Geodesic
On the asymptotics of the fundamental solution of a parabolic equation in the critical case
Sbornik. Mathematics, Tome 67 (1990) no. 2, pp. 581-594 
E. F. Lelikova. On the asymptotics of the fundamental solution of a parabolic equation in the critical case. Sbornik. Mathematics, Tome 67 (1990) no. 2, pp. 581-594. http://geodesic.mathdoc.fr/item/SM_1990_67_2_a13/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The behavior as $t\to\infty$ of the fundamental solution $G(x,s,t)$ of the Cauchy problem for the equation $u_t=u_{xx}-a(x)u$ $(x\in\mathbf R^1$, $t>0)$ is studied in the case when the decay rate of the coefficient $a(x)$ as $x\to\pm\infty$ is critical: $$ a(x)=a_2^\pm x^{-2}+\sum_{i=3}^\infty a_i^\pm x^{-i}\qquad(x\to\pm\infty). $$ The asymptotic expansion of $G(x,s,t)$ as $t\to\infty$ is constructed and established for all $x,s\in\mathbf R^1$. The fundamental solution decays like a power, and the decay rate is determined by the quantities $a_2^\pm$. Bibliography: 8 titles.

[1] Glazman I. M., Pryamye metody kachestvennogo spektralnogo analiza singulyarnykh differentsialnykh operatorov, F. M., M., 1963 | MR

[2] Yafaev D. R., “Sverkhstepennoe ubyvanie po vremeni reshenii uravneniya Shredingera”, DAN SSSR, 258:4 (1984), 850–853 | MR

[3] Lelikova E. F., “Asimptotika fundamentalnogo resheniya parabolicheskogo uravneniya pri $t\to\infty$”, Matem. sb., 132(174) (1987), 322–344 | MR

[4] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[5] Van-Daik M., Metody vozmuschenii v mekhanike zhidkosti, Mir, M., 1967

[6] Babich V. M., Kirpichnikova N. N., Metod pogranichnogo sloya v zadachakh difraktsii, Izd-vo Leningr. un-ta, L., 1974 | MR

[7] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[8] Lebedev N. N., Spetsialnye funktsii i ikh prilozheniya, F. M., M., L., 1960