[Paper Review] A higher speed type II blowup for the five dimensional energy critical heat equation

Junichi Harada|arXiv (Cornell University)|Jun 10, 2019

Nonlinear Partial Differential EquationsMathematics13 references3 citations

TL;DR

This paper establishes the first rigorous existence proof of type II blowup solutions for the five-dimensional energy critical heat equation with blowup speed faster than previously known, specifically achieving ‖u(t)‖∞ ∼ (T−t)⁻³ᵏ for integers k≥2. Using a refined inner-outer gluing method with a non-constant particular solution Θ(x,t) ∼ (T−t)ˡ, it constructs radial solutions that blow up at a higher rate than the k=1 case, extending prior formal asymptotic results by Filippas, Herrero, and Velázquez.

ABSTRACT

This paper is concerned with blow-up solutions of the five dimensional energy critical heat equation $u_t=Δu+|u|^\frac{4}{3}u$. A goal of this paper is to show the existence of type II blowup solutions which behave as $\|u(t)\|_\infty\sim(T-t)^{-3k}$ ($k=2,3,\cdots$). Our solutions are the same one formally derived by Filippas, Herrero and Velázquez \cite{Filippas}.

Motivation & Objective

• To rigorously prove the existence of type II blowup solutions for the 5D energy critical heat equation with blowup rates faster than previously established.
• To extend the formal asymptotic results of Filippas, Herrero, and Velázquez, who predicted blowup rates of the form (T−t)⁻³ᵏ for k≥2 in five dimensions.
• To construct solutions with blowup speed (T−t)⁻³ᵏ for k≥2 by modifying the outer solution framework in the inner-outer gluing method.
• To demonstrate that higher blowup speeds are achievable by choosing a non-constant particular solution Θ(x,t) that scales as (T−t)ˡ for |x|∼√(T−t), enabling faster blowup dynamics.

Proposed method

• The solution is constructed using the inner-outer gluing method, decomposing the solution as u = Qλ + Θχ_out + λ⁻³ε(y,t)χ_in + w, where y = x/λ.
• A non-constant particular solution Θ(x,t) is introduced that behaves as (T−t)ˡ for |x|∼√(T−t), which enables the higher blowup speed (T−t)⁻³ᵏ with k=l+1.
• The outer solution w(x,t) is controlled via a comparison argument and decay estimates derived from the heat equation, using techniques adapted from Herrero and Velázquez's work on type II blowup.
• A fixed-point argument in a weighted function space Xσ is employed to solve the outer problem, ensuring existence and regularity of the solution up to T.
• The inner solution ε(y,t) is treated similarly to prior works, relying on spectral analysis and linearization around the ground state Qλ.
• The construction is completed by taking σ→0, yielding a global solution in C(R⁵×[0,T))∩C²,¹(R⁵×(0,T)) with the desired blowup rate.

Experimental results

Research questions

• RQ1 Can type II blowup solutions with blowup speed (T−t)⁻³ᵏ for k≥2 be rigorously constructed for the 5D energy critical heat equation?
• RQ2 What role does the choice of the particular solution Θ(x,t) play in achieving higher blowup speeds in the inner-outer gluing framework?
• RQ3 How can the outer solution w(x,t) be estimated with sufficient decay to close the fixed-point argument in the gluing method?
• RQ4 Is it possible to extend the blowup rate beyond the k=1 case previously proven by Cortázar, del Pino, and Musso?
• RQ5 What modifications to the gluing method are required to achieve blowup rates faster than (T−t)⁻³?

Key findings

• The paper constructs radial type II blowup solutions for the 5D energy critical heat equation with blowup rate ‖u(t)‖∞ ∼ (T−t)⁻³ᵏ for any integer k≥2.
• The blowup rate is achieved by choosing a particular solution Θ(x,t) that scales as (T−t)ˡ for |x|∼√(T−t), leading to λ(t)∼(T−t)²ˡ⁺² and thus λ(t)⁻³∼(T−t)⁻³⁽ˡ⁺¹⁾.
• This construction provides the first rigorous existence result for k≥2 in the blowup rate (T−t)⁻³ᵏ, confirming the formal prediction of Filippas, Herrero, and Velázquez.
• The outer solution w(x,t) is shown to satisfy the decay estimate |w(x,t)|≲ T³ˡ⁺³/|x|³ + δ₀²/|x|² for |x|>1, which is crucial for the fixed-point argument.
• The solution is obtained via a Schauder fixed-point theorem applied to a compact, continuous mapping in a weighted function space Xσ, with σ→0 yielding the final solution.
• The result establishes that higher blowup speeds are dynamically possible in the energy-critical setting, beyond the previously known k=1 case.