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Since we make no further assumptions on the behaviour of the diffusion, our main result can be seen as being about the competition between the diffusion represented by <jats:italic>L<\/jats:italic> and the reaction represented by <jats:italic>f<\/jats:italic> in a general setting. We apply our result to Laplacians on manifolds, graphs, and, more generally, metric measure spaces with a heat kernel. 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