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.

arXiv comment21 pages. The revised version is accepted for publication in Nonlinear Differential Equations and Applications (NoDEA)