May  2020, 13(5): 1543-1551. doi: 10.3934/dcdss.2020087

Mean periodic solutions of a inhomogeneous heat equation with random coefficients

1. 

Voronezh State University, Universitetskaya pl., 1, Voronezh, 394018, Russia

2. 

Institute of Law and Economics, Leninskii pr., 119-A, Voronezh, 394042, Russia

3. 

Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Vavilova ul., 44/2, Moscow, 119333, Russia

* Corresponding author

Received  February 2018 Revised  September 2018 Published  June 2019

Fund Project: The first author is supported by the Russian Science Foundation project No. 17-11-01220

We present conditions ensuring the periodicity of the mathematical expectation of a solution of a scalar linear inhomogeneous heat equation with random coefficients where the coefficient in front of the unknown functions is Gaussian or it is uniformly distributed. The obtained results may be treated as finding a control ensuring the periodicity of the mathematical expectation of a solution of the heat equation.

Citation: Galina Kurina, Vladimir Zadorozhniy. Mean periodic solutions of a inhomogeneous heat equation with random coefficients. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1543-1551. doi: 10.3934/dcdss.2020087
References:
[1]

H. Amann, Periodic solutions for semi-linear parabolic equations, in Nonlinear Analysis: A Collection of Papers in Honor of Erich Rothe, Academic Press, (1978), 1–29.

[2]

N. Hirano, Existence of multiple periodic solutions for a semilinear evolution equations, Proc. Amer. Math. Soc., 106 (1989), 107-114.  doi: 10.1090/S0002-9939-1989-0953007-5.

[3]

R. Z. Khasminskii, Ustoychivost' Sistem Differencial'nyh Uravnenii pri Sluchainykh Vozmushcheniyakh ikh Parametrov, (Russian) [Stability of Systems of Differential Equations under Random Perturbations of Their Parameters], Nauka, Moscow, 1969.

[4]

Yu. S. Kolesov, O nekotorykh priznakakh sushchestvovaniya ustoichivykh periodicheskikh reshenii u kvasilineinykh parabolicheskikh uravnenii, (Russian) [Some of the signs of existence of stable periodic solutions for quasilinear parabolic equations], Dokl. AN SSSR, 157 (1964), 1288-1290. 

[5]

I. I. Shmulev, Periodicheskie resheniya pervoi kraevoi zadachi dlya parabolicheskikh uravnenii, (Russian) [Periodic solutions of the first boundary problem for pabolic equations], Matem. sb., 66 (1965), 398-410. 

[6]

A. N. Tikhonov and A. A. Samarskii, Uravneniya Matematičesko$\check{i}$ Fiziki, (Russian) [Equations of Mathematical Physics], Nauka, Moscow, 1953.

[7]

V. A. Yakubovich and V. M. Starzhinskii, Lineinye Differencial'nye Uravneniya s Periodicheskimi Koefficientami i ikh Prilozheniya, (Russian) [Linear Differential Equations with Periodic Coefficients and Their Applications], Nauka, Moscow, 1972.

[8]

V. G. Zadorozhniy, Metody Variatsionnogo Analiza, (Russian) [Methods of Variational Analysis], NIC "Regulyarnaya i Khaoticheskaya Dinamika", Institut Kompyuternyh Issledovanii, Moscow-Izhevsk, 2006.

[9]

V. G. Zadorozhniy and G. A. Kurina, Periodicheskie v srednem resheniya lineinogo differencial'nogo uravneniya pervogo poryadka, (Russian) [Mean-periodic solutions of a first-order linear differential equation], Dokl. Akad. Nauk, 450 (2013), 505-510 (Engl. transl.: Dokl. Math., 87 (2013), 325-330.) doi: 10.1134/s1064562413030277.

[10]

V. G. Zadorozhniy and G. A. Kurina, Periodicheskie v srednem resheniya lineinogo neodnorodnogo differencial'nogo uravneniya pervogo poryadka so sluchainymi koefficientami, (Russian) [Mean periodic solutions of a linear inhomogeneous first-order differential equation with random coefficients], Differentcial'nye Uravneniya, 50 (2014), 726-744 (English transl.: Differential Equations, 50 (2014), 722-741.

show all references

References:
[1]

H. Amann, Periodic solutions for semi-linear parabolic equations, in Nonlinear Analysis: A Collection of Papers in Honor of Erich Rothe, Academic Press, (1978), 1–29.

[2]

N. Hirano, Existence of multiple periodic solutions for a semilinear evolution equations, Proc. Amer. Math. Soc., 106 (1989), 107-114.  doi: 10.1090/S0002-9939-1989-0953007-5.

[3]

R. Z. Khasminskii, Ustoychivost' Sistem Differencial'nyh Uravnenii pri Sluchainykh Vozmushcheniyakh ikh Parametrov, (Russian) [Stability of Systems of Differential Equations under Random Perturbations of Their Parameters], Nauka, Moscow, 1969.

[4]

Yu. S. Kolesov, O nekotorykh priznakakh sushchestvovaniya ustoichivykh periodicheskikh reshenii u kvasilineinykh parabolicheskikh uravnenii, (Russian) [Some of the signs of existence of stable periodic solutions for quasilinear parabolic equations], Dokl. AN SSSR, 157 (1964), 1288-1290. 

[5]

I. I. Shmulev, Periodicheskie resheniya pervoi kraevoi zadachi dlya parabolicheskikh uravnenii, (Russian) [Periodic solutions of the first boundary problem for pabolic equations], Matem. sb., 66 (1965), 398-410. 

[6]

A. N. Tikhonov and A. A. Samarskii, Uravneniya Matematičesko$\check{i}$ Fiziki, (Russian) [Equations of Mathematical Physics], Nauka, Moscow, 1953.

[7]

V. A. Yakubovich and V. M. Starzhinskii, Lineinye Differencial'nye Uravneniya s Periodicheskimi Koefficientami i ikh Prilozheniya, (Russian) [Linear Differential Equations with Periodic Coefficients and Their Applications], Nauka, Moscow, 1972.

[8]

V. G. Zadorozhniy, Metody Variatsionnogo Analiza, (Russian) [Methods of Variational Analysis], NIC "Regulyarnaya i Khaoticheskaya Dinamika", Institut Kompyuternyh Issledovanii, Moscow-Izhevsk, 2006.

[9]

V. G. Zadorozhniy and G. A. Kurina, Periodicheskie v srednem resheniya lineinogo differencial'nogo uravneniya pervogo poryadka, (Russian) [Mean-periodic solutions of a first-order linear differential equation], Dokl. Akad. Nauk, 450 (2013), 505-510 (Engl. transl.: Dokl. Math., 87 (2013), 325-330.) doi: 10.1134/s1064562413030277.

[10]

V. G. Zadorozhniy and G. A. Kurina, Periodicheskie v srednem resheniya lineinogo neodnorodnogo differencial'nogo uravneniya pervogo poryadka so sluchainymi koefficientami, (Russian) [Mean periodic solutions of a linear inhomogeneous first-order differential equation with random coefficients], Differentcial'nye Uravneniya, 50 (2014), 726-744 (English transl.: Differential Equations, 50 (2014), 722-741.

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