The blow-up rate for a log non-scaling invariant semilinear wave equation in the conformal regime

1. Problem Statement

The paper investigates the blow-up behavior of solutions to the semilinear wave equation with a logarithmic perturbation in the conformal regime:
$$\partial_t^2 u - \Delta u = |u|^{p-1}u \ln^a(u^2 + 2), \quad (x, t) \in \mathbb{R}^n \times [0, T)$$
where:

• $n \geq 2$.
• The exponent is critical: $p = p_c = 1 + \frac{4}{n-1}$.
• The parameter $a < 0$ (the "log-subconformal" case).
• Initial data $(u_0, u_1)$ belong to local energy spaces $H^1_{loc,u} \times L^2_{loc,u}$.

Context:

• For the standard power nonlinearity ($a=0$), Merle and Zaag established that solutions blow up at a Type I rate (governed by the associated ODE) at non-characteristic points in both subconformal and conformal regimes.
• For $a > 0$, the problem becomes "log-superconformal," shifting the structure significantly.
• For $a < 0$ in the subconformal regime ($p < p_c$), previous work by the author showed Type I blow-up persists.
• The Gap: The behavior in the conformal regime ($p = p_c$) with $a < 0$ was unknown. The conformal case is notoriously difficult because the dissipation of the standard Lyapunov functional degenerates at the boundary of the similarity ball, and the logarithmic term breaks the scaling invariance, preventing direct application of previous methods.

2. Methodology

The core strategy involves transforming the problem into similarity variables and constructing a Lyapunov functional to control the solution's growth near the blow-up time.

A. Similarity Variables

For a non-characteristic point $x_0$ and blow-up time $T(x_0)$, the authors introduce:
$$y = \frac{x - x_0}{T_0 - t}, \quad s = -\ln(T_0 - t), \quad u(x, t) = \psi_{T_0}(t) w(y, s)$$
where $\psi_{T_0}(t) = (T_0 - t)^{-\frac{2}{p-1}} (-\ln(T_0 - t))^{-\frac{a}{p-1}}$ is the explicit asymptotic profile of the ODE solution.
The function $w(y, s)$ satisfies a perturbed wave equation in the unit ball $B \subset \mathbb{R}^n$:
$$\partial_s^2 w = \frac{1}{\rho} \text{div}(\rho \nabla w - \rho (y \cdot \nabla w)y) - \dots + \frac{1}{s^a} |w|^{p-1}w \ln^a(\phi^2 w^2 + 2)$$
where $\rho(y, s) = (1-|y|^2)^{\alpha(s)}$ and $\alpha(s) = -\frac{a}{(p_c-1)s} > 0$.

B. Construction of Lyapunov Functionals

The proof proceeds through a multi-stage construction of functionals to handle the weak dissipation and the logarithmic perturbation:

1. Exponential Bounds (Section 2):

   • The authors first construct a functional $H_\eta$ (for small $\eta > 0$) to derive exponential bounds for the time-average of the $H^1 \times L^2$ norm of $w$.
   • This step utilizes Pohozaev-type identities and Hardy-type inequalities to control singular weights near the boundary $|y|=1$.

2. Almost Linear Bounds (Section 3):

   • Using the exponential bounds, they construct a functional $P(w, s)$ to refine the estimates to almost linear bounds (specifically $O(s^{1+\epsilon})$).
   • This involves a delicate analysis of the time derivative of the energy, exploiting the fact that $a < 0$ allows certain terms to be controlled.

3. The Main Lyapunov Functional (Section 4 - Theorem 1):

   • The authors define a functional $L(w, s) = \exp\left(\frac{p_c+3}{\sqrt{s}}\right) G(w, s) + \frac{\theta_1}{s}$, where $G = E + J$.
   • Key Innovation: Unlike the pure power case where dissipation is supported only on the boundary, the presence of the logarithmic term ($a < 0$) creates a weak dissipation term over the entire ball, but with a coefficient decaying as $O(1/s)$.
   • They prove that $L(w, s)$ is a decreasing functional for large $s$, establishing a Lyapunov structure.

4. Refinement to Boundedness (Section 5):

   • The weak dissipation initially yields only linear bounds ($O(s)$) for the time-averaged norms.
   • To obtain the sharp Type I blow-up rate, the authors must prove the norms are bounded (independent of $s$).
   • They introduce a new energy functional $E_0(w, s)$ associated with the unperturbed conformal equation. By analyzing the difference $E - E_0$ and using the boundedness of $L(w, s)$, they prove that the time-average of the $L^2$ norm of $\partial_s w$ on the boundary $\partial B$ is bounded.
   • This allows them to upgrade the linear bounds to logarithmic and finally to uniform bounds.

3. Key Contributions

1. First Result for Log-Subconformal Critical Case: This is the first work establishing the blow-up rate for a non-scaling invariant semilinear wave equation in the critical (conformal) regime.
2. Novel Lyapunov Functional: The construction of $L(w, s)$ handles the specific difficulty of "log-subconformal" dynamics where dissipation is weak and degenerate. The functional includes a specific exponential weight $\exp(\frac{p_c+3}{\sqrt{s}})$ to counteract the slow decay of the damping term.
3. Overcoming Degenerate Dissipation: The paper addresses the technical obstacle where the standard dissipation term vanishes at the boundary in the conformal case. The authors show that the logarithmic perturbation ($a < 0$) actually provides dissipation over the whole domain, albeit weak, which is sufficient to close the estimates.
4. Extension of Roy-Zaag Results: The work extends the results of Roy and Zaag (who treated double-logarithmic perturbations in the subconformal regime) to the critical conformal case.

4. Main Results

Theorem 1 (Existence of Lyapunov Functional):
For a non-characteristic point $x_0$, there exists a time $t_1$ such that for $s$ large enough, the functional $L(w(s), s)$ is non-negative and satisfies a differential inequality indicating decay (up to a small error term). This establishes the stability of the solution in similarity variables.

Theorem 2 (Sharp Blow-up Rate):
Let $u$ be a solution blowing up at $T(x_0)$ at a non-characteristic point $x_0$. Then, for $t$ close to $T(x_0)$:
$$0 < \epsilon_0 \leq \frac{\|u(t)\|_{L^2(B(x_0, T(x_0)-t))}}{(T(x_0)-t)^{n/2} \psi_{T(x_0)}(t)} + \dots \leq K$$
where $\psi_{T(x_0)}(t) = (T(x_0)-t)^{-\frac{2}{p-1}} (-\ln(T(x_0)-t))^{-\frac{a}{p-1}}$.

Interpretation:
The solution blows up with a Type I rate, governed by the ODE solution $v'' = |v|^{p-1}v \ln^a(v^2+2)$. The logarithmic factor modifies the standard power-law blow-up rate by a factor of $(-\ln(T-t))^{-a/(p-1)}$.

5. Significance

• Theoretical Advancement: It resolves a major open problem regarding the stability of Type I blow-up in the presence of non-homogeneous, non-scaling invariant perturbations at the critical exponent.
• Methodological Robustness: The techniques developed (specifically the handling of weak dissipation via refined energy functionals and the use of covering arguments adapted to the singular weights) provide a framework for studying other critical problems where scaling symmetry is broken.
• Physical Relevance: Logarithmic nonlinearities appear in various physical models (nuclear physics, optics). Understanding their blow-up dynamics in critical regimes is essential for predicting singularity formation in these systems.
• Limitations: The method relies on $a < 0$. The case $a > 0$ (log-superconformal) remains open and likely requires different techniques due to the shift in the problem's structure. The results also apply only to non-characteristic points; the behavior at characteristic points remains an open question.