The Blow-up solutions for fractional heat equations on torus and Euclidean space

We produce a finite time blow-up solution for nonlinear fractional heat equation (${\mathrm{\partial }}_{t}u+(-\mathrm{\Delta }{)}^{\beta \mathrm{/}2}u=u^k$) in modulation and Fourier amalgam spaces on the torus ${\mathbb{T}}^d$ and the Euclidean space $\mathbb R^d.$ This complements several known local and small data global well-posedness results in modulation spaces on ${\mathbb{R}}^d\mathrm{.}$ Our method of proof rely on the formal solution of the equation. This method should be further applied to other non-linear evolution equations.