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Ray asymptotics of the Green function for the heat equation with a small parameter. (English. Russian original) Zbl 0599.35078
Math. USSR, Sb. 52, 315-329 (1985); translation from Mat. Sb., Nov. Ser. 124(166), No. 3, 320-334 (1984).
We consider the heat equation $$\partial u/\partial t=F_ 0\Delta u$$ with a small parameter $$F_ 0\to 0$$. As is known, the Green function for this heat equation, which we denote by $$\Gamma$$ (x,t,$$\xi$$,$$\tau)$$, is a solution of the following boundary value problem: $\Gamma (x,t,\xi,\tau)=F_ 0\Delta \Gamma (x,t,\xi,\tau),$ $\Gamma (x,t,\xi,\tau)\to \delta (x-\xi)\quad as\quad t\to \tau,\quad \Gamma (x,t,\xi,\tau)\to 0\quad as\quad x\to S,$ where $$(x,t)\in \Omega_ T=\Omega \times [\tau,T]$$, $$0\leq \tau <t<T$$, $$\Omega \in R^ n$$, S is the boundary of $$\Omega$$, and $$\Delta$$ is the Laplace operator in $$R^ n$$. In our case $$\Omega$$ is a bounded, simply connected domain of $$R^ n$$ with a sufficiently smooth boundary S. The object of this paper is to obtain and justify an asymptotic expansion as $$F_ 0\to 0$$ of the function $$\Gamma$$ (x,t,$$\xi$$,$$\tau)$$.

##### MSC:
 35K05 Heat equation 35B40 Asymptotic behavior of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations
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