Unbounded solutions to defocusing parabolic systems

Abstract

We give existence theorems for the Cauchy problem of a large class of semi-linear parabolic equations in $L^{\infty}$, $L^{\infty} \cap L^p$ or $L^{\infty} \cap \dot W^{1,p}$, using a contracting map argument. We then construct integral solutions to parabolic equations with data growing at infinity and defocusing nonlinearity, and give an example of instantaneous blow up when the nonlinearity is focusing and the initial data has tame growth.