Finite time blow-up for a nonlinear parabolic equation with smooth coefficients

TL;DR

This paper proves that solutions to a specific nonlinear parabolic PDE with smooth coefficients can blow up in finite time, addressing a previously open question in the mathematical analysis of such equations.

Contribution

The authors demonstrate finite-time blow-up for smooth solutions of a nonlinear parabolic PDE, using a virial-type estimate, thus resolving an open problem from prior research.

Findings

Finite-time blow-up of solutions established

Virial-type estimate effectively used

Addresses an open question from previous work

Abstract

In this article, we consider an n-dimensional parabolic partial differential equation with a smooth coefficient term in the nonlinear gradient term. This equation was first introduced and analyzed in [E. Issoglio, On a non-linear transport-diffusion equation with distributional coefficients, Journal of Differential Equations, Volume 267, Issue 10 (2019)], where one of the main open questions is the possible finite-time blow-up of solutions. Here, leveraging a virial-type estimate, we provide a positive answer to this question within the framework of smooth solutions.