Complex time blow-up of the nonlinear heat equation

Abstract

This paper investigates the connection between blow-up solutions of scalar reaction-diffusion equations, in particular of $u_t = u_{xx} + u^2, $ and its counterpart - eternally existing solutions like heteroclinic orbits - by complex time. We prove that heteroclinic orbits in one-dimensional unstable manifolds are accompanied by blow-up solutions. Furthermore we show, that we can continue blow-up solutions into a slit complex time and eventually back to the real axis. The solution picks up an imaginary factor after continuation which is related to the eigenvalue relations of the linearizations at the source and the sink of the heteroclinic orbit.