Abstract: We study the existence versus blow-up for solutions to semilinear heat equations both on certain classes of Riemannian manifolds and on infinite graphs. On the manifold side, we show that when the bottom $\Lambda$ of the $L^2$ spectrum of the Laplace-Beltrami operator is strictly positive, as is the case for hyperbolic space $\mathbb{H}^N$, a Fujita-type phenomenon re-emerges for nonlinearities different from the classical power-type ones, with $\Lambda$ playing the role of the critical exponent. For standard power nonlinearities $f(u) = u^p$, we prove that sufficiently small initial data yield global solutions for all $p\ge1$. In parallel, we consider the semilinear heat equation on infinite graphs. Again a crucial role is played by the first eigenvalue $\lambda_1$ of the discrete Laplacian. We establish that if the nonlinearity in the reaction term $f(u)$, among other assumptions, satisfies $f'(0) > \lambda_1(G)$, then all positive nontrivial solutions blow up in finite time, while if $f'(0) < \lambda_1(G)$ then global in time solutions exist. These topics are the content of two papers in collaboration with G. Grillo and F. Punzo.
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