Open access peer-reviewed chapter - ONLINE FIRST

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

By Vladyslav Antonovich Litovchenko

Submitted: September 20th 2020Reviewed: November 13th 2020Published: December 12th 2020

DOI: 10.5772/intechopen.95024

Downloaded: 26

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:

Gtτx=4πatτnex24atτ,t>τ0,xRnE1

(here a– is the coefficient of thermal conductivity, and – is the Euclidean norm in Rn). 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 pis 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 pand hin such systems produces a peculiar “dissipation” effect, the measure of which may be a special characteristic of the system – its genus μ: 1phμ1. 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 μ=1, while for the systems with ph, generally speaking, the genus is μ<1. Besides, the more the parabolicity index hdeviates from the order of the system p, the more its genus μ, 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 Gtτ, the analytic properties in the complex space Cn[15] are getting worse, and the order of exponential behavior on the real hyperplane Rnchanges [16]:

xkGtτxAktτn+γ+k+heδ0xtτμ/pppμ,0<μ1,eδ0xtτμ/hhhμ,μ0,γ0.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 Gtτxparabolic systems with the positive genus μon the set τ+×Rnshows the behavior typical for Gtτx: it decreases exponentially and has a peculiarity at only one point tx=τ0. This fact allowed us to successfully develop the classical theory of the Cauchy problem for parabolic systems with variable coefficients and non-negative genus μin [25, 26, 27, 28]. However, according to these estimates, in the case of μ<0the function Gtτxmay have a peculiarity on the entire hyperplane t=τ, xRn. 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 μ<0?

The answer to this question is given in this paper. A method for studying the function Gtτxfor parabolic Shilov-type systems of genus μ<0, which allows us to more accurately describe the behavior of this function in the vicinity of the point tx=τ0is 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 N– be the set of all natural numbers; Nm=1m; Rnand Cn– real and complex space of n1dimension respectively; Z+n– the set of all n-dimensional multi-indices; RR1,CC1, Z+Z+1; i– imaginary unit; – scalar product in the space Rn; x+iyx2+y212, if xyR; zlz1l1znln, zlz1l1znln, z+hz1h++znh, z+z+1, if zz1znCn, ll1lnZ+n, hR; ξ– is the partial derivative with the variable ξ.

Let us fix mpN, T0+arbitrarily and consider the system of partial differential equations of porder

tutx=Atixutx,txΠ0T,E3

in which Π0T0T×Rn, utxcolu1txum(tx)– is an unknown vector-function and

Atixk+pakjltik+xkj,l=1mE4

matrix differential expression with coefficients akjl.

Let us denote by Athe matrix symbol of the differential expression Atix:

Ats=k+pakjltskj,l=1m,t0T,sCn.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 akjlare constant, i.e., when

AtixAix,AtA,E6

the system (3) on the set Π0Tis referred to as Shilov-type parabolic system with the parabolicity index h, 0<hp, if [15]

δ0>0δ0ξRn:maxjNmReλjξδ0ξh+δ,E7

where λjs- characteristic numbers of the matrix symbol As, sCn.

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

tvtξ=Atξvtξ,0τ<tT,ξRn.E8

The matriciant of the system (8) is such a matrix solution of the system Θτt,0τ<tT,that

Θτtt=τ=Eτ0TE9

(here E– a single matrix of morder).

Under the condition of continuity of the coefficients of the system (3), the matriciant Θτthas the structure [29]

Θτt=E+r=1τtτt1τtr1j=1rAtjdtrdt2dt1.E10

The system (3) with continuous coefficients on 0Tis called a Shilov-type parabolic system on the set Π0Twith parabolicity index h, 0<hp, if for the matriciant Θτt,0τ<tT,of the corresponding dual by Fourier system (8) the following estimation is performed [15]

Θτtξc1+ξγeδtτξh,tξΠτT,E11

with some positive constants cand δ. Here

γphm1,ajlj=1,l=1k,mmaxjlajl.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 ph, 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 Θτtinto the complex space Cnto the complete analytical function. Taking into account the estimation

Atsc1+sp,0tT,sCn,E13

we find that

Θτtsc0eδ0tτsp,0τ<tT,sCnE14

(here, a c0and δ0are positive constants independent of τ, tand s).

The smoothness of the matriciant Θτttogether 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

Kν=ξ+Cn:ηK1+ξνE15

from νwith 1ph1, in which the following estimate is performed

Θτtξ+c11+ξγeδ1tτξh,0τ<tT.E16

The genusμof the Shilov-type parabolic system (3) is the exact upper boundary of the indices ν, with which in the domain Kνfor the matriciant Θτtthe estimate (16) is performed [15]

Similarly to 2b-parabolicity, it is convenient to call the Shilov-type parabolicity a ph-parabolicity.

It should be noted that the fundamental solution of the Cauchy problem for ph-parabolic system (3) is represented by the function [15]

Gtτx=2πnRneixξΘτtξ.E17

The following section gives an example of a ph-parabolic system and defines a class of systems with dissipative parabolicity, each of which is a ph-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 phbased 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 Atixallows an image

Atix=A0ix+A1tix,E18

where

A0ixk+pakljik+xkl,j=1m,A1tixk+p1akljtik+xkl,j=1m.E19

Let us assume that the corresponding system

tutx=A0ixutx,txΠ0T,E20

is ph-parabolic on the set ΠτT, and the coefficients of the differential expression A1tixare continuous complex-valued functions defined on 0T, while the values p, p1and hsatisfy the condition

(A): 0p1+phm1<h.

Example of system (3) with condition (A). Let n=1, m=2, a>0and cj, jN5, are some continuous on 0Tcomplex-valued functions. Then the system

tu1=ax4+c1tx2u1+x5x3+c2txu2,tu2=c3tx2x3u1ax4c4tx2c5tu2,E21

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

A0ix=ax4x5x3x3ax4,E22
A1tix=c1tx2c2txc3tx2c4tx2+c5tE23

and solving the appropriate equation

detA0sλE=0,sCn,E24

we obtain that λ1,2s=as4±is8+s6, p=5, p1=2and h=4. For these values p,p1and h, 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 anph-parabolic system with variable coefficients.

Proof. According to the definition of ph-parabolicity for the system (3) with variable coefficients, it is enough to show that for the matrix Θτtof the corresponding dual by Fourier system (8) on the set ΠτT, τ0T, the estimate (11) is performed.

On condition of continuity of the coefficients, the matriciant Θτtis the only solution of the Cauchy problem for the system (8) with the initial condition

vtt=τ=E.E25

Thus, the correct equality

tΘτtξ=A0ξΘτtξ+Qτtξ,E26

in which

QτtξA1tξΘτtξ.E27

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

Θτtξ=etτP0ξ+τtetβP0ξQτβξ,tξΠτT,τ0T.E28

It should be noted that etτP0is the matriciant of the dual by Fourier system to ph-parabolic system (20), therefore, the estimate (11) is performed for it. Hence, considering the inequality

Qτtξc01+ξp1Θτtξ,tξΠτT,τ[0;T)E29

(here the positive constant c0in independent of τ,tand ξ), the next estimate is obtained

Θτtξc1+ξγeδtτξh+c11+ξγ1+ξp1τteδtβξhΘτβξ,E30

from which we come to the ratio

Θτtξeδtτξh1+ξγc+c11+ξγ1+ξp1τtΘτβξeδβτξh1+ξγ.E31

Using now the classic Gro¨nwall’s lemma [4], we get

Θτtξc1+ξγetτδξhc11+ξγ1+ξp1,tξΠτT,τ[0;T).E32

This inequality, in combination with condition (A), ensures the existence of positive constants cand δ, with which for all tξΠτT,τ0T,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) for2b-parabolic systems with the coefficients continuously depending ont. Therefore, analyzing this proof, especially its last part, we can understand why, in contrast to the2b-parabolicity, in the case ofphthe difficulties in establishing thecondition (11).

The study of the properties of the matriciant Θτtfor systems with a negative genus μwill 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 μ<0.

Theorem 2 Let the system (3)phbe parabolic with the negative genusμ, and letl0andα0be such arbitrarily fixed numbers thatl1+αhandαhlμαh. Then

cδAB0+kZ+nqZ+xRn\0τ0TtτT:
xkGtτxcAqBk+xqq1μhqkkhtτl+μqnk+l0γheδx+tτl+μ/h11μ/h,E33

wherel0max1l.

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

Let us consider the functional matrix

ltξΘ0ttl/hξ,l0,t0T,ξRn,E34

for which, according to the definition of the genus μof the system (3), on the set

Kμ=ξ+Cn:tl/hηK01+tl/hξμE35

the estimate is performed

ltξ+c1+tl/hξγeδt1lξ+h,t(0;T],E36

with positive values cand δ, independent of t, ξand η.

To estimate the derivatives ξqlwe use the Cauchy integral formula

ξqltξ=j=1nqj!2πiΓRjltσdσjσjξjqj+1,qZ+n,ξRn,t0T,E37

in which ΓRj– is a circle with the center in the point ξj+i0of the radius

Rj=K01+tl/hξjμ,0<K0<<1.E38

Let us put ΓRΓR1××ΓRnand fix a fairly small positive K0so that ΓRKμ(the existence of such K0is substantiated in ([30], p. 287) when proving the theorem 4 of the Phragmén-Lindelöf type in the case of nindependent variables). Then, according to the estimate (36), we have

ξqltξc1+tl/hξ̂γeδt1lξˇ+hj=1nqj!Rjqj,E39

where ξ̂ξˇRn– fixed points with such coordinates

ξ̂jξˇjξjRjξj+Rj,jNn,E40

that

ξ̂j2=maxξjRjξj+Rjxj2,ξˇj=minξjRjξj+Rjxj,E41

that is

ξ̂j=ξj+χjRj,ξˇj=ξj+ζjRj,E42

at some χjζj11.

First of all it should be noted that

Rj=K01+tl/hξjμK0,ξjR,t0T.E43

Since

ξnξ+,ξRn,E44

then

ξ̂nj=1nξj+χjRjnj=1nξj+Rjnj=1nξj+K0n1+ξ+,K01/n,ξRn,t0T.E45

Now let us estimate the value eδt1lξˇ+h.

Let us start with the simpler case when t1T.

We assume that ξj2K0, then

ξˇjh=ξjRjhξjK0h2hξjh.E46

If ξj<2K0, then

eδt1lξˇjh1=eδ0t1lξjheδ0t1lξjheδ0t1lξjh+aδ0>0,E47

where a=δ02K0hmaxt1Tt1l.

Therefore, for each δ>0there are such positive constants c0and δ0that for all ξjRand t1Tthe estimate is performed

eδt1lξˇjhc0eδ0t1lξjh.E48

We show that the statement (48) is also true in the case of t01.

We shall fix arbitrarily α0and further consider that l1+αh. Then for ξj<tα, we have:

eδt1lξˇjheδ0t1lξjhδ0t1lξjheδ0t1lξjht1+αhleδ0t1lξjh1δ0>0.E49

Now let tαξj, and αbe such that the condition: lαhμαhis satisfied. Taking into consideration that

RjK01+tαl/hμK0tl/hαμK0tα,E50

we obtain:

ξˇjh=ξjRjhξjK0tαh=ξjh1K0tα/ξjhξjh1K0h2hξjh.E51

Hence we arrive at performing (48) at t01.

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

supy0yβeδy=ββ,β>0,δ>0,E52

we find:

c011+tl/hξ̂γeδ3t1lξˇ+h2nγt/h1+ξ+γeδ0t1lξ+h2nγt/h1+ξ+eδ0γt1lξ+hγ2nγt/h1+nγtl1heδ01/hγ;c01Rjqjeδ3nt1lξˇ+hRjqjeδ0t1lξjh=K0qj1+tl/hξjμqjeδ0t1lξjh2μK0qj1+tμlqj/hξjμqjeδ0t1lξjh2μK0qj1+μqjheδ0tμqj/h.E53

Together with (39), these estimates ensure the existence of such positive constants c, Aand δthat for all ξRn, t0Tand qZ+nthe following inequality is true

ξqltξcAq+q1μhqtμq+l0γheδt1lξ+h,E54

in which l0=max1l.

Next, we shall use the image

Gt0x=2πntnl/hRneixtl/hξtξ,txΠ0T.E55

Identity

tl/hLξ;xeixtl/hξ=eixtl/hξ,E56

in which

Lξ;x=ix2j=1nxjξj,E57

at x0allows to write the previous equality in the form

Gt0x=2πntlqn/hRnLξ;xqeixtl/hξtξqZ+.E58

Hence, after integrating by parts qtimes, we arrive at the relation

Gt0x=1q2πntlqn/hRneixtl/hξLξ;xqtξqZ+,E59

from which we obtain that

xrxkGt0x2πntlqnkr+hRnξkξrLξ;xqtξ,E60

for all rkZ+nand qZ+.

Having considered the estimate (54), for tξΠ0Tand x0we find:

ξrLξ;xqtξcAq+r+xqtμq+r+l0γhr1μhrq1μhqeδt1lξ+h.E61

Then

xrxkGt0xc1Aq+r+xqtl+μq+r+ln+k+l0γhr1μhrq1μhq××Rnξkeδl1lξ+hc1Aq+r+xqtl+μq+r+nk+l0γhr1μhrq1μhq××j=1nsupy0ykjheδ2yRneδ2ζh+c2Aq+r+Bk+xqtl+μq+r+nk+l0γh××r1μhrq1μhqkkh,t0T,x0,qZ+,krZ+nE62

(here positive values c2, Aand Bdo not depend on t, x, q, kand r).

Thus, for all t0T, xRn\0, qZ+and kZ+nthe correct estimates are

xkGt0xc2AqBk+xqtl+μq+r+nk+l0γhq1μhqkkh××j=1ninfrjtl+μhArjrjrj1μhxjrjcAqBk+xqtl+μqnk+l0γhq1μhqkkheδx+tl+μ/h11μ/h,E63

in which the values c>0, A>0, B>0and δ>0do not depend on k, q, tand x.

The theorem is proved.

Remark 2 Zhitomirskii’s estimates (2) are obtained from (33) forq=0,l=0andα=0.

Given that l=1+αh, αhlμ=αhand q=0, from the theorem 2 we arrive at the following statement.

Corollary 1 Forph-parabolic system (3) with genusμ<0there are such positive constantsc,Bandδthat for allkZ+n,xRn,τ0TandtτTthe next estimate is performed

xkGtτxcBk+kkhtτn+γ+k+heδx+tτ1/h11μ/h.E64

Therefore, according to the corollary 1, the fundamental solution Gin the case of negative genus μalso has a singularity only at the point tx=τ0.

Corollary 2 Let (3)phbe a parabolic system with negative genusμ, then for alltτT,τ0T,xRn\0andkZ+nestimate is performed

xkGtτxcBk+kkhk+1μhk+xn+γ+1+k+tτ1γheδx+tτ1/h11μ/h,E65

in which the positive values c, δand Bdo not depend on t, τ, xand k; and are integer and fractional parts of the number respectively.

Proof. Estimates (65) are obtained directly from (33) at l=1+αh, αhlμ=αhand q=n+γ+1+k+.

The established estimates (65) provide exponential decrease when changing tτ+0on the set Rn\0derivatives of the function Gtτin case μ<0. Similarly to the case μ0considered in [25, 26, 27, 28], this will allow us to successfully study the Cauchy problem for wide classes of ph-parabolic systems (3) with negative genus μand variable coefficients akjltx. Moreover, this will also allow us to describe in a similar way the sets of classical solutions of such systems with generalized limit values fon the initial hyperplane and to study the local behavior of these solutions when changing tτ+0on that part of Rnwhere the functional fhas 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 akjltis 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 μthe function Gtτxon the set τT×Rnhas only one singular point tx=τ0. Similarly to the case μ0, these estimates allow to perform the expansion of the Shilov class ph-parabolic systems by supplementing it with the systems with negative genus μand 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 μ<0at the approximation of the initial hyperplane.

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Vladyslav Antonovich Litovchenko (December 12th 2020). Peculiarities of the Fundamental Solution of Parabolic Systems with a Negative Genus [Online First], IntechOpen, DOI: 10.5772/intechopen.95024. Available from:

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