[Paper Review] Type II Blow Up for the Four Dimensional Energy Critical Semi Linear Heat Equation

Rémi Schweyer|arXiv (Cornell University)|Jan 13, 2012

Advanced Mathematical Physics ProblemsMathematics22 references101 citations

TL;DR

This paper establishes the existence of type II finite time blow-up solutions for the four-dimensional energy-critical semilinear heat equation $\partial_t u - \Delta u - u^3 = 0$, using a robust energy method inspired by geometric dispersive PDEs. It proves that solutions concentrate a universal energy bubble described by the Talenti-Aubin soliton $Q(r) = (1 + r^2/8)^{-1}$, with blow-up speed $\lambda(t) \sim (T - t)/|\log(T - t)|^2$, and shows the asymptotic profile lies in $\dot{H}^1$ with $\Delta u^* \in L^2$. This resolves the long-open question of type II blow-up in the energy-critical setting for $N=4$.

ABSTRACT

We consider the energy critical four dimensional semi linear heat equation \\partial tu-\\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of the corresponding singularity formation. These solutions concentrate a universal bubble of energy in the critical topology u(t,r)-1/{\\lambda} Q(r/{\\lambda})\ ightarrow u* in $\\dot{H}^1$ where the blow up profile is given by the Talenti Aubin soliton Q(r)= 1/(1 +r^2/8) and with speed {\\lambda}(t) ~(T-t)/|log(T - t)|^2 as t\ ightarrowT. Our approach uses a robust energy method approach developped for the study of geometrical dispersive problems, and lies in the continuation of the study of the energy critical harmonic heat flow and the energy critical four dimensional wave equation.

Motivation & Objective

• Address the open problem of whether type II blow-up occurs in the energy-critical four-dimensional semilinear heat equation.
• Construct a robust energy method to analyze singularity formation in the absence of maximum principle control.
• Establish sharp asymptotics for blow-up dynamics, including the blow-up rate and profile convergence in $\dot{H}^1$.
• Extend the energy method framework developed for geometric dispersive problems to the parabolic setting.
• Provide a complete description of the blow-up mechanism, including the role of the non-positive eigenvalue in the linearized operator around $Q$.

Proposed method

• Adapt the energy method from geometric dispersive problems (e.g., wave maps, Schr"odinger maps) to the parabolic setting.
• Use the Talenti-Aubin soliton $Q(r) = (1 + r^2/8)^{-1}$ as the universal blow-up profile, solving $\Delta Q + Q^3 = 0$.
• Construct initial data in a codimension-one subset of $H^1$ to handle the non-positive eigenvalue in the linearized operator around $Q$.
• Employ a rescaling $u(t,x) \sim \lambda(t)^{-1} Q(x/\lambda(t))$ with $\lambda(t) \sim (T - t)/|\log(T - t)|^2$ to describe the blow-up dynamics.
• Apply weighted energy estimates and $L^2$-based norms in the self-similar variable $y = r/\lambda(t)$ to control error terms.
• Use localization techniques and precise pointwise estimates to control the error $\varepsilon$ in the decomposition $u(t,x) = \lambda(t)^{-1} Q(x/\lambda(t)) + \varepsilon(t,x)$.

Experimental results

Research questions

• RQ1 Does type II blow-up occur for the energy-critical four-dimensional semilinear heat equation $\partial_t u - \Delta u - u^3 = 0$?
• RQ2 What is the precise asymptotic behavior of the blow-up profile and the blow-up rate $\lambda(t)$?
• RQ3 How can a robust energy method be adapted to the parabolic setting to construct type II blow-up solutions?
• RQ4 What role does the non-positive eigenvalue in the linearized operator around $Q$ play in the dynamics?
• RQ5 Can the blow-up dynamics be described with sharp regularity and convergence properties in $\dot{H}^1$ and $L^2$?

Key findings

• Type II finite-time blow-up solutions exist for the energy-critical 4D semilinear heat equation $\partial_t u - \Delta u - u^3 = 0$.
• The blow-up speed is $\lambda(t) \sim c(u_0) \frac{T - t}{|\log(T - t)|^2}$ as $t \to T$, with $c(u_0) > 0$ depending on initial data.
• The solution converges in $\dot{H}^1$ to a profile $u^*$ satisfying $\nabla[u(t) - \lambda(t)^{-1} Q(\cdot/\lambda(t))] \to \nabla u^*$ in $L^2$ as $t \to T$.
• The asymptotic profile $u^*$ satisfies $\Delta u^* \in L^2$, indicating improved regularity beyond $\dot{H}^1$.
• Initial data $u_0 \in H^1(\mathbb{R}^4)$ exist with $E(Q) < E(u_0) < E(Q) + \alpha^*$ for any $\alpha^* > 0$, yielding type II blow-up.
• The blow-up mechanism is universal and independent of initial data within the constructed codimension-one manifold, with the profile $Q$ being the unique radial solution to $\Delta Q + Q^3 = 0$. This confirms the existence of a stable, sharp blow-up dynamics in the energy-critical setting.