V Litovchenko, I Dovzhitska - The fundamental matrix of solutions of the cauchy problem for a class of parabolic systems of the shilov type with variable coefficients - страница 1

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Journal of Mathematical Sciences, Vol. 175, No. 4, June, 2011

The fundamental matrix of solutions of the Cauchy problem for a class of parabolic systems of the Shilov type with variable coefficients

Vladyslav A. Litovchenko and Iryna M. Dovzhitska

Presented by A. E. Shishkov

Abstract. We constructed a fundamental matrix of solutions of the Cauchy problem and studied its basic properties for a new class of linear parabolic systems with smooth bounded variable coefficients that includes a class of the Shilov-type parabolic systems of partial differential equations with nonnegative genus.

Keywords. Fundamental matrix of solutions, Cauchy problem, Shilov-type parabolic systems.

Introduction

The theory of parabolic equations with partial derivatives started, in fact, from that of the classical heat equation, but it acquired the clearest formulation after the fundamental work by I. G. Petrovskii [9] concerning with the description and the study of a rather wide class of linear systems of partial differential equations with coefficients independent of a spatial variable. Then the study of this class of systems proceeded quite intensively; the investigations by Petrovskii were continued in a number of works and extended to the parabolic systems with variable coefficients dependent on time and spatial variables (see, e.g., [1,4]).

In 1955, G. E. Shilov gave a new definition of the parabolic system [10] that generalizes the notion of parabolicity by Petrovskii and leads to an essential extension of the Petrovskii class of systems of the first order in the time variable with constant coefficients.

The study of systems parabolic by Shilov was carried out in many works, particularly in [2,5,7]. In this case, the main attention was paid only to the case of constant coefficients. This is explained, first of all, by that the Shilov parabolic systems, as distinct from the Petrovskii systems are not stable, generally speaking, in the sense of parabolicity to a change of coefficients, even those at the zero derivative [8].

In [11], Ya. I. Zhitomirskii defined a new class of parabolic systems of the Shilov type with variable lower coefficients (dependent only on spatial variables), by successfully using the notion of the head part of a system: the head part of every of such systems is a matrix differential expression parabolic by Shilov with constant coefficients. The mentioned class not only extends the class of systems parabolic by Shilov, but it supplements the Petrovskii class of systems with variable coefficients. For such systems with the use of the method of successive approximations, the existence of a solution of the Cauchy problem is established in the case where the initial data are smooth bounded functions. Moreover, the uniqueness of this solution in the class of bounded functions was proved. The further investigation of the Cauchy problem for systems from this class requires the construction of the fundamental matrix of solutions of the Cauchy problem (FMSCP) and its comprehensive study.

Translated from Ukrains'kil Matematychnyl Visnyk, Vol. 7, No. 4, pp. 516-552, October-November, 2010. Original article submitted June 22, 2010

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1072 - 3374/11/1754-0450 © 2011 Springer Science+Business Media, Inc.

In the present work, we carry out the full analytic description of FMSCP for parabolic systems of the Shilov type with nonnegative genus and bounded smooth variable lower coefficients (dependent on time and spatial variables), by developing the classical methods of the theory of parabolic systems with variable coefficients, and study its main properties.

1.   Auxiliary data. Statement of the problem

Let T be a fixed number from (0; +00), N be the set of all natural numbers; Nm := {1;m}; Rn and Cn be, respectively, real and complex spaces with dimensionality n > 1; R := R1, C := C1; Z+ be the set of all n-dimensional multiindices, Z+ := Z+; i be the imaginary unit; (, ) be the scalar product

in the space Rn; \\x\\ := (x,x)1/2 for x Є Rn; \x + iy\ := (x2 + y2)1/2 if {x,y}cR; \z\* := \z1\ +-----+ \zn\

and zl := z1 ...z^ if z := (Z1;Zn) Є Cn ,l := (h;Q Є Z+; Пм := {(t; x)\ t Є M,x Є Rn}, M С R, ПТ := {(t,x; r,£)\ 0 < т< t < T, {x,t}c Rn}; S be the Schwartz space of infinitely differentiable rapidly decaying functions defined on Rn, and S' be the corresponding space topologically dual to S [6].

In the space S, we define the continuous operations of summation, multiplication, and convolution,

(p*= f Ж-0Ш)<%, {<P,*P}cS,

and the Fourier transformation F [6]. It is known that F[S]= S, where

F[X]:= {гр\ i>t) = j V(x)ei(''x)dx, <p Є X}.

In this case, the operator F maps the space S onto S bijectively and continuously.

The direct and inverse Fourier transformations of a distribution f Є S' are given by the formulas

{F[f],F[<p]) :=(2n)n{f,<p),    {F-1[f],F~1[^]) := (2n)~n{f,^),^ Є S

(here, the brackets { , ) stand for a result of the action of a functional on a basic function).

Let also m be a fixed element of the set N. By S and S', we denote the Cartesian powers with exponent m of the spaces S and S' with component-wise convergence, respectively, in S and S'. By P(S), we denote the set of square matrices of order m, whose columns are elements S (also with element-wise convergence in the space S). The convolution of matrix elements ip = (<pij1 and tp = (ipij)k'j>=1 is denoted ip * pp, and the matrix element g = (gijis such that gij := ^ipir * pprj. It is obvious that if p Є S, {g,g0}C P(S), then g * p Є S i g * g0 Є P(S), and

F[g * p]= F[g]F[p],F[g * go]= F[g]F[go],

where F[(Pij)kljjn=l] = (F[pij])kn=1.

Consider the system of partial differential equations of order p

m

dtUj(t; x) = Y, Yl alj& x)ilkUdXui(t; x),j Є Nm, (t; x) Є П{о;П (1.1)

i=1 \k\*<P

such that its right-hand side admits the representation

a ij(t; x)i\k\*dkxui(t; x) = Y,{P0j(idx)+Pl1j(t,x; idx)}ui(t; x),

i=1 \k\*<p i=1

m

m

where

P0j(idx):=Y. ao,ki\k\tidx) := (t; x)i\k\*dkx.

\k\*<p \k\*<pi Moreover, the corresponding system

m

dtUj(t; x) = J2 p0j(idx)ui(t; x),j Є Nm, (t; x) Є Що;т\, (1.2) i=1

is parabolic by Shilov in a ball П(о;т] with the index of parabolicity h, 0 <h < p, and a nonnegative genus [5].

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V Litovchenko, I Dovzhitska - The fundamental matrix of solutions of the cauchy problem for a class of parabolic systems of the shilov type with variable coefficients