Geodesic
On the behaviour at infinity of solutions to nonlocal parabolic type problems
The Bulletin of Irkutsk State University. Series Mathematics, Tome 30 (2019), pp. 99-113
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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.
Keywords: nonlocal operators, fundamental solution, Markov jump process with independent increments.
Mots-clés : parabolic equations 
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E. A. Zhizhina; A. L. Piatnitski. On the behaviour at infinity of solutions to nonlocal parabolic type problems. The Bulletin of Irkutsk State University. Series Mathematics, Tome 30 (2019), pp. 99-113. http://geodesic.mathdoc.fr/item/IIGUM_2019_30_a7/

[1] D. G. Aronson, “Bounds for the fundamental solution of a parabolic equation”, Bull. Amer. Math. Soc., 73 (1967), 890–896 | DOI | MR | Zbl

[2] R. F. Bass, D. A. Levin, “Transition probabilities for symmetric jump processes”, Trans. Amer. Math. Soc., 354:7 (2002), 2933–2953 | DOI | MR | Zbl

[3] R. N. Bhattacharia, R. R. Rao, Normal Approximations and Asymptotic Expansions, John Wiley and Sons, 1976 | MR

[4] C. Brandle, E. Chasseigne, R. Ferreira, “Unbounded solutions of the nonlocal heat equation”, Comm Pure and Appl. Analysis, 10:6 (2011), 1663–1686 | DOI | MR | Zbl

[5] A. Bendikov, “Asymptotic formulas for symmetric stable semigroups”, Expo. Math., 12 (1994), 381–384 | MR | Zbl

[6] A. Grigor'yan, Yu. Kondratiev, A. Piatnitski, E. Zhizhina, “Pointwise estimates for heat kernels of convolution type operators”, Proc. London Math. Soc., 117:4 (2018), 849–880 | DOI | MR | Zbl

[7] Yu. Kondratiev, O. Kutoviy, S. Pirogov, “Correlation functions and invariant measures in continuous contact model”, Ininite Dimensional Analysis, Quantum Probability and Related Topics, 11:2 (2008), 231–258 | DOI | MR | Zbl

[8] Yu. Kondratiev, S. Molchanov, S. Pirogov, E. Zhizhina, “On ground state of some non local Schrodinger operator”, Applicable Analysis, 96:8 (2017), 1390–1400 | DOI | MR | Zbl

[9] S. Pirogov, E. Zhizhina, “A quasispecies continuous contact model in a subcritical regime”, Moscow Mathematical Journal, 19:1 (2019), 121–132 | DOI | MR | Zbl

[10] M. Shubin, Invitation to Partial Differential Equations, AMS Publ. EPUB online, 2010