (3) with continuous coefficients on $\left[0;T\right]$ is called a Shilov-type parabolic system on the set ${\mathrm{\Pi }}_{\left[0;T\right]}$ with parabolicity index $h$, $0<h\le p$, if for the matriciant ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right),0\le \tau <t\le T,$ of the corresponding dual by Fourier system (8) the following estimation is performed [15]

$$\mid {\mathrm{\Theta }}_{\tau }^t\left(\xi \right)\mid \le c\left(1+\parallel \xi {\parallel }^{\gamma }\right)e^{-\delta \left(t-\tau \right)\parallel \xi {\parallel }^h},\left(t;\xi \right)\in {\mathrm{\Pi }}_{\left(\tau ;T\right]},$$E11

with some positive constants $c$ and $\delta$. Here

$$\gamma ≔\left(p-h\right)\left(m-1\right),\mid {\left(a_{\mathit{jl}}\right)}_{j=1,l=1}^{k,m}\mid ≔\underset{jl}{max}\mid a_{\mathit{jl}}\mid .$$E12

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 $t$-dependent coefficients at $p\ne h$, 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 ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)$ into the complex space ${\mathbb{C}}^n$ to the complete analytical function. Taking into account the estimation

$$\mid \mathcal{A}\left(t;s\right)\mid \le c\left(1+\parallel s{\parallel }^p\right),0\le t\le T,s\in {\mathbb{C}}^n,$$E13

we find that

$$\mid {\mathrm{\Theta }}_{\tau }^t\left(s\right)\mid \le c_0{e}^{{\delta }_0\left(t-\tau \right)\parallel s{\parallel }^p},0\le \tau <t\le T,s\in {\mathbb{C}}^n$$E14

(here, a $c_0$ and ${\delta }_0$ are positive constants independent of $\tau$, $t$ and $s$).

The smoothness of the matriciant ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)$ 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

$${\mathbb{K}}_{\nu }=\left\{\xi +\mathit{i\eta }\in {\mathbb{C}}^n:\parallel \eta \parallel \le K{\left(1+\parallel \xi \parallel \right)}^{\nu }\right\}$$E15

from $\nu$ with $\left[1-\left(p-h\right);1\right]$, in which the following estimate is performed

$$\mid {\mathrm{\Theta }}_{\tau }^t\left(\xi +\mathit{i\eta }\right)\mid \le c_1\left(1+\parallel \xi {\parallel }^{\gamma }\right)e^{-{\delta }_1\left(t-\tau \right)\parallel \xi {\parallel }^h},0\le \tau <t\le T.$$E16

The genus$\mu$ of the Shilov-type parabolic system (3) is the exact upper boundary of the indices $\nu$, with which in the domain ${\mathbb{K}}_{\nu }$ for the matriciant ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)$ the estimate (16) is performed [15]

Similarly to $2b$-parabolicity, it is convenient to call the Shilov-type parabolicity a $\left\{p,h\right\}$-parabolicity.

It should be noted that the fundamental solution of the Cauchy problem for $\left\{p,h\right\}$-parabolic system (3) is represented by the function [15]

$$G\left(t,\tau ;x\right)={\left(2\pi \right)}^{-n}\underset{{\mathbb{R}}^n}{\int }{e}^{-i\left(x,\xi \right)}{\mathrm{\Theta }}_{\tau }^t\left(\xi \right)\mathit{d\xi }.$$E17

The following section gives an example of a $\left\{p,h\right\}$-parabolic system and defines a class of systems with dissipative parabolicity, each of which is a $\left\{p,h\right\}$-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 [4] that the corresponding condition (11) is satisfied for $2b$-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 $p\ne h$ 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 $A\left(t;i{\partial }_x\right)$ allows an image

$$A\left(t;i{\partial }_x\right)=A_0\left(i{\partial }_x\right)+A_1\left(t;i{\partial }_x\right),$$E18

where

$$A_0\left(i{\partial }_x\right)≔{\left(\sum _{{\left|k\right|}_{+}\le p}{a}_k^{\mathit{lj}}{i}^{{\left|k\right|}_{+}}{\partial }_x^k\right)}_{l,j=1}^m,A_1\left(t;i{\partial }_x\right)≔{\left(\sum _{{\left|k\right|}_{+}\le p_1}{a}_k^{\mathit{lj}}\left(t\right)i^{{\left|k\right|}_{+}}{\partial }_x^k\right)}_{l,j=1}^m.$$E19

Let us assume that the corresponding system

$${\partial }_{t}u\left(t;x\right)=A_0\left(i{\partial }_x\right)u\left(t;x\right),\left(t;x\right)\in {\mathrm{\Pi }}_{\left(0;T\right]},$$E20

is $\left\{p,h\right\}$-parabolic on the set ${\mathrm{\Pi }}_{\left(\tau ;T\right]}$, and the coefficients of the differential expression $A_1\left(t;i{\partial }_x\right)$ are continuous complex-valued functions defined on $\left[0;T\right]$, while the values $p$, $p_1$ and $h$ satisfy the condition

(A): $0\le p_1+\left(p-h\right)\left(m-1\right)<h.$

Example of system (3) with condition (A). Let $n=1$, $m=2$, $a>0$ and $c_j\left(\cdot \right)$, $j\in {\mathbb{N}}_5$, are some continuous on $\left[0;T\right]$ complex-valued functions. Then the system

$$\left\{\begin{array}{l}{\partial }_t{u}_1=\left\{-a{\partial }_x^4+c_1\left(t\right){\partial }_x^2\right\}{u}_1+\left\{{\partial }_x^5-{\partial }_x^3+c_2\left(t\right){\partial }_x\right\}{u}_2,\\ {\partial }_t{u}_2=\left\{c_3\left(t\right){\partial }_x^2-{\partial }_x^3\right\}{u}_1-\left\{a{\partial }_x^4-c_4\left(t\right){\partial }_x^2-c_5\left(t\right)\right\}{u}_2,\end{array}\right.$$E21

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

$$A_0\left(i{\partial }_x\right)=\left(\begin{array}{cc}-a{\partial }_x^4& {\partial }_x^5-{\partial }_x^3\\ -{\partial }_x^3& -a{\partial }_x^4\end{array}\right),$$E22

$$A_1\left(t;i{\partial }_x\right)=\left(\begin{array}{cc}{c}_1\left(t\right){\partial }_x^2& c_2\left(t\right){\partial }_x\\ c_3\left(t\right){\partial }_x^2& c_4\left(t\right){\partial }_x^2+c_5\left(t\right)\end{array}\right)$$E23

and solving the appropriate equation

$$det\left({\mathcal{A}}_0\left(s\right)-\mathit{\lambda E}\right)=0,s\in {\mathbb{C}}^n,$$E24

we obtain that ${\lambda }_{1,2}\left(s\right)=-{\mathit{as}}^4\pm i\sqrt{s^8+s^6}$, $p=5$, $p_1=2$ and $h=4$. For these values $p,p_1$ and $h$, obviously the condition (A) holds.

Theorem 1Let (3) be a system with continuous coefficients, for which the conditions formulated in this clause are satisfied. Then it is an$\left\{p,h\right\}$-parabolic system with variable coefficients.

Proof. According to the definition of $\left\{p,h\right\}$-parabolicity for the system (3) with variable coefficients, it is enough to show that for the matrix ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)$ of the corresponding dual by Fourier system (8) on the set ${\mathrm{\Pi }}_{\left[\tau ;T\right]}$, $\tau \in \left[0;T\right)$, the estimate (11) is performed.

On condition of continuity of the coefficients, the matriciant ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)$ is the only solution of the Cauchy problem for the system (8) with the initial condition

$${\left.v\left(t;\cdot \right)\right|}_{t=\tau }=E.$$E25

Thus, the correct equality

$${\partial }_t{\mathrm{\Theta }}_{\tau }^t\left(\xi \right)=A_0\left(\xi \right){\mathrm{\Theta }}_{\tau }^t\left(\xi \right)+Q\left(\tau ,t;\xi \right),$$E26

in which

$$Q\left(\tau ,t;\xi \right)≔A_1\left(t;\xi \right){\mathrm{\Theta }}_{\tau }^t\left(\xi \right).$$E27

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

$${\mathrm{\Theta }}_{\tau }^t\left(\xi \right)=e^{\left(t-\tau \right)P_0\left(\xi \right)}+\underset{\tau }{\overset{t}{\int }}{e}^{\left(t-\beta \right)P_0\left(\xi \right)}Q\left(\tau ,\beta ;\xi \right)\mathit{d\beta },\left(t;\xi \right)\in {\mathrm{\Pi }}_{\left(\tau ;T\right]},\tau \in \left[0;T\right).$$E28

It should be noted that $e^{\left(t-\tau \right)P_0\left(\cdot \right)}$ is the matriciant of the dual by Fourier system to $\left\{p,h\right\}$-parabolic system (20), therefore, the estimate (11) is performed for it. Hence, considering the inequality

$$\mid Q\left(\tau ,t;\xi \right)\mid \le c_0\left(1+\parallel \xi {\parallel }^{p_1}\right)\mid {\mathrm{\Theta }}_{\tau }^t\left(\xi \right)\mid ,\left(t;\xi \right)\in {\mathrm{\Pi }}_{\left(\tau ;T\right]},\tau \in [0;T)$$E29

(here the positive constant $c_0$ in independent of $\tau ,t$ and $\xi$), the next estimate is obtained

$$\mid {\mathrm{\Theta }}_{\tau }^t\left(\xi \right)\mid \le c\left(1+\parallel \xi {\parallel }^{\gamma }\right)e^{-\delta \left(t-\tau \right)\parallel \xi {\parallel }^h}+c_1\left(1+\parallel \xi {\parallel }^{\gamma }\right)\left(1+\parallel \xi {\parallel }^{p_1}\right)\underset{\tau }{\overset{t}{\int }}{e}^{-\delta \left(t-\beta \right)\parallel \xi {\parallel }^h}\mid {\mathrm{\Theta }}_{\tau }^{\beta }\left(\xi \right)\mid \mathit{d\beta },$$E30

from which we come to the ratio

$$\frac{\mid {\mathrm{\Theta }}_{\tau }^t\left(\xi \right)\mid e^{\delta \left(t-\tau \right)\parallel \xi {\parallel }^h}}{\left(1+\parallel \xi {\parallel }^{\gamma }\right)}\le c+c_1\left(1+\parallel \xi {\parallel }^{\gamma }\right)\left(1+\parallel \xi {\parallel }^{p_1}\right)\underset{\tau }{\overset{t}{\int }}\frac{\mid {\mathrm{\Theta }}_{\tau }^{\beta }\left(\xi \right)\mid e^{\delta \left(\beta -\tau \right)\parallel \xi {\parallel }^h}}{\left(1+\parallel \xi {\parallel }^{\gamma }\right)}\mathit{d\beta }.$$E31

Using now the classic Gr$\ddot{\mathrm{o}}$nwall’s lemma [4], we get

$$\mid {\mathrm{\Theta }}_{\tau }^t\left(\xi \right)\mid \le c\left(1+\parallel \xi {\parallel }^{\gamma }\right)e^{-\left(t-\tau \right)\left(\delta \parallel \xi {\parallel }^h-c_1\left(1+\parallel \xi {\parallel }^{\gamma }\right)\left(1+\parallel \xi {\parallel }^{p_1}\right)\right)},\left(t;\xi \right)\in {\mathrm{\Pi }}_{\left(\tau ;T\right]},\tau \in [0;T).$$E32

This inequality, in combination with condition (A), ensures the existence of positive constants $c$ and $\delta$, with which for all $\left(t;\xi \right)\in {\mathrm{\Pi }}_{\left(\tau ;T\right]},\tau \in \left[0;T\right),$ the estimate (11) is performed.

The theorem is proved.

Remark 1The proof of Theorem 1 is based on the classical idea of establishing an estimate (11) for$2b$-parabolic systems with the coefficients continuously depending on$t$. Therefore, analyzing this proof, especially its last part, we can understand why, in contrast to the$2b$-parabolicity, in the case of$p\ne h$the difficulties in establishing thecondition (11).

The study of the properties of the matriciant ${\mathrm{\Theta }}_{\tau }^t\left(\cdot \right)$ for systems with a negative genus $\mu$ 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 $\mu <0$.

Theorem 2Let the system (3)$\left\{p,h\right\}$be parabolic with the negative genus$\mu$, and let$l\ge 0$and$\alpha \ge 0$be such arbitrarily fixed numbers that$l\le 1+\mathit{\alpha h}$and$\left(\mathit{\alpha h}-l\right)\mu \ge \mathit{\alpha h}$. Then

$$\exists \left\{c,\delta ,A,B\right\}\subset \left(0;+\infty \right)\forall k\in {\mathbb{Z}}_{+}^n\forall q\in {\mathbb{Z}}_{+}\forall x\in {\mathbb{R}}^n\backslash \left\{0\right\}\forall \tau \in \left[0;T\right)\forall t\in \left(\tau ;T\right]:$$

$$\mid {\partial }_x^{k}G\left(t,\tau ;x\right)\mid \le \frac{{\mathit{cA}}^q{B}^{{\left|k\right|}_{+}}}{\parallel x{\parallel }^q}{q}^{\left(1-\frac{\mu }{h}\right)q}{k}^{\frac{k}{h}}{\left(t-\tau \right)}^{\frac{\left(l+\mu \right)q-n-{\left|k\right|}_{+}-l_0\gamma }{h}}{e}^{-\delta {\left(\frac{{\left|x\right|}_{+}}{{\left(t-\tau \right)}^{\left(l+\mu \right)/h}}\right)}^{\frac{1}{1-\mu /h}}},$$E33

where$l_0≔max\left\{1;l\right\}$.

Proof. To simplify the calculations, we put $\tau =0$. The general case $\tau >0$ is realized similarly.

Let us consider the functional matrix

$${\Im }_l\left(t;\xi \right)≔{\mathrm{\Theta }}_0^t\left(t^{-l/h}\xi \right),l\ge 0,t\in \left(0;T\right],\xi \in {\mathbb{R}}^n,$$E34

for which, according to the definition of the genus $\mu$ of the system (3), on the set

$${\mathbb{K}}_{\mu }=\left\{\xi +\mathit{i\eta }\in {\mathbb{C}}^n:t^{-l/h}\parallel \eta \parallel \le K_0{\left(1+t^{-l/h}\parallel \xi \parallel \right)}^{\mu }\right\}$$E35

the estimate is performed

$$\mid {\Im }_l\left(t;\xi +\mathit{i\eta }\right)\mid \le c{\left(1+t^{-l/h}\parallel \xi \parallel \right)}^{\gamma }{e}^{-\delta t^{1-l}{\left|\xi \right|}_{+}^h},t\in (0;T],$$E36

with positive values $c$ and $\delta$, independent of $t$, $\xi$ and $\eta$.

To estimate the derivatives ${\partial }_{\xi }^q{\Im }_l$ we use the Cauchy integral formula

$${\partial }_{\xi }^q{\Im }_l\left(t;\xi \right)=\prod _{j=1}^n\frac{q_j!}{2\mathit{\pi i}}\underset{{\mathrm{\Gamma }}_{R_j}}{\int }\frac{{\Im }_l\left(t;\sigma \right)d{\sigma }_j}{{\left({\sigma }_j-{\xi }_j\right)}^{q_j+1}},q\in {\mathbb{Z}}_{+}^n,\xi \in {\mathbb{R}}^n,t\in \left(0;T\right],$$E37

in which ${\mathrm{\Gamma }}_{R_j}$ – is a circle with the center in the point ${\xi }_j+i0$ of the radius

$$R_j=K_0{\left(1+t^{-l/h}|{\xi }_j|\right)}^{\mu },0<K_0<<1.$$E38

Let us put ${\mathrm{\Gamma }}_R≔{\mathrm{\Gamma }}_{R_1}\times \dots \times {\mathrm{\Gamma }}_{R_n}$ and fix a fairly small positive $K_0$ so that ${\mathrm{\Gamma }}_R\subset {\mathbb{K}}_{\mu }$ (the existence of such $K_0$ is substantiated in ([30], p. 287) when proving the theorem 4 of the Phragmén-Lindelöf type in the case of $n$ independent variables). Then, according to the estimate (36), we have

$$\mid {\partial }_{\xi }^q{\Im }_l\left(t;\xi \right)\mid \le c{\left(1+t^{-l/h}\parallel \widehat{\xi }\parallel \right)}^{\gamma }{e}^{-\delta t^{1-l}{\left|\check{\xi }\right|}_{+}^h}\prod _{j=1}^n\frac{q_j!}{R_j^{q_j}},$$E39

where $\left\{\widehat{\xi };\check{\xi }\right\}\subset {\mathbb{R}}^n$ – fixed points with such coordinates

$$\left\{{\widehat{\xi }}_j;{\check{\xi }}_j\right\}\subset \left[{\xi }_j-R_j;{\xi }_j+R_j\right],j\in {\mathbb{N}}_n,$$E40

that

$${{\widehat{\xi }}_j}^2=\underset{\left[{\xi }_j-R_j;{\xi }_j+R_j\right]}{max}{x}_j^2,\mid {\check{\xi }}_j\mid =\underset{\left[{\xi }_j-R_j;{\xi }_j+R_j\right]}{min}\mid x_j\mid ,$$E41

that is

$${\widehat{\xi }}_j={\xi }_j+{\chi }_j{R}_j,{\check{\xi }}_j={\xi }_j+{\zeta }_j{R}_j,$$E42

at some $\left\{{\chi }_j,{\zeta }_j\right\}\subset \left[-1;1\right]$.

First of all it should be noted that

$$R_j=\frac{K_0}{{\left(1+t^{-l/h}|{\xi }_j|\right)}^{\mid \mu \mid }}\le K_0,{\xi }_j\in \mathbb{R},t\in \left(0;T\right].$$E43

Since

$$\parallel \xi \parallel \le \sqrt{n}{\left|\xi \right|}_{+},\xi \in {\mathbb{R}}^n,$$E44

then

$$\begin{array}{ll}\parallel \widehat{\xi }\parallel & \le \sqrt{n}\sum _{j=1}^n\mid {\xi }_j+{\chi }_j{R}_j\mid \le \sqrt{n}\sum _{j=1}^n\left(|{\xi }_j|+R_j\right)\le \sqrt{n}\sum _{j=1}^n\left(|{\xi }_j|+K_0\right)\le \\ & \le \sqrt{n}\left(1+{\left|\xi \right|}_{+}\right),K_0\le 1/n,\xi \in {\mathbb{R}}^n,t\in \left(0;T\right].\end{array}$$E45

Now let us estimate the value $e^{-\delta t^{1-l}{\left|\check{\xi }\right|}_{+}^h}$.

Let us start with the simpler case when $t\in \left[1;T\right]$.

We assume that $\mid {\xi }_j\mid \ge 2K_0$, then

$${\left|{\check{\xi }}_j\right|}^h={\left(|{\xi }_j|-R_j\right)}^h\ge {\left(|{\xi }_j|-K_0\right)}^h\ge 2^{-h}{\left|{\xi }_j\right|}^h.$$E46

If $\mid {\xi }_j\mid <2K_0$, then

$$e^{-\delta t^{1-l}{\left|{\check{\xi }}_j\right|}^h}\le 1=e^{{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h}{e}^{-{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h}\le e^{-{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h+a}\left(\forall {\delta }_0>0\right),$$E47

where $a={\delta }_0{\left(2K_0\right)}^h\underset{t\in \left[1;T\right]}{max}{t}^{1-l}$.

Therefore, for each $\delta >0$ there are such positive constants $c_0$ and ${\delta }_0$ that for all ${\xi }_j\in \mathbb{R}$ and $t\in \left[1;T\right]$ the estimate is performed

$$e^{-\delta t^{1-l}{\left|{\check{\xi }}_j\right|}^h}\le c_0{e}^{-{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h}.$$E48

We show that the statement (48) is also true in the case of $t\in \left(0;1\right)$.

We shall fix arbitrarily $\alpha \ge 0$ and further consider that $l\le 1+\mathit{\alpha h}$. Then for $\mid {\xi }_j\mid <t^{\alpha }$, we have:

$$e^{-\delta t^{1-l}{\left|{\check{\xi }}_j\right|}^h}\le e^{{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h-{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h}\le e^{-{\delta }_0\left(t^{1-l}{\left|{\xi }_j\right|}^h-t^{1+\mathit{\alpha h}-l}\right)}\le e^{-{\delta }_0\left(t^{1-l}{\left|{\xi }_j\right|}^h-1\right)}\left(\forall {\delta }_0>0\right).$$E49

Now let $t^{\alpha }\le \mid {\xi }_j\mid$, and $\alpha$ be such that the condition: $\left(l-\mathit{\alpha h}\right)\mid \mu \mid \ge \mathit{\alpha h}$ is satisfied. Taking into consideration that

$$R_j\le \frac{K_0}{{\left(1+t^{\alpha -l/h}\right)}^{\mid \mu \mid }}\le K_0{t}^{\left(l/h-\alpha \right)\mid \mu \mid }\le K_0{t}^{\alpha },$$E50

we obtain:

$$\begin{array}{ll}{\left|{\check{\xi }}_j\right|}^h& ={\left(|{\xi }_j|-R_j\right)}^h\ge {\left(|{\xi }_j|-K_0{t}^{\alpha }\right)}^h={\left|{\xi }_j\right|}^h{\left(1-K_0{t}^{\alpha }/|{\xi }_j|\right)}^h\ge \\ & \ge {\left|{\xi }_j\right|}^h{\left(1-K_0\right)}^h\ge 2^{-h}{\left|{\xi }_j\right|}^h.\end{array}$$E51

Hence we arrive at performing (48) at $t\in \left(0;1\right)$.

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

$$\underset{y\ge 0}{sup}\left\{y^{\beta }{e}^{-\mathit{\delta y}}\right\}={\left(\frac{\beta }{\mathit{e\delta }}\right)}^{\beta },\beta >0,\delta >0,$$E52

we find:

$$\begin{array}{l}{c}_0^{-1}{\left(1+t^{-l/h}\parallel \widehat{\xi }\parallel \right)}^{\gamma }{e}^{-\frac{\delta }{3}{t}^{1-l}{\left|\check{\xi }\right|}_{+}^h}\le {\left(2\sqrt{n}\right)}^{\gamma }{t}^{-\mathit{l\gamma }/h}{\left(1+{\left|\xi \right|}_{+}\right)}^{\gamma }{e}^{-{\delta }_0{t}^{1-l}{\left|\xi \right|}_{+}^h}\le \\ \le {\left(2\sqrt{n}\right)}^{\gamma }{t}^{-\mathit{l\gamma }/h}{\left(1+{\left|\xi \right|}_{+}{e}^{-\frac{{\delta }_0}{\gamma }{t}^{1-l}{\left|\xi \right|}_{+}^h}\right)}^{\gamma }\le {\left(2\sqrt{n}\right)}^{\gamma }{t}^{-\mathit{l\gamma }/h}{\left(1+n{\left(\frac{\gamma t^{l-1}}{\mathit{he}{\delta }_0}\right)}^{1/h}\right)}^{\gamma };\\ c_0^{-1}{R}_j^{-q_j}{e}^{-\frac{\delta }{3n}{t}^{1-l}{\left|\check{\xi }\right|}_{+}^h}\le R_j^{-q_j}{e}^{-{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h}=K_0^{-q_j}{\left(1+t^{-l/h}|{\xi }_j|\right)}^{\mid \mu \mid q_j}{e}^{-{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h}\le \\ \le {\left(2^{\mu }{K}_0\right)}^{-q_j}\left(1+t^{\mu {\mathit{lq}}_j/h}{\left|{\xi }_j\right|}^{\mid \mu \mid q_j}{e}^{-{\delta }_0{t}^{1-l}{\left|{\xi }_j\right|}^h}\right)\le {\left(2^{\mu }{K}_0\right)}^{-q_j}\left(1+{\left(\frac{\mid \mu \mid q_j}{\mathit{he}{\delta }_{0}t}\right)}^{\mid \mu \mid q_j/h}\right).\end{array}$$E53

Together with (39), these estimates ensure the existence of such positive constants $c$, $A$ and $\delta$ that for all $\xi \in {\mathbb{R}}^n$, $t\in \left(0;T\right]$ and $q\in {\mathbb{Z}}_{+}^n$ the following inequality is true

$$\mid {\partial }_{\xi }^q{\Im }_l\left(t;\xi \right)\mid \le {\mathit{cA}}^{{\left|q\right|}_{+}}{q}^{\left(1-\frac{\mu }{h}\right)q}{t}^{\frac{\mu {\left|q\right|}_{+}-l_0\gamma }{h}}{e}^{-\delta t^{1-l}{\left|\xi \right|}_{+}^h},$$E54

in which $l_0=max\left\{1;l\right\}$.

Next, we shall use the image

$$G\left(t,0;x\right)={\left(2\pi \right)}^{-n}{t}^{-\mathit{nl}/h}\underset{{\mathbb{R}}^n}{\int }{e}^{-i\left(x,t^{-l/h}\xi \right)}\Im \left(t;\xi \right)\mathit{d\xi },\left(t;x\right)\in {\mathrm{\Pi }}_{\left(0;T\right]}.$$E55

Identity

$$t^{l/h}{L}_{\xi ;x}\left[e^{-i\left(x,t^{-l/h}\xi \right)}\right]=e^{-i\left(x,t^{-l/h}\xi \right)},$$E56

in which

$$L_{\xi ;x}=i\parallel x{\parallel }^{-2}\sum _{j=1}^n{x}_j{\partial }_{{\xi }_j},$$E57

at $x\ne 0$ allows to write the previous equality in the form

$$G\left(t,0;x\right)={\left(2\pi \right)}^{-n}{t}^{l\left(q-n\right)/h}\underset{{\mathbb{R}}^n}{\int }{L}_{\xi ;x}^q\left[e^{-i\left(x,t^{-l/h}\xi \right)}\right]\Im \left(t;\xi \right)\mathit{d\xi }\left(\forall q\in {\mathbb{Z}}_{+}\right).$$E58

Hence, after integrating by parts $q$ times, we arrive at the relation

$$G\left(t,0;x\right)={\left(-1\right)}^q{\left(2\pi \right)}^{-n}{t}^{l\left(q-n\right)/h}\underset{{\mathbb{R}}^n}{\int }{e}^{-i\left(x,t^{-l/h}\xi \right)}{L}_{\xi ;x}^q\left[\Im \left(t;\xi \right)\right]\mathit{d\xi }\left(\forall q\in {\mathbb{Z}}_{+}\right),$$E59

from which we obtain that

$$\mid x^r{\partial }_x^{k}G\left(t,0;x\right)\mid \le {\left(2\pi \right)}^{-n}{t}^{\frac{l\left(q-n-{\left|k-r\right|}_{+}\right)}{h}}\underset{{\mathbb{R}}^n}{\int }{\left|\xi \right|}^k\mid {\partial }_{\xi }^r\left(L_{\xi ;x}^q\left[\Im \left(t;\xi \right)\right]\right)\mid \mathit{d\xi },$$E60

for all $\left\{r,k\right\}\subset {\mathbb{Z}}_{+}^n$ and $q\in {\mathbb{Z}}_{+}$.

Having considered the estimate (54), for $\left(t;\xi \right)\in {\mathrm{\Pi }}_{\left(0;T\right]}$ and $x\ne 0$ we find:

$$\mid {\partial }_{\xi }^r\left(L_{\xi ;x}^q\left[\Im \left(t;\xi \right)\right]\right)\mid \le {\mathit{cA}}^{q+{\left|r\right|}_{+}}\parallel x{\parallel }^{-q}{t}^{\frac{\mu \left(q+{\left|r\right|}_{+}\right)-l_0\gamma }{h}}{r}^{\left(1-\frac{\mu }{h}\right)r}{q}^{\left(1-\frac{\mu }{h}\right)q}{e}^{-\delta t^{1-l}{\left|\xi \right|}_{+}^h}.$$E61

Then

$$\begin{array}{ll}\mid x^r{\partial }_x^{k}G\left(t,0;x\right)\mid & \le c_1{A}^{q+{\left|r\right|}_{+}}\parallel x{\parallel }^{-q}{t}^{\frac{\left(l+\mu \right)\left(q+{\left|r\right|}_{+}\right)-l\left(n+{\left|k\right|}_{+}\right)-l_0\gamma }{h}}{r}^{\left(1-\frac{\mu }{h}\right)r}{q}^{\left(1-\frac{\mu }{h}\right)q}\times \\ & \times \underset{{\mathbb{R}}^n}{\int }{\left|\xi \right|}^k{e}^{-\delta l^{1-l}{\left|\xi \right|}_{+}^h}\mathit{d\xi }\le c_1{A}^{q+{\left|r\right|}_{+}}\parallel x{\parallel }^{-q}{t}^{\frac{\left(l+\mu \right)\left(q+{\left|r\right|}_{+}\right)-n-{\left|k\right|}_{+}-l_0\gamma }{h}}{r}^{\left(1-\frac{\mu }{h}\right)r}{q}^{\left(1-\frac{\mu }{h}\right)q}\times \\ & \times \left(\prod _{j=1}^n\underset{y\ge 0}{sup}\left\{y^{\frac{k_j}{h}}{e}^{-\frac{\delta }{2}y}\right\}\right)\underset{{\mathbb{R}}^n}{\int }{e}^{-\frac{\delta }{2}{\left|{\zeta }^h\right|}_{+}}\mathit{d\zeta }\le c_2{A}^{q+{\left|r\right|}_{+}}{B}^{{\left|k\right|}_{+}}\parallel x{\parallel }^{-q}{t}^{\frac{\left(l+\mu \right)\left(q+{\left|r\right|}_{+}\right)-n-{\left|k\right|}_{+}-l_0\gamma }{h}}\times \\ & \times r^{\left(1-\frac{\mu }{h}\right)r}{q}^{\left(1-\frac{\mu }{h}\right)q}{k}^{\frac{k}{h}},t\in \left(0;T\right],x\ne 0,q\in {\mathbb{Z}}_{+},\left\{k,r\right\}\subset {\mathbb{Z}}_{+}^n\end{array}$$E62

(here positive values $c_2$, $A$ and $B$ do not depend on $t$, $x$, $q$, $k$ and $r$).

Thus, for all $t\in \left(0;T\right]$, $x\in {\mathbb{R}}^n\backslash \left\{0\right\}$, $q\in {\mathbb{Z}}_{+}$ and $k\in {\mathbb{Z}}_{+}^n$ the correct estimates are

$$\begin{array}{ll}\mid {\partial }_x^{k}G\left(t,0;x\right)\mid & \le c_2{A}^q{B}^{{\left|k\right|}_{+}}\parallel x{\parallel }^{-q}{t}^{\frac{\left(l+\mu \right)\left(q+{\left|r\right|}_{+}\right)-n-{\left|k\right|}_{+}-l_0\gamma }{h}}{q}^{\left(1-\frac{\mu }{h}\right)q}{k}^{\frac{k}{h}}\times \\ & \times \left(\prod _{j=1}^n\underset{r_j}{inf}\left\{{\left(t^{\frac{l+\mu }{h}}A\right)}^{r_j}{r}_j^{r_j\left(1-\frac{\mu }{h}\right)}{\left|x_j\right|}^{-r_j}\right\}\right)\le \\ & \le {\mathit{cA}}^q{B}^{{\left|k\right|}_{+}}\parallel x{\parallel }^{-q}{t}^{\frac{\left(l+\mu \right)q-n-{\left|k\right|}_{+}-l_0\gamma }{h}}{q}^{\left(1-\frac{\mu }{h}\right)q}{k}^{\frac{k}{h}}{e}^{-\delta {\left(\frac{{\left|x\right|}_{+}}{t^{\left(l+\mu \right)/h}}\right)}^{\frac{1}{1-\mu /h}}},\end{array}$$E63

in which the values $c>0$, $A>0$, $B>0$ and $\delta >0$ do not depend on $k$, $q$, $t$ and $x$.

The theorem is proved.

Remark 2Zhitomirskii’s estimates (2) are obtained from (33) for$q=0$,$l=0$and$\alpha =0$.

Given that $l=1+\mathit{\alpha h}$, $\left(\mathit{\alpha h}-l\right)\mu =\mathit{\alpha h}$ and $q=0$, from the theorem 2 we arrive at the following statement.

Corollary 1For$\left\{p,h\right\}$-parabolic system (3) with genus$\mu <0$there are such positive constants$c$,$B$and$\delta$that for all$k\in {\mathbb{Z}}_{+}^n$,$x\in {\mathbb{R}}^n$,$\tau \in \left[0;T\right)$and$t\in \left(\tau ;T\right]$the next estimate is performed

$$\mid {\partial }_x^{k}G\left(t,\tau ;x\right)\mid \le {\mathit{cB}}^{{\left|k\right|}_{+}}{k}^{\frac{k}{h}}{\left(t-\tau \right)}^{-\frac{n+\gamma +{\left|k\right|}_{+}}{h}}{e}^{-\delta {\left(\frac{{\left|x\right|}_{+}}{{\left(t-\tau \right)}^{1/h}}\right)}^{\frac{1}{1-\mu /h}}}.$$E64

Therefore, according to the corollary 1, the fundamental solution $G$ in the case of negative genus $\mu$ also has a singularity only at the point $\left(t;x\right)=\left(\tau ;0\right)$.

Corollary 2Let (3)$\left\{p,h\right\}$be a parabolic system with negative genus$\mu$, then for all$t\in \left(\tau ;T\right]$,$\tau \in \left[0;T\right)$,$x\in {\mathbb{R}}^n\backslash \left\{0\right\}$and$k\in {\mathbb{Z}}_{+}^n$estimate is performed

$$\mid {\partial }_x^{k}G\left(t,\tau ;x\right)\mid \le \frac{{\mathit{cB}}^{{\left|k\right|}_{+}}{k}^{\frac{k}{h}}{\left|k\right|}_{+}^{\left(1-\frac{\mu }{h}\right){\left|k\right|}_{+}}}{\parallel x{\parallel }^{n+\left[\gamma \right]+1+{\left|k\right|}_{+}}}{\left(t-\tau \right)}^{\frac{1-\left\{\gamma \right\}}{h}}{e}^{-\delta {\left(\frac{{\left|x\right|}_{+}}{{\left(t-\tau \right)}^{1/h}}\right)}^{\frac{1}{1-\mu /h}}},$$E65

in which the positive values $c$, $\delta$ and $B$ do not depend on $t$, $\tau$, $x$ and $k$; $\left[\cdot \right]$ and $\left\{\cdot \right\}$ are integer and fractional parts of the number respectively.

Proof. Estimates (65) are obtained directly from (33) at $l=1+\mathit{\alpha h}$, $\left(\mathit{\alpha h}-l\right)\mu =\mathit{\alpha h}$ and $q=n+\left[\gamma \right]+1+{\left|k\right|}_{+}$.

The established estimates (65) provide exponential decrease when changing $t\to \tau +0$ on the set ${\mathbb{R}}^n\backslash \left\{0\right\}$ derivatives of the function $G\left(t,\tau ;\cdot \right)$ in case $\mu <0$. Similarly to the case $\mu \ge 0$ considered in [25, 26, 27, 28], this will allow us to successfully study the Cauchy problem for wide classes of $\left\{p,h\right\}$-parabolic systems (3) with negative genus $\mu$ and variable coefficients $a_k^{\mathit{jl}}\left(t;x\right)$. Moreover, this will also allow us to describe in a similar way the sets of classical solutions of such systems with generalized limit values $f$ on the initial hyperplane and to study the local behavior of these solutions when changing $t\to \tau +0$ on that part of ${\mathbb{R}}^n$ where the functional $f$ has good properties etc.

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 $a_k^{\mathit{jl}}\left(t\right)$ is quite broad and cannot be confined to the class of $2b$-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 $\mu$ the function $G\left(t,\tau ;x\right)$ on the set $\left(\tau ;T\right]\times {\mathbb{R}}^n$ has only one singular point $\left(t;x\right)=\left(\tau ;0\right)$. Similarly to the case $\mu \ge 0$, these estimates allow to perform the expansion of the Shilov class $\left\{p,h\right\}$-parabolic systems by supplementing it with the systems with negative genus $\mu$ 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 $\mu <0$ at the approximation of the initial hyperplane.