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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2019. Vol. 30

On the Behaviour at Infinity of Solutions to Nonlocal Parabolic Type Problems

Author(s)
Е. А. Zhizhina, А. L. Piatnitski
Abstract
The paper deals with possible behaviour at infinity of solutions to the Cauchy problem for a parabolic type equation whose elliptic part is the generator of a Markov jump process , i.e. a nonlocal diffusion operator. The analysis of the behaviour of the solutions at infinity is based on the results on the asymptotics of the fundamental solutions of nonlocal parabolic problems. It is shown that such fundamental solutions might have different asymptotics and decay rates in the regions of moderate, large and super-large deviations. The asymptotic formulae for the said fundamental solutions are then used for describing classes of unbounded functions in which the studied Cauchy problem is well-posed. We also consider the question of uniqueness of a solution in these functional classes.
About the Authors

Elena Zhizhina, Dr. Sci. (Phys.–Math.), Leading Scientific Researcher, Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny per., 19, build.1, Moscow, 127051, Russian Federation, tel.: +7(495)6504225, e-mail: ejj@iitp.ru

Andrey Piatnitski, Dr. Sci. (Phys.–Math.), Prof., Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny per., 19, build.1, Moscow, 127051, Russian Federation, The Arctic University of Norway, Campus Narvik, P. O. Box 385, 8505 Narvik, Norway, tel.: +7(495)6504225, e-mail: apiatnitski@gmail.com

For citation

Zhizhina E.A., Piatnitski A.L. On the Behaviour at Infinity of Solutions to Nonlocal Parabolic Type Problems. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 30, pp. 99-113. https://doi.org/10.26516/1997-7670.2019.30.99

Keywords
nonlocal operators, parabolic equations, fundamental solution, Markov jump process with independent increments
UDC
517.956.4, 517.956.8
MSC
35K08, 45E10
DOI
https://doi.org/10.26516/1997-7670.2019.30.99
References
  1. Aronson D.G. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc., 1967, vol. 73, pp. 890-896.
  2. Bass R.F., Levin D.A. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc., 2002, vol. 354, no. 7, pp. 2933-2953. https://doi.org/10.1090/S0002-9947-02-02998-7.
  3. Bhattacharia R.N., Rao R.R. Normal Approximations and Asymptotic Expansions. John Wiley and Sons, 1976.
  4. Brandle C., Chasseigne E., Ferreira R. Unbounded solutions of the nonlocal heat equation. Comm Pure and Appl. Analysis, 2011, vol. 10, no. 6, pp. 1663-1686. https://doi.org/10.3934/cpaa.2011.10.1663.
  5. Bendikov A. Asymptotic formulas for symmetric stable semigroups, Expo. Math., 1994, vol. 12, pp. 381-384.
  6. Grigor’yan A., Kondratiev Yu., Piatnitski A. and Zhizhina E. Pointwise estimates for heat kernels of convolution type operators, Proc. London Math. Soc., 2018, vol. 117, no. 4, pp. 849-880. https://dx.doi.org/10.1112/plms.12144
  7. Kondratiev Yu., Kutoviy O., Pirogov S. Correlation functions and invariant measures in continuous contact model, Ininite Dimensional Analysis, Quantum Probability and Related Topics, 2008, vol. 11, no. 2, pp. 231-258. https://doi.org/10.1142/S0219025708003038.
  8. Kondratiev Yu., Molchanov S., Pirogov S., Zhizhina E. On ground state of some non local Schrodinger operator, Applicable Analysis, 2017, vol. 96, no. 8, pp. 1390-1400. http://dx.doi.org/10.1080/00036811.2016.1192138.
  9. Pirogov S., Zhizhina E. A quasispecies continuous contact model in a subcritical regime, Moscow Mathematical Journal, 2019, vol. 19, no. 1, pp. 121-132. https://doi.org/10.17323/1609-4514-2019-19-1-121-132.
  10. Shubin M. Invitation to Partial Differential Equations, 2010, AMS Publ. EPUB online.

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