Suppose \((M,g)\) is a compact Riemannian manifold. Then a solution of

\[ -\Delta_g u = \lambda u \]

behaves like \(t^{\sqrt{\lambda}}\) [1].

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  1. H. Donnelly and C. Fefferman, Growth and geometry of eigenfunctions of the Laplacian, in Analysis and partial differential equations, Dekker, New York, 1990, vol. 122, p. 635–655.