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January  2016, 15(1): 185-196. doi: 10.3934/cpaa.2016.15.185

One Class of Sobolev Type Equations of Higher Order with Additive "White Noise"

Angelo Favini 1, , Georgy A. Sviridyuk 2, and  Alyona A. Zamyshlyaeva 2,

1. 

University of Bologna, Department of Mathematics, Piazza di Porta San Donato, 5, Bologna, Italy
2. 

South Ural State University, Dep. of Mathematics, Mechanics and Computer science, Lenin avenue, 76, Chelyabinsk, Russian Federation, Russian Federation

Received  September 2015 Revised  October 2015 Published  December 2015

Sobolev type equation theory has been an object of interest in recent years, with much attention being devoted to deterministic equations and systems. Still, there are also mathematical models containing random perturbation, such as white noise; these models are often used in natural experiments and have recently driven a large amount of research on stochastic differential equations. A new concept of ``white noise", originally constructed for finite dimensional spaces, is extended here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher-order Sobolev type equation theory and provide some practical applications. The main idea is to construct ``noise" spaces using the Nelson -- Gliklikh derivative. Abstract results are applied to the Boussinesq -- Lòve model with additive ``white noise" within Sobolev type equation theory. Because of their usefulness, we mainly focus on Sobolev type equations with relatively p-bounded operators. We also use well-known methods in the investigation of Sobolev type equations, such as the phase space method, which reduces a singular equation to a regular one, as defined on some subspace of the initial space.
Keywords: propagator, Sobolev type equation, ``white noise", K-Wiener process..
Mathematics Subject Classification: Primary: 35G05, 35G16; Secondary: 47D09, 60H3.
Citation: Angelo Favini, Georgy A. Sviridyuk, Alyona A. Zamyshlyaeva. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure and Applied Analysis, 2016, 15 (1) : 185-196. doi: 10.3934/cpaa.2016.15.185
References:
[1]

A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, Series in Nonlinear Analysis and Applications, 15, De Gruyter, 2011. doi: 10.1515/9783110255294.

[2]

G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative, N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 2003. doi: 10.1201/9780203911433.

[3]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 1999.

[4]

Yu. E. Gliklikh, Global and Stochastic Analysis with Applications to Mathematical Physics, London, Dordrecht, Heidelberg, N.Y., Springer, 2011. doi: 10.1007/978-0-85729-163-9.

[5]

Yu. E. Gliklikh and E. Yu. Mashkov, Stochastic Leontieff type equations and mean derivatives of stochastic processes, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 6 (2013), 25-39.

[6]

M. Kovács and S. Larsson, Introduction to stochastic partial differential equations, in Proceedings of "New Directions in the Mathematical and Computer Sciences", National Universities Commission, Abuja, Nigeria, October 8-12, 2007 Publications of the ICMCS, 4 (2008), 159-232.

[7]

A. I. Kozhanov, Boundary Problems for Odd Ordered Equations of Mathematical Physics, Novosibirsk, NGU, 1990.

[8]

L. D. Landau and E. M. Lifshits, Theoretical Phisics, VII. Elasticity Theory, Mscow, Nauka, 1987.

[9]

I. V. Melnikova, A. I. Filinkov and M. A. Alshansky, Abstract stochastic equations II. Solutions in spaces of abstract stochastic distributions, Journal of Mathematical Sciences, 116 (2003), 3620-3656. doi: 10.1023/A:1024159908410.

[10]

A. L. Shestakov, A. V. Keller and E. I. Nazarova, Numerical solution of the optimal measurement problem, Automation and Remote Control, 73 (2012), 97-104. doi: 10.1134/S0005117912010079.

[11]

A. L. Shestakov and G. A. Sviridyuk, On a new conception of white noise, Obozrenie Prikladnoy i Promyshlennoy Matematiki, 19 (2012), 287-288.

[12]

A. L. Shestakov and G. A. Sviridyuk, On the measurement of the "white noise", Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 286 (2012), 99-108.

[13]

A. L. Shestakov and G. A. Sviridyuk, Optimal measurement of dynamically distorted signals, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 234 (2011), 70-75.

[14]

A. L. Shestakov, G. A. Sviridyuk and Yu. V. Hudyakov, Dynamic measurement in spaces of "noise", Bulletin of the South Ural State University. Series: Computer Technologies, Automatic Control and Radioelectronics, 13 (2013), 4-11.

[15]

R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Pitman, London, San Francisco, Melbourne, 1977.

[16]

N. Sidorov, B. Loginov, A. Sinithyn and M. Falaleev, Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications, Dordrecht, Boston, London, Kluwer Academic Publishers, 2002. doi: 10.1007/978-94-017-2122-6.

[17]

G. A. Sviridyuk and T. V. Apetova, The phase spaces of linear dynamic Sobolev type equations, Doklady Akademii Nauk, 330 (1993), 696-699.

[18]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Utrecht, Boston, Köln, Tokyo, VSP, 2003. doi: 10.1515/9783110915501.

[19]

G. A. Sviridyuk and N. A. Manakova, The Dynamical Models of Sobolev Type with Showalter - Sidorov Condition and Additive 'Noise', Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 7 (2014), 90-103.

[20]

G. A. Sviridyuk and O. V. Vakarina, Linear Sobolev type equations of higher order, Doklady Akademii Nauk, 393 (1998), 308-310.

[21]

G. A. Sviridyuk and A. A. Zamyshlyaeva, The phase spaces of a class of linear higher-order Sobolev type equations, Differential Equations, 42 (2006), 269-278. doi: 10.1134/S0012266106020145.

[22]

G. Uizem, Linear and Nonlinear Waves, Mscow, Mir, 1977.

[23]

S. Wang and G. Chen, Small amplitude solutions of the generalized IMBq equation, Mathematical Analysis and Applications, 274 (2002), 846-866. doi: 10.1016/S0022-247X(02)00401-8.

[24]

S. A. Zagrebina and E. A. Soldatova, The linear Sobolev-type equations with relatively p-bounded operators and additive white noise, The Bulletin of Irkutsk State University. Series "Mathematics", 6 (2013), 20-34.

[25]

A. A. Zamyshlyaeva, The higher-order Sobolev-type models, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 7 (2014), 5-28.

[26]

A. A. Zamyshlyaeva, Stochastic incomplete linear Sobolev type high-ordered equations with additive white noise, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 299 (2012), 73-82.

show all references

References:
[1]

A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, Series in Nonlinear Analysis and Applications, 15, De Gruyter, 2011. doi: 10.1515/9783110255294.

[2]

G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative, N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 2003. doi: 10.1201/9780203911433.

[3]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 1999.

[4]

Yu. E. Gliklikh, Global and Stochastic Analysis with Applications to Mathematical Physics, London, Dordrecht, Heidelberg, N.Y., Springer, 2011. doi: 10.1007/978-0-85729-163-9.

[5]

Yu. E. Gliklikh and E. Yu. Mashkov, Stochastic Leontieff type equations and mean derivatives of stochastic processes, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 6 (2013), 25-39.

[6]

M. Kovács and S. Larsson, Introduction to stochastic partial differential equations, in Proceedings of "New Directions in the Mathematical and Computer Sciences", National Universities Commission, Abuja, Nigeria, October 8-12, 2007 Publications of the ICMCS, 4 (2008), 159-232.

[7]

A. I. Kozhanov, Boundary Problems for Odd Ordered Equations of Mathematical Physics, Novosibirsk, NGU, 1990.

[8]

L. D. Landau and E. M. Lifshits, Theoretical Phisics, VII. Elasticity Theory, Mscow, Nauka, 1987.

[9]

I. V. Melnikova, A. I. Filinkov and M. A. Alshansky, Abstract stochastic equations II. Solutions in spaces of abstract stochastic distributions, Journal of Mathematical Sciences, 116 (2003), 3620-3656. doi: 10.1023/A:1024159908410.

[10]

A. L. Shestakov, A. V. Keller and E. I. Nazarova, Numerical solution of the optimal measurement problem, Automation and Remote Control, 73 (2012), 97-104. doi: 10.1134/S0005117912010079.

[11]

A. L. Shestakov and G. A. Sviridyuk, On a new conception of white noise, Obozrenie Prikladnoy i Promyshlennoy Matematiki, 19 (2012), 287-288.

[12]

A. L. Shestakov and G. A. Sviridyuk, On the measurement of the "white noise", Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 286 (2012), 99-108.

[13]

A. L. Shestakov and G. A. Sviridyuk, Optimal measurement of dynamically distorted signals, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 234 (2011), 70-75.

[14]

A. L. Shestakov, G. A. Sviridyuk and Yu. V. Hudyakov, Dynamic measurement in spaces of "noise", Bulletin of the South Ural State University. Series: Computer Technologies, Automatic Control and Radioelectronics, 13 (2013), 4-11.

[15]

R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Pitman, London, San Francisco, Melbourne, 1977.

[16]

N. Sidorov, B. Loginov, A. Sinithyn and M. Falaleev, Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications, Dordrecht, Boston, London, Kluwer Academic Publishers, 2002. doi: 10.1007/978-94-017-2122-6.

[17]

G. A. Sviridyuk and T. V. Apetova, The phase spaces of linear dynamic Sobolev type equations, Doklady Akademii Nauk, 330 (1993), 696-699.

[18]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Utrecht, Boston, Köln, Tokyo, VSP, 2003. doi: 10.1515/9783110915501.

[19]

G. A. Sviridyuk and N. A. Manakova, The Dynamical Models of Sobolev Type with Showalter - Sidorov Condition and Additive 'Noise', Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 7 (2014), 90-103.

[20]

G. A. Sviridyuk and O. V. Vakarina, Linear Sobolev type equations of higher order, Doklady Akademii Nauk, 393 (1998), 308-310.

[21]

G. A. Sviridyuk and A. A. Zamyshlyaeva, The phase spaces of a class of linear higher-order Sobolev type equations, Differential Equations, 42 (2006), 269-278. doi: 10.1134/S0012266106020145.

[22]

G. Uizem, Linear and Nonlinear Waves, Mscow, Mir, 1977.

[23]

S. Wang and G. Chen, Small amplitude solutions of the generalized IMBq equation, Mathematical Analysis and Applications, 274 (2002), 846-866. doi: 10.1016/S0022-247X(02)00401-8.

[24]

S. A. Zagrebina and E. A. Soldatova, The linear Sobolev-type equations with relatively p-bounded operators and additive white noise, The Bulletin of Irkutsk State University. Series "Mathematics", 6 (2013), 20-34.

[25]

A. A. Zamyshlyaeva, The higher-order Sobolev-type models, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 7 (2014), 5-28.

[26]

A. A. Zamyshlyaeva, Stochastic incomplete linear Sobolev type high-ordered equations with additive white noise, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 299 (2012), 73-82.

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