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

Abstract

We produce a finite time blow-up solution for nonlinear fractional heat equation (\(\partial _t u + (-\Delta )^{\beta /2}u=u^k\)) 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 \({\mathbb {R}}^d.\) Our method of proof rely on the formal solution of the equation. This method should be further applied to other non-linear evolution equations.