Self-Similar Blow-Up Profiles

• Self-similar blow-up profiles are solutions to nonlinear PDEs that develop finite-time singularities through a spatiotemporal scaling ansatz.
• They are derived via reduction to a nonlinear ODE and classified based on parameters affecting multiplicity, oscillation counts, and interface localization.
• Analytical, dynamical systems, and numerical methods (including neural networks) provide insights into the existence, stability, and asymptotic behavior of these profiles.

Self-similar blow-up profiles describe solutions to nonlinear PDEs and fluid models that become singular in finite time via self-similar scaling—that is, where the solution exhibits specific spatiotemporal scaling collapse near the blow-up time. These profiles are central for classifying singularity formation, characterizing local structure near singularities, and determining rates and stability of blow-up across a broad range of nonlinear dispersive, diffusive, and hydrodynamic systems.

1. Self-Similar Ansatz, Scaling, and Fundamental ODEs

A self-similar blow-up profile is derived from ansatz

$$u(x,t) = (T-t)^{-\alpha} f\left( \xi \right), \qquad \xi = |x| (T-t)^{-\beta}$$

where $T$ is the blow-up time, and $(\alpha,\beta)$ are scaling exponents determined via balancing temporal and spatial scaling in the nonlinear PDE. For example, in the weighted quasilinear diffusion equation with source: $u_t = \Delta u^m + |x|^\sigma u^p,$ substitution of the self-similar form leads to exponents (for $L = \sigma(m-1) + 2(p-1)>0$)

$$\alpha = \frac{\sigma+2}{L},\qquad \beta = \frac{p-m}{L}.$$

The profile $f(\xi)$ then solves a nonlinear ODE (radially symmetric case in $\mathbb{R}^N$): $$\left( f^m \right)'' + \frac{N-1}{\xi} (f^m)' -\alpha f - \beta \xi f' + \xi^\sigma f^p = 0.$$ Boundary behavior at $\xi=0$ and decay as $T$0 are governed by possible types of profiles: positive-peak, source-type, or dead-core, each with precise asymptotics. Analogous reduction underlies profile construction for models ranging from parabolic-elliptic chemotaxis (Nguyen et al., 4 Mar 2025) to nonlocal fluid models (Huang et al., 2023, Chen, 14 May 2026).

2. Existence, Multiplicity, and Classification

The number and types of self-similar blow-up profiles depend delicately on the system’s parameters and symmetries:

• Multiplicity and oscillation counts: For the quasilinear diffusion equation above, for $T$1 and parameters $T$2 near $T$3, $T$4 near zero, there exist at least $T$5 distinct positive-peak and $T$6 distinct dead-core profiles, each specified by their number of local minima (Iagar et al., 2024). This multiplicity is established via topological shooting in the associated dynamical system of the profile ODE.
• Sobolev subcriticality: For source term exponent $T$7 strictly less than the Sobolev critical exponent, at least one nontrivial radially symmetric blow-up profile exists for $T$8 and any $T$9 (Iagar et al., 2024), generalizing classical semilinear theory.
• Non-existence thresholds: There exists a sharp threshold $(\alpha,\beta)$0 so that for any $(\alpha,\beta)$1, all such profiles are destroyed—no radially symmetric self-similar blow-up solution exists (Iagar et al., 2024).
• Localization and support: In critical-weight regimes, particularly when $(\alpha,\beta)$2 for weighted reaction-diffusion, all self-similar profiles are compactly supported and completely localized—the support lies within a universal radius depending on $(\alpha,\beta)$3 (Iagar et al., 2020). For $(\alpha,\beta)$4, no self-similar blow-up occurs; for large $(\alpha,\beta)$5 above certain threshold, only profiles that exhibit singularity formation at spatial infinity persist (Iagar et al., 2021).
• Matched asymptotic structures: In systems near criticality (e.g., thin film equations or slightly supercritical NLS), matched asymptotic expansions yield composite profiles composed of inner (peak or singular core), intermediate (matching layer), and outer (algebraic or exponential tail) regions, with precise parameter selection via global consistency (Dallaston, 2018, Bahri et al., 2019).

Table: Existence and Multiplicity Influences

Equation Type	Parameter Regimes	Existence/Multiplicity
Quasilinear diffusion	$(\alpha,\beta)$6 near $(\alpha,\beta)$7, $(\alpha,\beta)$8	Infinitely many positive-peak and dead-core profiles for each $(\alpha,\beta)$9 (Iagar et al., 2024)
Keller-Segel chemotaxis	$u_t = \Delta u^m + |x|^\sigma u^p,$0	Infinitely many radial self-similar profiles via matching at two scales (Nguyen et al., 4 Mar 2025)
Weighted reaction	large $u_t = \Delta u^m + |x|^\sigma u^p,$1	Only compactly-supported “grow-up at infinity” profiles survive (Iagar et al., 2019, Iagar et al., 2020)

3. Dynamical Systems, Fixed-Point, and Analytical Techniques

Detailed construction and classification of blow-up profiles is achieved via the following:

• Reduction to Autonomous Systems: The profile ODE is recast as a finite- or infinite-dimensional autonomous dynamical system in phase space—e.g., introducing variables $u_t = \Delta u^m + |x|^\sigma u^p,$2 tied to $u_t = \Delta u^m + |x|^\sigma u^p,$3 and its derivatives converts the ODE into a three-dimensional system, whose critical points and connecting orbits encode all admissible behaviors (Iagar et al., 2024, Iagar et al., 2021).
• Shooting and Bifurcation Arguments: By tracking unstable/stable manifolds of critical points, and sign-changing across invariant surfaces, one counts and classifies the number of profile solutions for given parameter values (Iagar et al., 2024). Bifurcation and matching at multiple spatial scales yield countable families of profiles (see Keller-Segel (Nguyen et al., 4 Mar 2025), NLS (Bahri et al., 2019)).
• Analytic Fixed-Point and Regularity: In nonlocal and higher-order equations, profiles are established as fixed points in weighted $u_t = \Delta u^m + |x|^\sigma u^p,$4 or Banach function spaces. Properties such as monotonicity, convexity, compactness are proved by carefully controlling integral operators (e.g., Hilbert transform, fractional Biot-Savart) and bootstrapping regularity (Huang et al., 2023, Chen, 14 May 2026).
• Numerical and Computational Methods: For complex multi-dimensional or nonlocal profile equations lacking closed-form, neural-network-based solvers (PINNs) provide high-precision numerical construction, robustly enforcing PDE, symmetry, and boundary conditions simultaneously and capturing even unstable branches (Wang et al., 2022).

4. Profile Behavior, Interface Types, and Asymptotics

Self-similar profiles may exhibit various local and global behaviors:

• Origin Behavior: Profiles may exhibit a positive-peak $u_t = \Delta u^m + |x|^\sigma u^p,$5, source-type vanishing $u_t = \Delta u^m + |x|^\sigma u^p,$6, or dead-core behavior (profile identically zero up to some interface $u_t = \Delta u^m + |x|^\sigma u^p,$7), with the regime determined by the equation parameters (Iagar et al., 2024, Iagar et al., 2020).
• Interface Structure: Near the edge of support (the “interface”), profiles display either
  • Type I (diffusive): $u_t = \Delta u^m + |x|^\sigma u^p,$8,
  • Type II (reactive): $u_t = \Delta u^m + |x|^\sigma u^p,$9,
  • with associated kinetic relationships for interface motion in the original variables (Iagar et al., 2020, Iagar et al., 2021).
• Decay and Spectral Properties: At infinity, blow-up profiles decay algebraically according to driven balance in the profile ODE, e.g., $L = \sigma(m-1) + 2(p-1)>0$0. In higher dimensions or with nonlocality, the decay may be non-integrable, corresponding to infinite-energy blow-up (Hou et al., 11 Mar 2026, Chen, 14 May 2026).
• Finite/Infinite Energy: Classical self-similar blow-up often presumes finite energy, yet for certain nonlocal or singular regimes, the natural profiles are infinite-energy, corresponding to profiles not decaying fast enough at infinity but still capturing the singular nature of physical observables (Hou et al., 11 Mar 2026).
• Stability and Attractivity: For several evolution equations with, e.g., time-derivative nonlinearities, generalized self-similar blow-up profiles form finite-codimension attractors in suitable function spaces, as shown via Lyapunov–Perron and spectral gap arguments (Raees et al., 17 Nov 2025).

5. Recent Developments and Application to Nonlocal and Fluid Systems

Contemporary research extends the self-similar blow-up machinery to new frontiers, including:

• Incompressible Euler and Axisymmetric Flows: Rigorous constraints and sharp lower bounds on the similarity scaling exponent $L = \sigma(m-1) + 2(p-1)>0$1 have been established—e.g., any finite-energy self-similar profile for 3D Euler must have $L = \sigma(m-1) + 2(p-1)>0$2, and stronger, $L = \sigma(m-1) + 2(p-1)>0$3 under outgoing or axisymmetric structure, effectively ruling out “violent” scaling and constraining possible singularity scenarios (Constantin et al., 19 Feb 2026).
• 1D and 3D Nonlocal Fluid Models: Exact and asymptotically self-similar profiles with precise regularity, compactness, or singular behavior have been constructed for nonlocal models like the generalized Constantin-Lax-Majda and Hou-Luo equations, elucidating blow-up scenarios in relation to axisymmetric Euler mechanics. In particular, exact $L = \sigma(m-1) + 2(p-1)>0$4 and $L = \sigma(m-1) + 2(p-1)>0$5 blow-up profiles for 1D/3D models are built via analytic contraction mapping arguments and fixed-point theorems, even when the 3D flows lack traditional strong decay (Chen, 14 May 2026, Chen, 14 May 2026).
• Robust Neural Network Frameworks: Physics-informed neural networks have been employed to construct precise 2D/3D self-similar blow-up profiles in boundary value formulation, automatically capturing the scaling exponent, symmetries, and ensuring smoothness, opening avenues for computer-assisted proofs of blow-up (Wang et al., 2022).
• Singular Multiscale and Two-Stage Blow-up: New numerical and theoretical results for degenerate initial data in nonlocal models show self-similar blow-up may proceed in two stages: initial smooth $L = \sigma(m-1) + 2(p-1)>0$6 blow-up followed by weak extension and $L = \sigma(m-1) + 2(p-1)>0$7 blow-up with singular self-similar profile formation, revealing richer structures than classical scenarios (Chen et al., 2 Apr 2026, Huang et al., 26 Mar 2026).

6. Broader Implications and Connections

Self-similar blow-up profiles are fundamental for:

• Classifying blow-up versus global regularity regimes and identifying critical thresholds (exponents, weights, Sobolev bounds) where new behavior emerges, such as multiplicity transitions, interface localization, or the annihilation of profiles as weights/parameters are varied (Iagar et al., 2024, Iagar et al., 2021).
• Guiding computational and analytic singularity searches in Navier-Stokes/Euler and related fluid models, providing benchmarks for numerics—e.g., constraining possible $L = \sigma(m-1) + 2(p-1)>0$8 singularity rates and excluding fast singularity formation for finite-energy flows (Constantin et al., 19 Feb 2026, Chen, 14 May 2026).
• Informing matched asymptotic and multi-scale analysis, allowing systematic construction of global blow-up structure from local model equations, and underpinning stability/attractivity studies for nonlinear evolution equations.

These advances collectively demonstrate that self-similar blow-up profiles are not only classification tools but constructive objects central to the theory and computation of singularities in nonlinear PDEs and fluid dynamics.