Note for global existence of semilinear heat equation in weighted $L^\infty$

TL;DR

This paper investigates the conditions for global existence of solutions to semilinear heat equations with small initial data in weighted $L^ abla$ spaces, using a contraction mapping approach based on weighted uniform estimates.

Contribution

It introduces a simple contraction argument for establishing global existence in weighted $L^ abla$ spaces, extending previous results to a broader class of initial data.

Findings

Global existence results for semilinear heat equations with small data.

A new contraction method based on weighted uniform control.

Applicability to initial data with negative power.

Abstract

The local and global existence of the Cauchy problem for semilinear heat equations with small data is studied in the weighted $L^\infty (\mathbb R^n)$ framework by a simple contraction argument. The contraction argument is based on a weighted uniform control of solutions related with the free solutions and the first iterations for the initial data of negative power.