[Paper Review] Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

Tej‐Eddine Ghoul, Van Tien Nguyen+1 more|arXiv (Cornell University)|Nov 8, 2016

Nonlinear Partial Differential Equations41 references20 citations

TL;DR

This paper constructs a finite-time blowup solution for a nonlinear heat equation with a critical quadratic gradient term ($\alpha |\nabla U|^2$) and exponential reaction ($e^U$), proving that both the solution and its gradient blow up simultaneously at the origin. The final blowup profile differs from the $\alpha=0$ case due to the critical nature of the gradient term, and the solution's stability under initial data perturbations is established via a finite-dimensional reduction and index theory.

ABSTRACT

We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\\partial_t U = \\Delta U + \\alpha|\ abla U|^2 + e^U,\\quad (x, t)\\in\\mathbb{R}^N\ imes[0,T), \\quad \\alpha > -1.$$ 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 \ extit{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.

Motivation & Objective

• To construct a solution that blows up in finite time for the nonlinear heat equation $\partial_t U = \Delta U + \alpha |\nabla U|^2 + e^U$ with $\alpha > -1$.
• To characterize the precise final blowup profile of the solution and its gradient at the origin.
• To show that the quadratic gradient term is critical, altering the blowup profile compared to the case $\alpha = 0$.
• To establish the stability of the constructed blowup solution under small perturbations of the initial data.
• To overcome the challenge of controlling the outer region where the linearized operator spectrum cannot be made negative, requiring new analytical techniques.

Proposed method

• Uses a similarity variable transformation to convert the blowup problem into a large-time asymptotic analysis of a rescaled equation.
• Employs a finite-dimensional reduction method inspired by Merle and Zaag (1997), reducing the infinite-dimensional PDE to a finite-dimensional dynamical system.
• Applies index theory to control the behavior of the solution near the blowup time and point.
• Introduces a geometrical interpretation of the finite-dimensional parameters in terms of the blowup time and blowup location to prove stability.
• Uses weighted energy estimates and Gronwall-type inequalities with singular kernels to control the solution in different spatial regions.
• Applies Lemma B.1 and variants of Gronwall's inequality (Lemma B.3) combined with integral estimates (Lemma B.2) to bound the solution and its gradient in the outer region.

Experimental results

Research questions

• RQ1 How does the presence of a critical quadratic gradient term $\alpha |\nabla U|^2$ affect the blowup profile of the equation $\partial_t U = \Delta U + \alpha |\nabla U|^2 + e^U$?
• RQ2 What is the precise final blowup profile of the solution and its gradient when $\alpha > -1$ and $r=2$ is critical?
• RQ3 Why does the blowup profile differ from the case $\alpha = 0$, and how does the critical exponent $r=2$ alter the dynamics?
• RQ4 Can the constructed blowup solution be shown to be stable under small perturbations of the initial data?
• RQ5 What new analytical techniques are required to control the outer region where the linearized operator spectrum fails to be negative?

Key findings

• The solution and its gradient blow up simultaneously at the origin in finite time $T>0$, with a precisely described final blowup profile.
• The blowup profile differs from the $\alpha = 0$ case due to the critical nature of the quadratic gradient term, which alters the asymptotic structure.
• The constructed solution is stable under small perturbations of the initial data, as shown through a geometrical interpretation of the finite-dimensional parameters.
• The outer region control is achieved via novel estimates using weighted $L^\infty$ norms and integral inequalities with singular kernels.
• For small $\epsilon$, the solution satisfies $e^{u(\xi,\tau)} \leq C\epsilon (1-\tau)^{-\epsilon}$ and $|\nabla u(\xi,\tau)| \leq C\epsilon$ in the inner region $|\xi| < 1/8$, uniformly in $\tau \in [0,1)$.
• The blowup rate satisfies $\lim_{t \to T} \left[ U(a + z\sqrt{(T-t)|\ln(T-t)|}, t) + \ln(T-t) \right] = \Phi_0(z) = -\ln\left(1 + \frac{|z|^2}{4}\right)$, confirming the intermediate-scale profile.