Self-Similar Singular Solutions in Nonlinear PDEs

• Self-similar singular solutions are defined as scaling invariant profiles that capture the asymptotic behavior near singularities in nonlinear PDEs.
• They reduce complex PDEs to tractable ODE systems using similarity variables, allowing precise characterization of blow-up rates and profile matching.
• These solutions provide practical insights into phenomena like fluid dynamics and reaction-diffusion, offering a unified approach to understanding singularity formation.

Self-similar singular solutions are a fundamental class of solutions to nonlinear partial differential equations (PDEs) and related nonlinear problems, in which the solution exhibits invariance under a one-parameter scaling group near a singularity (such as blow-up, formation of sharp interfaces, or loss of regularity). These solutions often capture the leading-order asymptotic structure of singularity formation in various models across fluid dynamics, statistical physics, geometric flows, reaction-diffusion, and mathematical biology. Their construction and classification rely on an overview of dynamical systems techniques, singular boundary value problems, and phase-plane reductions, and their rigorous analysis illuminates both the mechanisms and limitations of singularity formation in nonlinear evolution equations.

1. Mathematical Formulation of Self-Similar Singular Solutions

A self-similar singular solution typically takes the form

$$u(x, t) = (T-t)^{-\alpha} F\left(\frac{x - x_0}{(T-t)^\beta}\right)$$

as $t \nearrow T$ (finite time blow-up), or

$$u(x, t) = t^{-\alpha} F\left(\frac{x}{t^\beta}\right)$$

for $t \to \infty$ (global attractor profiles), where $F$ is a multi-dimensional profile determined by a nonlinear ODE or an integro-differential equation. The exponents $\alpha, \beta$ are determined by the underlying scaling invariance of the equation. This reduction typically transforms the PDE into a stationary problem for $F$, whose analysis involves boundary (or matching) conditions at the singular point (e.g., $|x| = 0$ or $\xi = 0$) and at infinity (or an interface).

The paradigm is exemplified in the monograph-level survey "The role of self-similarity in singularities of PDE's" (0812.1339), which formalizes the reduction of singularity formation to the study of fixed points and their attractivity in a dynamical system for the profile.

2. Canonical Examples and Physical Origin

Self-similar singular solutions arise in a broad array of PDEs:

• Parabolic Equations (Semilinear Heat Equation): For $u_t - \Delta u = u^p$, singular self-similar blow-up profiles are characterized by ODEs in the similarity variable and exhibit power-law singularities at the origin. Under suitable conditions, such as $p \geq (n+2)/(n-2)$, uniqueness and explicit representations are attained (Quittner, 2016).
• Boundary Layer Equations in Fluid Mechanics: The classical Prandtl boundary-layer system admits self-similar reductions, yielding third-order ODEs with singular data at infinity, with rigorous analyticity of stable- or center-manifolds near singular points (e.g., matching at $\tau\to-\infty$). A canonical example is the Diesperov–Schlichting reduction for zero pressure gradient boundary layers, leading to mixing layer and jet-type profiles (Konyukhova et al., 2018).
• Reaction-Diffusion with Potentials: Self-similar grow-up (or absorption) profiles appear in models such as $u_t = \Delta u^m + |x|^\sigma u^p$, where singular self-similar solutions can prevent finite-time blow-up due to the interplay of diffusion and singular potential growth (Iagar et al., 2021, Iagar et al., 1 Jun 2024).
• Geometric Flows: Mean curvature flow and curve shortening flow admit self-shrinkers and self-expanders as central singular solutions, with associated ODEs for curvature and explicit classification of planar and space curves (Adames, 2015).
• Fluid Models and Vorticity Stretching: Singular solutions to model vorticity equations of the form $\omega_t = Z_{11} \omega \, \omega$, with $Z_{11}$ a Calderón-Zygmund operator, show self-similar finite-time blow-up and require delicate analysis of nonlocal operator degeneracy (Du et al., 17 Dec 2025).

3. Analytical Framework and Dynamical Systems Approach

Singular self-similar reduction produces highly nontrivial boundary value problems:

• The profile equation is typically an autonomous or non-autonomous ODE (or ODE system), often third-order or higher.
• Singularities at one end (e.g., $\tau \to -\infty$) may correspond to equilibrium points with complicated Jordan block structures, requiring center-stable (or unstable) manifold theory and local Lyapunov series expansions for the precise asymptotic matching (Konyukhova et al., 2018).
• Nontrivial matching conditions at both ends (e.g., origin and infinity, or finite interface points for compactly supported solutions) necessitate shooting methods and scaling arguments to obtain unique profiles.

This structure is highlighted in the reduction of PDEs to infinite-dimensional dynamical systems in $\tau = -\log(T-t)$, where fixed points (type I singularities), center manifolds (type II, with logarithmic corrections), limit cycles (discrete self-similarity), and even more complex attractors arise (0812.1339).

A crucial feature is the robust existence of singular, sometimes explicit, similarity profiles under critical exponent regimes (e.g., the fast-diffusion equation for $0 < m < (n-2)/n$ (Hui, 2023), or parabolic-elliptic Keller–Segel for $d\geq 3$ (Biler et al., 2020)) and uniqueness in suitable function spaces (Quittner, 2016).

4. Classification, Uniqueness, and Asymptotics

A central theme is the classification of self-similar singular solutions:

• Uniqueness and Singular Manifolds: For the semilinear heat equation with $p > (n+2)/(n-2)$, there is a unique, positive, singular similarity profile $U_*(r) = L r^{-2/(p-1)}$ (Quittner, 2016).
• Phase Space Topology: For the porous medium equation with absorption $u_t = \Delta u^m - |x|^{\sigma}u^p$, the existence and uniqueness of very singular, compactly supported self-similar profiles is established in the regime $m < p < m + (\sigma+2)/N$, with other regimes admitting non-compactly supported, fat-tailed profiles (Iagar et al., 1 Jun 2024).
• Asymptotics: Detailed analysis near singular points, such as expansion in exponential Lyapunov series, yields very precise control over admissible profile behaviors. For example, solutions to the boundary layer equation (Konyukhova et al., 2018) admit expansions

$$\Phi(\tau) = -a + d e^{a \tau} + \sum_{k=2}^\infty h_k d^k e^{k a \tau}$$

on a one-dimensional stable manifold, with existence and uniqueness selected by boundary data at other endpoints.

• Role of Non-decaying and Fat-tailed Profiles: In several contexts, admissible profiles may not decay at infinity, instead exhibiting power-law or even slower growth, as in self-similar coagulation equations for Smoluchowski’s kernel. In such settings, uniqueness or non-existence is determined by careful shell-by-shell energy inequalities and refined Calderón–Zygmund representations for the pressure or nonlocal terms.

5. Numerical and Computational Schemes

Rigorous computational approaches play a pivotal role in the analysis and construction of singular self-similar solutions:

• Shooting and Scaling: The boundary value problem is often reduced to shooting from an approximated stable (or unstable) manifold at a large, negative starting point, with iterative adjustment of free parameters (such as derivative values) to enforce matching conditions (Konyukhova et al., 2018).
• High-order Integrators and Series Truncation: Effective use of adaptive, high-order Runge–Kutta and stiff solvers, combined with truncation of series expansions, enables precise tracking of solution behavior up to matching points, particularly when the singular nature forbids standard initial value methods (Konyukhova et al., 2018).
• Analytic-Numeric Matching: In cases with sharply separated domains (e.g., near the origin and infinity), hybrid approaches combine analytic expansions and rigorous interval arithmetic to join local and global solution behaviors (Dahne et al., 7 Oct 2024).

These methods are essential in multidimensional and strongly nonlinear cases, where explicit solutions are unavailable.

6. Physical Interpretations and Applications

Self-similar singular solutions have interpretation as universal asymptotics for singularity formation, interface blow-up, jet formation, and profile selection in a range of settings:

• Jet and Mixing Layer Flows: The boundary-layer self-similar solutions describe classical planar jets, semi-jets, near-wall jets, and mixing layers in terms of the parameter $m$ controlling the nature of the outer flow (Konyukhova et al., 2018).
• Reaction-Diffusion and Growth Models: In reaction-diffusion with singular potentials, self-similar solutions determine the threshold between global grow-up and finite-time blow-up, providing exact global attractors and demonstrating the suppression of singularities in certain parameter regimes (Iagar et al., 2021).
• Pattern Formation and Fat-tails: In coalescence models such as Smoluchowski’s coagulation equation with singular kernels, self-similar fat-tailed solutions characterize the persistent distribution of large clusters (Niethammer et al., 2014).
• Chemoattractant Collapse: In Keller–Segel type models, self-similar singular solutions describe the aggregation and possible blow-up of mass distributions (Biler et al., 2020).

Their utility lies in both providing explicit descriptions of singular behaviors and serving as templates for the large-time or near-singularity evolution in more general, non-symmetric solutions.

7. Open Problems and Limitations

Despite the comprehensive classification in several model settings, limitations remain:

• Rigorous exclusion of singular self-similar solutions: In certain systems, for example, Euler and electron-MHD, powerful nonexistence and structure theorems restrict or rule out self-similar singularities—except possibly for highly non-decaying, infinite-energy, or subtly nonlocal profiles (Pomeau, 2016, Xue, 2014, Dai et al., 1 May 2024, Xue, 2014). The algebraic decay of profiles often leads to divergence of the total energy, lying outside the range of standard nonexistence theorems, but physically relevant solutions must reconcile this.
• Discretely and Locally Self-Similar Solutions: There exist classes of solutions showing self-similarity only under discrete scaling or in local neighborhoods, with partial classification and nonexistence results depending on precise decay or integrability conditions (Xue, 2014, Xue, 2014).
• Type II/Anomalous Scaling and Multimode Attractors: Beyond pure scaling (type I), additional complexity arises with center-manifold driven logarithmic corrections, periodic "echoes" in log-time (discrete self-similarity), or even chaotic behavior in the similarity variables (0812.1339). The full genericity and universality of these more exotic regimes in physical PDEs remain open.
• Infinite Dimensional Dynamics: The reduction to finite-dimensional slow manifolds is established in many cases, but high-dimensional or infinite-dimensional phenomena can occur, especially in coupled systems and with nonlocal nonlinearities.

A broad direction remains to rigorously connect the local or formal singular similarity solutions to global (initial value) PDE dynamics, and to fully understand their role as universal singularity attractors or as obstacles in certain settings.

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The theory of self-similar singular solutions thus provides a rigorous framework for classifying and analyzing the mechanisms and universality of singularity formation in nonlinear PDEs, informed by an overview of dynamical system, analytic, and computational methods across diverse physical and mathematical contexts (0812.1339, Konyukhova et al., 2018, Quittner, 2016, Iagar et al., 2021, Iagar et al., 1 Jun 2024, Adames, 2015, Du et al., 17 Dec 2025, Hui, 2023).