For \( \left\{\overset{\sim }{p};\overrightarrow{h}\right\} \)-parabolic equations with continuous coefficients, we study the problem of finding classical solutions satisfying modified initial conditions with generalized data in the form of Gelfand- and Shilovtype distributions. This condition linearly combines the values of the solution at the initial time and at a certain intermediate time point. We establish the conditions for the correct solvability of this problem and deduce the formula for its solution. By using the obtained results, we solve the corresponding problem with impulsive action.
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References
S. M. Alekseeva and N. I. Yurchuk, “Method of quasiinversion for the problem of control of initial conditions for the heat-conduction equation with integral boundary conditions,” Differents. Uravn., 34, No. 4, 495–502 (1998).
I. A. Belavin, S. P. Kapitsa, and S. P. Kurdyumov, “Mathematical model of global demographic processes with regard for the space distribution,” Zh. Vychisl. Mat. Mat. Fiz., 38, No. 6, 885–902 (1998).
I. M. Gelfand and G. E. Shilov, Some Problems of the Theory of Differential Equations [in Russian], Fizmatgiz, Moscow (1958).
I. M. Gelfand and G. E. Shilov, Spaces of Test and Generalized Functions [in Russian], Fizmatgiz, Moscow (1958).
V. V. Gorodetskii, “Cauchy problem for equations parabolic in Shilov’s sense in the classes of generalized periodic functions,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 5, 82–84 (1988).
V. V. Gorodetskii, “On the localization of solutions of the Cauchy problem for \( \overrightarrow{2b} \)-parabolic systems in the classes of generalized functions,” Differents. Uravn., 24, No. 2, 348–350 (1988).
V. V. Gorodetskii, “Some stabilization theorems for solutions of the Cauchy problem for Shilov-parabolic systems in classes of generalized functions,” Ukr. Mat. Zh., 40, No. 1, 43–48 (1988); English translation: Ukr. Math. J., 40, No. 1, 35–40 (1988).
V. V. Horodets’kyi, R. I. Kolisnyk, and O. V. Martynyuk, “Nonlocal problem for partial differential equations of the parabolic type,” Bukov. Mat. Zh., 8, No. 2, 24–39 (2020); 10.31861/bmj2020.02.03.
S. D. Éidel’man, Parabolic Systems [in Russian], Nauka, Moscow (1964).
S. D. Éidel’man, S. D. Ivasishen, and F. O. Porper, “Liouville theorems for systems parabolic in Shilov’s sense,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 6, 169–179 (1961).
Ya. I. Zhitomirskii, “Cauchy problem for some types of parabolic (in a sense of G. E. Shilov) systems of linear partial differential equations with continuous coefficients,” Izv. Akad. Nauk SSSR, Ser. Mat., 23, 925–932 (1959).
N. I. Ivanchov, “Boundary-value problems for a parabolic equation with integral conditions,” Different. Equat., 40, 591–609 (2004); 10.1023/B:DIEQ.0000035796.56467.44.
L. I. Korbut and M. I. Matiichuk, “Representations of solutions of nonlocal boundary-value problems for parabolic equations,” Ukr. Mat. Zh., 46, No. 7, 947–951 (1994); English translation: Ukr. Math. J., 46, No. 7, 1039–1044 (1994).
V. Litovchenko, “The Cauchy problem for parabolic equations by Shilov,” Sib. Mat. Zh., 45, No. 4, 809–821 (2004); DOI: https://doi.org/10.1023/B:SIMJ.0000035831.63036.bb.
V. Litovchenko, “Cauchy problem for \( \left\{\overset{\sim }{p};\overrightarrow{h}\right\} \)-parabolic equations with time-dependent coefficients,” Math. Notes, 77, No. 3-4, 364–379 (2005); DOI: 10.1007/s11006-005-0036-9.
V. A. Litovchenko, “One method for the investigation of fundamental solution of the Cauchy problem for parabolic systems,” Ukr. Mat. Zh., 70, No. 6, 801–811 (2018); English translation: Ukr. Math. J., 70, No. 6, 922–934 (2018); 10.1007/s11253-018-1542-8.
V. A. Litovchenko, “Fundamental solution of the Cauchy problem for \( \left\{\overset{\sim }{p};\overrightarrow{h}\right\} \)-parabolic systems with variable coefficients,” Nelin. Kolyv., 21, No. 2, 189–196 (2018); English translation: J. Math. Sci., 243, No. 2, 230–239 (2019); 10.1007/s10958-019-04537-x.
V. A. Litovchenko and I. M. Dovzhitska, “The fundamental matrix of solutions of the Cauchy problem for a class of parabolic systems of the Shilov type with variable coefficients,” J. Math. Sci., 175, No. 4, 450–476 (2011); DOI: https://doi.org/10.1007/s10958-011-0356-0.
V. Litovchenko and I. Dovzhytska, “Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients,” Cent. Europ. J. Math., 10, No. 3, 1084–1102 (2012); DOI: https://doi.org/10.2478/s11533-012-0025-7.
V. A. Litovchenko and I. M. Dovzhytska, “Stabilization of solutions to Shilov-type parabolic systems with nonnegative genus,” Sib. Mat. J., 55, No. 2, 276–283 (2014); https://doi.org/10.1134/S0037446614020104.
V. A. Litovchenko and G. M. Unguryan, “Conjugate Cauchy problem for parabolic Shilov-type systems with nonnegative genus,” Different. Equat., 54, 335–351 (2018); DOI: https://doi.org/10.1134/S0012266118030060.
V. Litovchenko and G. Unguryan, “Some properties of Green’s functions of Shilov-type parabolic systems,” Miskolc Math. Notes, 20, No. 1, 365–379 (2019); DOI: https://doi.org/10.18514/MMN.2019.2089.
L. P. Luo, Y. Q. Wang, and Z. G. Gong, “New criteria for oscillation of vector parabolic equations with the influence of impulse and delay,” Acta Sci. Natur. Univ. Sunyatseni, 51, No. 2, 45–48 (2012).
O. V. Martynyuk, Cauchy Problem for Nonlocal Multipoint Problem for Evolutionary Equations of the First Order with Respect to the Time Variable [in Ukrainian) Doctoral-Degree Thesis (Physics and Mathematics), Chernivtsi (2017).
M. I. Matiichuk and V. M. Luchko, “Cauchy problem for parabolic systems with pulse action,” Ukr. Mat. Zh., 58, No. 11, 1525–1535 (2006); English translation: Ukr. Math. J., 58, No. 11, 1734–1747 (2006); 10.1007/s11253-006-0165-7.
A. M. Nakhushev, Equations of Mathematical Biology [in Russian], Vysshaya Shkola, Moscow (1995).
I. G. Petrovskii, “On the Cauchy problem for systems of partial differential equations in the domain of nonanalytic functions,” Byull. MGU, Mat. Mekh., 1, No. 7, 1–72 (1938).
I. D. Pukal’skii and B. O. Yashan, “The Cauchy problem for parabolic equations with degeneration,” Adv. Math. Phys., 2020, Article ID 1245143 (2020), 7 p.; https://doi.org/10.1155/2020/1245143.
A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Pulse Action [in Russian], Vyshcha Shkola, Kiev (1987).
J. R. Cannon and J. van der Hoek, “Diffusion subject to the specification of mass,” J. Math. Anal. Appl., 115, No. 2, 517–529 (1986).
J. Chabrowski, “On the non-local problems with a functional for parabolic equation,” Funkcial. Ekvac., 27, 101–123 (1984).
V. E. Slyusarchuk, “General theorems on the existence and uniqueness of solutions of impulsive differential equations,”Ukr. Mat. Zh., 52, No. 7, 954–964 (2000); English translation: Ukr. Math. J., 52, No. 7, 1094–1106 (2000).
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (1964).
V. V. Shelukhin, “Nonlocal (in time) problem for the equations of dynamics of barotropic ocean,” Sib. Mat. Zh., 36, No. 3, 701–724 (1995).
G. E. Shilov, “On the conditions of correctness of the Cauchy problem for a system of partial differential equations with constant coefficients,” Usp. Mat. Nauk, 10, No. 4, 89–101 (1955).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 11, pp. 1532–1540, November, 2021. Ukrainian DOI: https://doi.org/10.37863/umzh.v73i11.6521.
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Unguryan, G.M. Nonlocal Problem with Impulsive Action for Parabolic Equations of the Vector Order. Ukr Math J 73, 1772–1782 (2022). https://doi.org/10.1007/s11253-022-02029-x
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DOI: https://doi.org/10.1007/s11253-022-02029-x