Abstract
We study the asymptotic behavior, off the diagonal, as t→0, of the fundamental solution of a second order parabolic equation, and obtain by analytic methods both local and global results. We treat both the fundamental solution of the Cauchy problem and that of some mixed initial boundary value problems. We also investigate the oscillatory behavior, off the diagonal, as λ→∞, of the spectral function of a higher order elliptic operator. Some open problems are also exhibited.
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