On the Generalized Cauchy Problem for One Class of Differential Equations of Infinite Order
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ON THE GENERALIZED CAUCHY PROBLEM FOR ONE CLASS OF DIFFERENTIAL EQUATIONS OF INFINITE ORDER V. V. Horodets’kyi,1 O. V. Martynyuk,2 and R. I. Petryshyn1
UDC 517.98
We establish the solvability of a nonlocal multipoint (in time) problem regarded as a generalization of the Cauchy problem for the evolution equation with pseudodifferential operator (differentiation operator of infinite order) with initial conditions from the space of generalized functions of the ultradistribution type.
Introduction A broad class of partial differential equations is formed by linear parabolic and B-parabolic equations. The theory of these equations originates from the investigations of the heat conduction equation. The classical theory of the Cauchy and boundary-value problems for these equations and systems of equations was developed in the ´ works by Petrovskii, Eidel’man, Ivasyshen, Matiichuk, Zhitarashu, Friedman, Teklind, Solonnikov, Krekhivs’kyi, and other researchers. The Cauchy problem with initial data from the spaces of generalized functions in the form of distributions and ultradistributions was studied by Shylov, Gurevich, M. Horbachuk, V. Horbachuk, Zhytomyrs’kyi, Ivasyshen, Horodets’kyi, Litovchenko, and other researchers. As a formal extension of the class of parabolic equations, we consider the set of evolutionary equations with pseudodifferential operators that can be represented in the form ⇥ ⇤ −1 a(t, x; σ)Jx!σ , A = Jσ!x
{x, σ} ⇢ Rn ,
t > 0,
where a is a function (symbol) satisfying certain conditions and J and J −1 are the direct and inverse Fourier or Bessel transforms. The class of pseudodifferential operators includes the differential operators, the operators of fractional differentiation and integration, the convolution operators, and the Bessel operator B⌫ =
d d2 + (2⌫ + 1)x−1 , dx2 dx
⌫ > −1/2,
which contains in its structure the expression 1/x and can formally be represented in the form ⇥ 2 ⇤ F −σ , B⌫ = FB−1 B ⌫ ⌫
where FB⌫ is the integral Bessel transformation, etc. At present, in the theory of Cauchy problem for evolutionary pseudodifferential equations, significant results were obtained by numerous domestic and foreign mathematicians both in the investigation of correct solvability 1 2
Yu. Fed’kovych Chernivtsi National University, Chernivtsi, Ukraine. Yu. Fed’kovych Chernivtsi National University, Chernivtsi, Ukraine; e-mail: [email protected].
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 886–902, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.2321. Original article submitted January 21, 2020. 1030
0041-5995/20/7207–1030
© 2020
Springer Science+Business Media, LLC
O N THE G ENERALIZED C AUCHY P ROBLEM FOR O NE C LASS OF D IFFERENTIAL E QUATIONS OF I NFINITE O RDER
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of the Cauchy problem and in the representation of its solution in the case where initial conditions are elements of various function spaces (and, in particular, of the spaces of generalized functions). In the investigation of the classes of uniqueness and well-posedness of the Cauchy problem f
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