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

TL;DR

This paper constructs finite-time blow-up solutions for nonlinear fractional heat equations on both the torus and Euclidean space, expanding understanding of solution behavior in modulation and Fourier spaces.

Contribution

It introduces a novel method based on formal solutions to demonstrate blow-up phenomena for fractional heat equations in modulation and Fourier spaces.

Findings

Finite-time blow-up solutions constructed

Method applicable to other nonlinear evolution equations

Complements existing well-posedness results

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.