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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
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[1] S. Agmon, Lectures on Exponential Decay of Solutions of Second Order elliptic equations: bounds on eigenfunctions of \(N\)-body Schrödinger operators , Math. Notes, vol. 29, Princeton Univ. Press, Princeton, 1982. · Zbl 0503.35001
[2] V. S. Buslaev, Continuum integrals and the asymptotic behavior of the solutions of parabolic equations at \(t\rightarrow0\), applications to diffraction , Topics in Mathematical Physics, Vol. 2, Spectral Theory and Problems in Diffraction, Plenum, New York, 1968, pp. 67-86.
[3] T. Hargé Thèse, Orsay. · JFM 33.0098.02
[4] P. Hsu, Short time asymptotics of the heat kernel on a concave boundary , SIAM J. Math. Anal. 20 (1989), no. 5, 1109-1127. · Zbl 0685.58035
[5] N. Ikeda and S. Kusuoka, Short time asymptotics for fundamental solutions of diffusion equations , Stochastic Analysis (Paris, 1987), Lecture Notes in Math., vol. 1322, Springer-Verlag, Berlin, 1988, pp. 37-49. · Zbl 0647.60085
[6] G. Lebeau, Régularité Gevrey \(3\) pour la diffraction , Comm. Partial Differential Equations 9 (1984), no. 15, 1437-1494. · Zbl 0559.35019
[7] J. Milnor, Morse Theory , Ann. of Math. Stud., vol. 51, Princeton Univ. Press, Princeton, 1963. · Zbl 0108.10401
[8] J. R. Norris and D. W. Stroock, Estimates on the fundamental solution to heat flows with uniformly elliptic coefficients , Proc. London Math. Soc. (3) 62 (1991), no. 2, 373-402. · Zbl 0694.35075
[9] M. van den Berg, A Gaussian lower bound for the Dirichlet heat kernel , Bull. London Math. Soc. 24 (1992), no. 5, 475-477. · Zbl 0801.35035
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