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

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.

We withdraw this manuscript because a similar argument has already been discussed by T. Cazenave, F. Dickstein, and F.B. Weissler, Universal solutions of a nonlinear heat equation on ${\mathbb{R}}^N$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 1, 77 -- 117