Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.26513

Blow-up bounds under profile approximation for superconformal wave equations

Firas Kaabi
View original source ↗Read source document ↗Export metadata ↓

Abstract

For the focusing semilinear wave equation in the superconformal, Sobolev-subcritical range, we prove an upper blow-up bound at the ODE rate near a non-characteristic point, assuming approximation by signed multiples of explicit stationary profiles in similarity variables. The comparison parameters may vary with time, and the error is measured only in a degenerate energy norm on a small extension of the unit ball. The main step is a quantitative exclusion of concentration at the boundary of the profile family. It follows from the nonnegativity of the superconformal Hamza--Zaag energy at admissible vertices and an upper perturbation estimate. We also exclude families with an isolated dominant profile and, under a noncancellation condition, clusters of comparable profiles. For the latter, we derive a limiting energy in $\R^{N+1}$ which accounts for the interactions between profiles.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Firas Kaabi. 2026-10-19. Blow-up bounds under profile approximation for superconformal wave equations. https://arxiv.org/abs/2610.26513

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the dispersive estimates for the discrete Schrödinger equation on a honeycomb lattice

The discrete Schrödinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of waves on graphene. For free evolution, we first show that the degenerate frequencies of the dispersion relation are completely characterized by three symmetric periodic curves (Theorem 2.1), and that the three curves meet at Dirac points where conical singularities appear (see Figure 2.1). Based on this observation, we prove the $L^1\to L^\infty$ dispersion estimates for the linear flow depending on the frequency localization (Theorem 2.3). Collecting all, we obtain the dispersion estimate with $O(|t|^{-2/3})$ decay as well as Strichartz estimates. As an application, we prove small data scattering for a nonlinear model (Theorem 2.10). The proof of the key dispersion estimates is based on the associated oscillatory integral estimates with degenerate phases and conical singularities at Dirac points. Our proof is direct and uses only elementary methods.

math.AP↗
arXiv2026

Well-posedness and Stability of Thermoelastic Suspension Bridges with Cattaneo Heat Conduction: The Role of Viscoelastic Memory

Thermoelastic suspension bridges operate under strong thermo-mechanical coupling, and the classical Fourier law fails to capture the finite speed of thermal propagation observed in low-conductivity materials such as concrete. This paper investigates a suspension bridge model in which the heat flux is governed by the Cattaneo law with relaxation time $τ$, and in which the bridge deck may or may not possess viscoelastic memory. Using operator semigroup theory, we establish well-posedness and derive sharp decay rates for both regimes. For the memoryless system ($δ=0$), we show that the semigroup is polynomially stable with rate $t^{-1/2}$ when the structural parameters satisfy $\frac{ρ_1}κ = \frac{ρ_2}{b}$, and that the rate deteriorates to $t^{-1/4}$ otherwise; moreover, exponential stability fails whenever $χ_0 \neq 0$, or $χ_0 = 0$ and $χ_1 = 0$, where $χ_0$ and $χ_1$ are parameter-dependent coefficients. In sharp contrast, when viscoelastic memory is present ($δ=1$), the semigroup becomes exponentially stable, regardless of the structural parameter ratio. This reveals a regularizing role of viscoelastic memory: it converts polynomial decay into exponential decay and removes the parameter-ratio obstruction. The results provide a theoretical basis for exploiting inherent viscoelasticity in long-span bridges and for optimizing deck parameters in memoryless structures.

math.AP↗