Özet:

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: [email protected] 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: [email protected]

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