Singularities of complex-valued solutions to linear parabolic equations

TL;DR

This paper constructs examples of blowup in smooth initial data for complex-valued solutions to linear parabolic equations, demonstrating the maximum possible irregularity consistent with energy estimates.

Contribution

It provides explicit examples showing blowup phenomena in complex-valued solutions, highlighting the limits of regularity for linear parabolic equations.

Findings

Examples of blowup from smooth data in complex solutions

Solutions exhibit irregularity matching energy estimate limits

Demonstrates maximal irregularity possible in such equations

Abstract

We construct examples of blowup from smooth data for complex-valued solutions to linear uniformly parabolic equations in dimension $n \geq 2$, which are exactly as irregular as parabolic energy estimates allow.