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On the asymptotics of the fundamental solution of a parabolic equation in the critical case. (English. Russian original) Zbl 0701.35022

Math. USSR, Sb. 67, No. 2, 581-594 (1990); translation from Mat. Sb. 180, No. 8, 1119-1131 (1989).
The author considers the asymptotic behavior of the fundamental solution G(x,s,t) of the Cauchy problem for the parabolic equation \[ G_ t=G_{xx}-a(x)G,\quad x\in R,\quad t>0,\quad G(x,s,0)=\delta (x-s),\quad x,s\in {\mathbb{R}}. \] Here the coefficient a(x) has the following asymptotic form \(a(x)=a^{\pm}_ 2x^{-2}+\sum^{\infty}_{j=3}a^{\pm}_ j| x|^{-\mu^{\pm}_ j}\) for \(x\to \pm \infty\), where \(\mu^{\pm}_ j>2\), \(\mu^{\pm}_ j\to \infty\) when \(j\to +\infty\). The author proves that in general the fundamental solution G(x,s,t) decreases polynomially for \(t\to \infty\), the speed of decrease is determined by the values of coefficients \(a^{\pm}_ 2\).
Reviewer: J.Wang

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
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