Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

TL;DR

This paper constructs and analyzes solutions that blow up in finite time for a nonlinear heat equation with a critical gradient term, revealing how the quadratic gradient influences the blowup profile and stability.

Contribution

It introduces a method to construct finite-time blowup solutions for a nonlinear heat equation with a critical gradient term, detailing their asymptotic behavior and stability.

Findings

Constructed solutions blow up at the origin in finite time.

The quadratic gradient term alters the blowup profile compared to the case without it.

The solutions are stable under initial data perturbations.

Abstract

We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\partial_t U = \Delta U + \alpha|\nabla U|^2 + e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad \alpha > -1.$$ We construct for this equation a solution which blows up in finite time $T > 0$ and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time $T$ simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some senses, resulting in the change of the final blowup profile in comparison with the case $\alpha = 0$. The proof of the construction inspired by the method of Merle and Zaag in 1997, relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the \textit{blowup region}, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations of the initial data.