Abstract
We consider a new class of systems of equations that combine the structures of Solonnikov and Éidel’man parabolic systems. We prove a theorem on the reduction of a general initial-value problem to a problem with zero initial data and a theorem on the correct solvability of an initial-value problem in a model case.
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I. G. Petrovskii, “On the Cauchy problem for systems of linear partial differential equations in the domain of nonanalytic functions,” Byul. Mosk. Univ., Ser. Mat. Mekh., 1, No. 7, 1–72 (1938).
S. D. Éidel’man, “On one class of parabolic systems,” Dokl. Akad. Nauk SSSR, 133, No. 1, 40–43 (1960).
S. D. Ivasishen and S. D. Éidel’man, “\(\overrightarrow {2b} \)-Parabolic systems,” in: Proceedings of the Seminar on Functional Analysis [in Russian], Issue 1, Institute of Mathematics, Ukrainian Academy of Sciences, pp. 3–175, 271–273 (1968).
S. D. Ivasyshen and O. S. Kondur, “On the Green matrix of the Cauchy problem and characterization of certain classes of solutions for \(\overrightarrow {2b} \)-parabolic systems of arbitrary order,” Mat. Stud., 14, No. 1, 73–84 (2000).
S. D. Éidel’man, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser, Berlin (2004).
V. A. Solonnikov, “On boundary-value problems for general parabolic systems,” Dokl. Akad. Nauk SSSR, 157, No. 1, 56–59 (1964).
V. A. Solonnikov, “On boundary-value problems for linear parabolic systems of differential equations of general form,” Tr. Mat. Inst. Akad. Nauk SSSR, 83, 3–163 (1965).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
H. P. Ivasyuk, “Initial-value problem for model Solonnikov parabolic systems with inhomogeneous structure,” Nauk. Visn. Cherniv. Univ., Issue 269, 49–52 (2005).
S. D. Éidel’man, Parabolic Systems [in Russian], Nauka, Moscow (1964).
H. P. Ivasyuk, “On properties of potentials of a model \(\overrightarrow {2b} \)-parabolic equation of arbitrary order,” Nauk. Visn. Cherniv. Univ., Issue 288, 51–56 (2006).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1501–1510, November, 2006.
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Ivasyshen, S.D., Ivasyuk, H.P. Solonnikov parabolic systems with quasihomogeneous structure. Ukr Math J 58, 1704–1716 (2006). https://doi.org/10.1007/s11253-006-0163-9
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DOI: https://doi.org/10.1007/s11253-006-0163-9