Complex time blow-up of the nonlinear heat equation

TL;DR

This paper explores the complex time behavior of blow-up solutions in nonlinear heat equations, revealing how they relate to heteroclinic orbits and can be analytically continued into complex time with specific imaginary factors.

Contribution

It establishes a connection between blow-up solutions and heteroclinic orbits in complex time, and demonstrates the continuation of blow-up solutions into complex time with eigenvalue-related factors.

Findings

Heteroclinic orbits are associated with blow-up solutions.

Blow-up solutions can be continued into complex time.

Continuation introduces an imaginary factor linked to eigenvalues.

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.