Asymptotics of~a~fundamental solution of~a~parabolic equation as~ $t\to\infty$

E. F. Lelikova

Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 315-337

Résumé

The author analyzes the behavior as $t\to\infty$ of the fundamental solution $G(x, s, t)$ of the Cauchy problem for the equation $v_t-v_{xx}-a(x)v_x-b(x)v=0$ with infinitely differentiable coefficients $a(x)$ and $b(x)$ decreasing as $|x|\to\infty$. For the case when the functions $a(x)$ and $b(x)$ can be expanded as $x\to\pm\infty$ on asymptotic series of the form \begin{gather*} a(x)=a_1|x|^{-\alpha_1}+\dots +a_i|x|^{-\alpha_i}+\dots , \\ b(x)=b_1|x|^{-\beta_1}+\dots +b_i|x|^{-\beta_i}+\dots , \end{gather*} where $\alpha_m$, $\beta_m\uparrow\infty$ as $m\to\infty$, $\alpha_1>1$, $\beta_1>2$, she constructs and justifies asymptotic expansion of the fundamental solution $G(x, s, t)$ to within any power of $G(x, s, t)$ uniformly with respect to all $x$ and $s$ in $\mathbf R^1$. Bibliography: 12 titles.