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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,\(\xi\),\(\tau)\).

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