Finite time blowup for parabolic systems in two dimensions

TL;DR

This paper constructs examples of finite time singularities in smooth solutions for linear and quasilinear parabolic systems in two dimensions, demonstrating blowup phenomena.

Contribution

It provides the first explicit constructions of finite time blowup for parabolic systems in two dimensions, including both linear and quasilinear cases.

Findings

Finite time singularities are possible in 2D parabolic systems.

Explicit examples of blowup from smooth initial data are constructed.

Results extend understanding of singularity formation in parabolic PDEs.

Abstract

We construct examples of finite time singularity from smooth data for linear uniformly parabolic systems in the plane. We obtain similar examples for quasilinear systems with coefficients that depend only on the solution.