6D Energy-Critical Heat Equation

• The 6D energy-critical heat equation is defined by a semilinear PDE with p=2 and an invariant energy under scaling, highlighting its borderline behavior.
• It exhibits type II blowup characterized by a slower-than-self-similar blowup rate with precise logarithmic corrections and inner–outer matched asymptotics.
• Soliton resolution and oscillatory dynamics are established through spectral analysis, modulation techniques, and concentration–compactness methods.

The six-dimensional energy-critical heat equation refers to the semilinear parabolic PDE:

$$u_t = \Delta u + |u|^{p-1}u \qquad\text{on }\; \mathbb{R}^6\times(0,T)$$

with critical exponent $p = \frac{n+2}{n-2} = 2$ for $n=6$. The problem is termed "energy-critical" because the conserved energy

$$E(u) = \frac{1}{2}\int_{\mathbb{R}^6}|\nabla u|^2\,dx - \frac{1}{3}\int_{\mathbb{R}^6}|u|^3\,dx$$

is invariant under the rescaling $$u(x,t) \mapsto \lambda^{-2}u(\lambda x, \lambda^2 t)$$. The 6D case is a subtle borderline for the emergence of "type II blowup," long-time oscillatory dynamics, and intricate threshold phenomena, with distinct features compared to lower or higher dimensions.

1. Equation, Scaling, and Ground States

The energy-critical heat flow in six dimensions is characterized by the critical nonlinearity $|u|u$, scaling-invariant energy, and the Aubin–Talenti bubble as ground state:

$Q(x) = (1 + |x|^2/24)^{-2}$

which solves the stationary elliptic equation $\Delta Q + Q^2 = 0$ on $\mathbb{R}^6$ (Harada, 5 Dec 2024, Aryan, 9 May 2024). All positive, decaying stationary solutions are given by the scaling family $Q_\lambda(x) = \lambda^{-2} Q(x/\lambda)$.

2. Type II Blowup and Borderline Dynamics

Type II blowup refers to solutions that blow up at a slower rate than the "type I" (or self-similar) rate. In 6D, unlike $n\ge7$ (where only type I is possible near the ground state), genuine type II blowup is constructed (Harada, 2020). Specifically, there exists a solution exhibiting:

• Localized blowup at the origin as $t\to T^-$,
• Profile decomposition:
  • Inner region ($|x|\sim\lambda(t)$):

  $u(x,t)\approx \lambda(t)^{-2} Q(x/\lambda(t))$ - Self-similar region ($|x|\sim \sqrt{T-t}$):

  $$u(x,t)\approx (T-t)^{-1}\Psi \big(x/\sqrt{T-t}\big)$$

  where $\Psi$ solves a nontrivial ODE.

• The blowup speed:

$\lambda(t) = (1+o(1))(T-t)^{5/4}|\ln(T-t)|^{-15/8}$

• The local energy diverges to $-\infty$:

$$E_{\rm loc}(u(t)) = \frac{1}{2}\int_{|x|<1}|\nabla u(x,t)|^2 dx - \frac{1}{3}\int_{|x|<1}|u(x,t)|^3 dx \rightarrow -\infty$$

A matched asymptotic expansion and inner–outer gluing yield the existence of this blowup, the profile $\Psi$, and correctors. The occurrence is borderline; existence or uniqueness for other blowup speeds remains open (Harada, 2020).

3. Soliton Resolution and Profile Decomposition

The soliton resolution conjecture holds for radial solutions: every finite-energy solution resolves into a finite sum of decoupled bubbles (ground-state profiles) and free radiation (Aryan, 9 May 2024). Specifically:

• Global regime: For $t\to\infty$, solutions are sums of ground states at well-separated scales plus heat flow of radiation, with scales decoupling as $\lambda_j(t)/\lambda_{j+1}(t)\to 0$.
• Blowup regime: Near finite blowup time, solutions are sums of bubbles at vanishing scales plus a body map remainder, with scales $\lambda_N(t) \ll \sqrt{T_+-t}$.

The decomposition is unique in terms of the number of bubbles, scales, and remainder. Modulation analysis and energy monotonicity are used to track and control the scales and behavior.

4. Global Dynamics, Thresholds, and Trichotomy

Sharp conditions on initial data lead to a dichotomy or trichotomy in behavior, as described in (Ikeda et al., 5 Dec 2024, Harada, 5 Dec 2024):

• Threshold defined by ground state: $E[u_0] \le E[Q]$
• Dissipation regime: If $\|\nabla u_0\|_{L^2}^2 < \|\nabla Q\|_{L^2}^2$ (Nehari functional $J(u_0)\ge 0$), solutions exist globally and decay:

$$\|u(t)\|_{H^1(\mathbb{R}^6)} \le C(1+t)^{-1}, \qquad E[u(t)] = O((1+t)^{-2})$$

• Blowup regime: If $\|\nabla u_0\|_{L^2}^2 > \|\nabla Q\|_{L^2}^2$, finite-time blowup occurs.
• Classification near ground state (Harada, 5 Dec 2024): Small perturbations yield three outcomes:
  1. Global convergence to a rescaled ground state bubble.
  2. Global dissipation to zero.
  3. Type I blowup (ODE-type): $$\|\nabla u(t)\|_{L^2} \sim \|\nabla Q\|_{L^2} (T-t)^{-1/2}$$.

A new proof ingredient in $n=6$ is the propagation of the $L^2$ norm of the remainder to close coercivity estimates and control explicit instability directions.

5. Oscillatory and Infinite-Time Solutions

Recent work provides explicit global solutions that oscillate in scale indefinitely in the homogeneous Sobolev space $\dot{H}^1(\mathbb{R}^6)$—a phenomenon absent in $H^1(\mathbb{R}^6)$ dynamics and unique to the critical 6D case (Harada, 22 Nov 2025). The constructed solution satisfies:

$$u(x,t) = \lambda(t)^{-2} Q(x / \lambda(t)) + \text{error}(x,t)$$

with $\lambda(t)$ such that

$$\liminf_{t\to\infty} \lambda(t) = 0, \quad \limsup_{t\to\infty} \lambda(t) = \infty$$

and the error decaying to zero in $\dot H^1$. The underlying mechanism is a sign-changing modulation ODE for $\lambda(t)$ driven by oscillatory radiation in the outer region. This behavior is constructed via inner–outer matched asymptotic expansions, orthogonality and spectral projections, and control of the zero and negative modes of the linearized problem.

6. Analytical Framework and Spectral Properties

All approaches use the following analytical ingredients, reflected throughout the literature (Harada, 2020, Harada, 22 Nov 2025, Harada, 5 Dec 2024, Aryan, 9 May 2024):

• Spectral analysis: The linearized operator $H_y = \Delta_y + 2Q(y)$ has one negative eigenvalue, a simple zero mode, and the remainder of its spectrum bounded away from zero.

• Orthogonality and modulation: Modulation equations and orthogonality conditions against unstable modes fix scale $\lambda(t)$ and absorb secular terms in the remainder.
• Coercivity estimates: Crucial to suppress instability, especially for six dimensions, where control of $L^2$-norm is needed for closure.
• Parabolic Schauder and $L^p$ estimates: Used in the outer region, replacing backward uniqueness in dispersive settings.
• Profile decompositions and concentration compactness: Underpin rigourous soliton resolution results.

7. Remarks, Open Questions, and Borderline Phenomena

• Six dimensions ($n=6$) represent the unique "borderline" for type II blowup: for $n\le 5$, type II appears generically; for $n\ge 7$, only type I is observed near the ground state (Harada, 2020).
• Uniqueness and stability of type II blowup remains open in six dimensions; the matched-asymptotics construction produces one solution with a precise logarithmic correction, but other blowup rates are not excluded.
• The non-radial case for soliton resolution and oscillatory solutions is unsolved; current proofs rely on radial symmetry and associated embeddings.
• Concentration–compactness and rigidity arguments adapt dispersive methods (Kenig–Merle strategy) for the parabolic, energy-critical setting (Ikeda et al., 5 Dec 2024).
• Oscillatory dynamics in $\dot H^1$ but not $H^1$ reveal a subtle dependence on the initial functional framework.

In conclusion, the six-dimensional energy-critical heat equation showcases unique interactions between scale invariance, threshold phenomena, type II blowup, soliton resolution, and oscillatory behavior, with recent rigorous results confirming previously formal predictions and opening new directions for dynamics near threshold and beyond (Harada, 2020, Aryan, 9 May 2024, Ikeda et al., 5 Dec 2024, Harada, 5 Dec 2024, Harada, 22 Nov 2025).