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 (partial derivative(t)u + (-Delta)(beta /2) u= u(k)) in modulation and Fourier amalgam spaces on the torus T-d and the Euclidean space R-d. This complements several known local and small data global well-posedness results in modulation spaces on R-d. Our method of proof rely on the formal solution of the equation. This method should be further applied to other non-linear evolution equations.