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Diffraction for the heat equation. (Diffraction pour l’équation de la chaleur.) (French) Zbl 0805.35039
Let $$p_ t (x,y)$$ be the kernel of the heat equation on $$\Omega$$ with the Dirichlet conditions on $$\partial \Omega$$: $$(\partial/ \partial t - \Delta_ x) p_ t = 0$$, $$p_ t |_{\partial \Omega_ x} = 0$$, $$p_{t=0} = \delta_{x=y}$$. In the paper is proved e.g. the following result: If $$w_ 0$$ is the first zero of the Airy function then there exist $$t_ 0>0$$ and $$C \in \mathbb{R}$$ such that $t \in] 0,t_ 0 [\Rightarrow p_ t (x_ 0,y_ 0) \leq \exp \left[ - {d^ 2_ 0 \over 4t} - w_ 0 {d_ 0^{1/3} \over (4t)^{1/3}} \int_{\gamma_ 0 \cap \Omega} {ds \over \rho (s)^{2/3} } + C \log t \right]$ where $$d_ 0 = d(x,y)$$ is a distance function and $$\gamma_ 0$$ is the geodesic minimum with respect to $$x_ 0$$, $$y_ 0$$.
Reviewer: J.Diblík (Brno)

MSC:
 35K05 Heat equation
Full Text:
References:
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