We introduce the fractional Laplacian operator (–Δ)^s, motivating its study by illustrating its effectiveness in modelling super-diffusion. We then consider the fractional Gelfand equation, a fractional analogue of the classical Gelfand equation describing an isothermal gas sphere in gravitational equilibrium. In one spatial dimension, we establish existence and uniqueness of solutions, and discuss blow-up phenomena as well as the long-time behaviour of the associated parabolic flow.

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