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

Abstract: In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t² u - \partial_x² u + \fracμ{1 + t} \partial_t u = |\partial_t u|^p \quad (p &gt; 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}<sup>1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($μ=0$).

• The paper rigorously establishes the existence and C¹ regularity of the blow-up curve using characteristic coordinates.
• It adapts an ODE-based method to overcome damping-induced complications, demonstrating that the blow-up rate remains ODE-like near singularities.
• The study employs precise comparison principles and scaling limits to exclude non-differentiable singular points on the blow-up set.

Regularity and Singularity of the Blow-up Curve for a Wave Equation with Derivative Nonlinearity and Scale-Invariant Damping

Introduction and Problem Setting

This work rigorously characterizes the blow-up geometry for a one-dimensional damped nonlinear wave equation with a time-derivative nonlinearity and scale-invariant damping: $$\partial_t^2 u - \partial_x^2 u + \frac{\mu}{1 + t} \partial_t u = |\partial_t u|^p, \quad p>1.$$ The primary objects of study are the regularity and structure of the blow-up curve, i.e., the locus in space-time where the temporal derivative becomes singular, and the asymptotic blow-up profile of the solution. The analysis proceeds under the assumption of sufficiently large, smooth initial data.

The novelty stems from the combination of derivative-type nonlinearity and time-dependent (scale-invariant) damping, which destroys variational structure and complicates classical energy-based approaches. The authors adapt and extend the characteristic ODE method, originally due to Caffarelli–Friedman and developed for pure-power and derivative-type equations by Sasaki, to handle the new analytic and geometric difficulties induced by the damping term.

Analytical Framework and Key Transformations

The equation is recast into a first-order system for the Riemann invariants: $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$ yielding: $$D_- \phi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},\qquad D_+ \psi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},$$ where $D_- = \partial_t - \partial_x$ and $D_+ = \partial_t + \partial_x$ are derivatives along the characteristic lines $x\mp t = \text{constant}$. This characteristic system allows the reduction of the original PDE’s singularity analysis to a study of blow-up in coupled ODEs along rays.

The damping term alters the scaling, equilibrium, and monotonicity structures, introducing technical obstacles overcome via careful ODE analysis, monotonicity iterations, and the imposition of sharp structural conditions on the initial data (enforced via explicit inequalities on the derivatives and amplitudes).

Main Results: Existence, Blow-up Rates, and Regularity

Existence and Nature of Blow-up

The system supports finite-time blow-up when the initial characteristics exceed a threshold,

$\gamma_1 + \gamma_2 > 2\mu^{1/(p-1)},$

with the blow-up time function $T(x)$ defined as the supremum over $t$ such that $\partial_t u(x, t)$ remains finite. The solution construction is via monotone, contractive successive approximations in the characteristic variables, leading to global-in-time well-posedness up to blow-up and uniqueness.

Blow-up Profile and Rates

The solution’s derivative near blow-up exhibits the singular asymptotics,

$$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$0

uniformly up to constants depending only on $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$1 and relevant norm bounds for the data. This scaling is identical to the conservative case, validating that the slowly varying damping coefficient has only a subdominant impact on the local blow-up profile—its leading-order behavior remains ODE-like. There are matching lower and upper bounds for $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$2 and their derivatives in terms of the distance to the blow-up curve, ensuring both lower and upper bisecting controls near singularity.

Lipschitz and $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$3 Regularity of the Blow-up Set

A principal result is the proof that the blow-up time function $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$4 is globally Lipschitz, in fact,

$$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$5

where $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$6 quantifies the strict separation of temporal and spatial derivative amplitudes in the monotonicity estimates. Moreover, by employing blow-up scaling limits and compactness (Arzelà-Ascoli), the authors show that the possible “defect measures” (non-differentiability or irregular points) are precluded: no singular points or corners form in the blow-up curve $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$7 under the a priori size and smoothness assumptions on the data.

The limiting rescaled blow-up surfaces converge (along subsequences) to affine functions orthogonal to the rays, whose slope is uniquely selected by the data. The method of exclusion of non-differentiability follows the Caffarelli–Friedman/Sasaki paradigm, exploiting analytic monotonicity and the structure of the limiting self-similar ODEs.

Limiting Dynamics and Self-Similar Structure

Critical to the proof is the rigorous analysis of the scaling limits of the solution sequence: $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$8 as $$\phi = \partial_t u + \partial_x u,\quad \psi = \partial_t u - \partial_x u,$$9, which, at leading order, decouple the spatial and temporal variables and yield limiting profiles of the exact form

$$D_- \phi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},\qquad D_+ \psi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},$$0

where the “blow-up direction” $$D_- \phi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},\qquad D_+ \psi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},$$1 encodes the tangent of the blow-up surface at the base point $$D_- \phi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},\qquad D_+ \psi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},$$2 and is shown to be unique and continuous in $$D_- \phi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},\qquad D_+ \psi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},$$3. This construction rules out the spontaneous generation of singular/characteristic points except through data or external constraint.

Technical Innovations and Analytical Challenges

While the approach closely follows the method of characteristics and scaling analysis established in [Sasaki 2018, 2019], the presence of scale-invariant damping introduces nontrivial analytic complications: the characteristic ODEs acquire nonautonomous terms destroying scaling symmetry and requiring the authors to develop intricate comparison principles, technical estimates for higher derivatives, and uniform contractivity and monotonicity bounds. The regularity results demand precise matching of leading- and higher-order expansion terms.

The uniqueness and regularity proof for the blow-up set leverages a two-scale approach: the local-in-space contraction-mapping properties of the solution iterations and the global geometric convexity properties of the limiting blow-up domain.

Implications and Future Directions

The demonstration that, for a large class of nonlinear wave equations with derivative source and scale-invariant damping, the blow-up set is necessarily a $$D_- \phi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},\qquad D_+ \psi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},$$4 curve, aligns the qualitative singularity structure of this damped system with that of the undamped and pure-power nonlinearities. This provides rigorous justification for the universality of the blow-up mechanism as encoded by the ODE model and self-similar profile, even in the presence of time-dependent dissipative forces.

On the theoretical side, these results significantly broaden the understanding of how time-dependent, non-variational perturbations interact with hyperbolic evolutions to structure singularity formation, and the method of proof is sufficiently robust to encompass a more general class of derivative and mixed nonlinearities.

Future work may analyze the persistence of higher regularity ($$D_- \phi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},\qquad D_+ \psi = 2^{-p}|\phi + \psi|^p - \frac{\mu}{1 + t} \frac{\phi + \psi}{2},$$5, analyticity) of the blow-up curve, the role of weaker/more singular damping, blow-up in multidimensional settings, non-radial geometries, or systems. The approach is also relevant for the study of stability/allocation of mass in numerical blow-up detection schemes for damped wave models.

Conclusion

This paper rigorously proves that for the nonlinear wave equation with time-derivative nonlinearity and scale-invariant damping, any solution with large enough, smooth data develops blow-up along a unique, continuously differentiable space-like curve. The blow-up profile is precisely characterized, and the rate is shown to be invariant under damping. The analytic techniques employed showcase how geometric and scaling analysis in characteristic coordinates can be extended to non-conservative, non-variational equations, yielding strong regularity and uniqueness properties of the blow-up set. No singular (nondifferentiable) points occur in the blow-up boundary under the stated assumptions.

Reference:

"Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping" (2604.03848)