On basin of zero-solutions to a semilinear parabolic equation with Ornstein-Uhlenbeck operator.

Fujita, Yasuhiro

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We characterize all subsets $M$ of $ℝ^{n} \times ℝ^{+}$ such that $sup_{X \in ℝ^{n} \times ℝ^{+}} u (X) = sup_{X \in M} u (X)$ for every bounded parabolic function $u$ on $ℝ^{n} \times ℝ^{+}$ . The closely related problem of representing functions as sums of Weierstrass kernels corresponding to points of $M$ is also considered. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References. As a by-product the question of representability of probability continuous distributions as sums of multiples of normal distributions is investigated. ...

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