Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

1. Problem Statement

The authors investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation with a derivative-type nonlinearity:
$$\partial_t^2 u - \partial_x^2 u + \frac{\mu}{1+t}\partial_t u = |\partial_t u|^p, \quad x \in \mathbb{R}, t > 0,$$
subject to sufficiently large and smooth initial data $u(x,0)=u_0(x)$ and $\partial_t u(x,0)=u_1(x)$, where $p > 1$ and $\mu > 0$.

Key Challenges:

• Derivative Nonlinearity: The source term depends on $\partial_t u$ rather than $u$. This breaks the variational structure typically used in semilinear wave equations (e.g., $\square u = |u|^{p-1}u$), rendering energy-based methods (like those of Merle and Zaag) inapplicable.
• Scale-Invariant Damping: The term $\frac{\mu}{1+t}\partial_t u$ introduces a time-dependent damping coefficient. While previous works (Sasaki [23, 24]) handled derivative nonlinearities without damping, the presence of $\mu$ complicates the characteristic analysis and destroys the simple comparison arguments used in the undamped case.
• Objective: The primary goal is to characterize the blow-up curve $T(x)$, defined as the time when the solution becomes singular, specifically proving its $C^1$-regularity (continuous differentiability) and determining its asymptotic profile.

2. Methodology

The authors adapt and extend the characteristic ODE method originally developed by Caffarelli and Friedman for semilinear equations and later refined by Sasaki for derivative nonlinearities.

Step 1: Riemann Invariants and First-Order System
The second-order PDE is transformed into a first-order system using Riemann invariants:
$$\phi = \partial_t u + \partial_x u, \quad \psi = \partial_t u - \partial_x u.$$
This yields the coupled system:
$$\begin{cases} D_- \phi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1+t}\frac{\phi + \psi}{2}, \\ D_+ \psi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1+t}\frac{\phi + \psi}{2}, \end{cases}$$
where $D_\pm = \partial_t \pm \partial_x$ are characteristic derivatives. This moves the nonlinearity from the derivative term to the undifferentiated variables $\phi$ and $\psi$.

Step 2: Iterative Construction and Monotonicity
To prove existence and regularity, the authors construct solutions via a successive approximation scheme (Picard iteration) along characteristic rays.

• They establish monotonicity of the iterative sequences $\{\phi_n, \psi_n\}$ under specific assumptions on the initial data (large enough amplitude).
• They derive gradient estimates showing that temporal derivatives dominate spatial derivatives ($\partial_t \phi \geq (1+\epsilon)|\partial_x \phi|$), which is crucial for controlling the geometry of the blow-up.

Step 3: Handling the Damping Term
The damping term $\frac{\mu}{1+t}$ introduces extra terms in the evolution equations that disrupt standard comparison principles.

• The authors introduce structural conditions (Assumptions A1–A5) on the initial data and parameters to ensure the nonlinear growth dominates the damping decay.
• They perform technical reductions to recreate a scenario comparable to the undamped case, allowing them to derive necessary bounds despite the extra $\mu$-dependent terms.

Step 4: Blow-up Limits and Self-Similarity
To analyze the regularity of the blow-up curve $T(x)$, the authors employ a blow-up scaling technique:
$$\phi_\lambda(y, s) = \lambda^q \phi(x_0 + \lambda y, T(x_0) + \lambda s), \quad q = \frac{1}{p-1}.$$
They analyze the limit of these rescaled functions as $\lambda \to 0$. The limiting system satisfies a simplified ODE system along rays, leading to explicit self-similar profiles $V_{\phi, \alpha}$ and $V_{\psi, \alpha}$.

3. Key Contributions

1. Extension to Damped Derivative Nonlinearities: This is the first work to establish the $C^1$-regularity of the blow-up curve for a wave equation with both a derivative nonlinearity ($|\partial_t u|^p$) and scale-invariant damping ($\frac{\mu}{1+t}\partial_t u$).
2. Overcoming the Lack of Variational Structure: The paper successfully adapts the Caffarelli-Friedman/Sasaki characteristic method to a setting where energy estimates fail, relying instead on the monotonicity of characteristic derivatives.
3. Characterization of the Blow-up Profile: The authors identify the precise asymptotic behavior of the solution near the blow-up time $T(x)$:
   $$\partial_t u(x, t) \sim \left[ \frac{1}{(p-1)(T(x)-t)} \right]^{\frac{1}{p-1}}.$$
   They show that the damping term $\frac{\mu}{1+t}$ is slowly varying near the blow-up time, meaning the blow-up rate is the same as in the constant damping case.
4. Proof of $C^1$-Regularity: The paper rigorously proves that the blow-up curve $T(x)$ is continuously differentiable, a non-trivial result given the potential for singular points in such systems.

4. Main Results

Theorem 3.2 (Existence and Regularity):
Under suitable assumptions on initial data (large amplitude, smoothness), there exists a unique blow-up time function $T(x) \in C^1(B_{R^*})$ and a unique solution $(\phi, \psi)$ that blows up as $t \nearrow T(x)$.

Theorem 3.4 (Blow-up Rates):
The solution satisfies precise estimates near the blow-up time:
$$C_1 (T(x)-t)^{-q} \leq \phi + \psi \leq C_2 (T(x)-t)^{-q},$$
$$C_1 (T(x)-t)^{-q-1} \leq \partial_t \phi, \partial_t \psi \leq C_2 (T(x)-t)^{-q-1},$$
where $q = \frac{1}{p-1}$.

Theorem 3.5 (Lipschitz Continuity):
The blow-up curve is Lipschitz continuous with a specific constant related to the initial data:
$$|T(x) - T(y)| \leq \frac{1}{1+\epsilon_0} |x - y|.$$

Theorem 3.6 (Linear Structure of Limits):
The limiting blow-up curve obtained via scaling is linear: $T_0(y) = \alpha y$. The authors prove that the slope $\alpha$ is unique (i.e., the left and right derivatives at any point coincide), which is the core of the differentiability proof.

Theorem 3.7 ($C^1$-Regularity):
The blow-up curve $x \mapsto T(x)$ is of class $C^1$ in the domain. The proof relies on a contradiction argument: if $T$ were not differentiable, one could construct two distinct limiting slopes $\alpha_1 \neq \alpha_2$, leading to a contradiction in the directional derivatives of the limiting profiles.

5. Significance

• Mathematical Rigor: The paper provides a robust framework for analyzing blow-up in damped hyperbolic systems with derivative nonlinearities, filling a gap in the literature where such combinations were previously uncharacterized.
• Geometric Insight: It confirms that despite the complexity introduced by time-dependent damping, the geometry of the blow-up set remains smooth ($C^1$) and the blow-up profile retains a self-similar structure.
• Methodological Advancement: The techniques developed (specifically the handling of the damping term within the characteristic ODE framework) offer a template for studying other damped wave equations with non-standard nonlinearities.
• Physical Relevance: Understanding the regularity of the blow-up curve is essential for predicting the formation of singularities in physical models involving wave propagation with dissipation and nonlinear sources.

In summary, this work successfully generalizes the theory of blow-up curves from the undamped, derivative-nonlinear case to the scale-invariant damped case, proving that the blow-up curve remains smooth and characterizing the precise nature of the singularity.