Quantitative Blow-up vs Scattering Dichotomy

• Quantitative blow-up versus scattering dichotomy is defined by sharp ground state thresholds and variational functionals that precisely dictate solution behavior in dispersive PDEs.
• The analysis employs virial identities, concentration-compactness, and scaling-invariant techniques to distinguish between global scattering and finite-time blow-up.
• Extensions to energy-supercritical, fractional, and coupled models demonstrate the robustness of these criteria across a wide spectrum of nonlinear dispersive systems.

A quantitative blow-up versus scattering dichotomy refers to the precise, threshold-governed separation between solutions to dispersive nonlinear PDEs that scatter (i.e., become asymptotically linear or decay) and those that blow up (i.e., experience norm divergence, singularity formation, or rapid energy concentration) in finite or infinite time. This dichotomy is established using a sharp division in terms of ground state quantities and associated variational functionals, and is now a central paradigm in nonlinear dispersive dynamics.

1. Ground State Thresholds and Variational Characterizations

A central feature in quantitative dichotomy results is the identification of a ground state—typically a nontrivial, positive, radial solution of an associated elliptic equation—which saturates a sharp Gagliardo–Nirenberg or Sobolev inequality. The mass, energy, and additional virial or Pohozaev functionals evaluated at the ground state determine the threshold for dynamics:

• For nonlinear Schrödinger equations (NLS), the ground state $Q$ is the unique minimizer of a variational problem related to the sharp Gagliardo–Nirenberg inequality. Scaling invariants such as

$\delta(u) = M(u)^{s_c} E(u)^{1-s_c},$

and kinetic thresholds such as $K(u) = M(u)^{1-s_c} \|\nabla u\|_{L^2}^{s_c}$, delineate the landscape.

• In models with potentials, inhomogeneities, or nonlocal nonlinearities (e.g., Chern–Simons–Schrödinger, Hartree, inhomogeneous NLS), the ground state is adapted by solving the corresponding constrained variational problem and using optimal constants dictated by the ground state profile (Georgiev et al., 2022, Miller, 9 Dec 2024, Guo et al., 2017).

2. Dichotomy Theorems: Sharp Division of Dynamics

A general form of the quantitative dichotomy is as follows:

• Below the threshold: If the initial data is below the ground state in both energy/mass and an auxiliary (often virial) functional—e.g.,

$\delta(u_0) < \delta(Q), \quad K(u_0) < K(Q),$

then the solution is global and scatters in $H^1$ or the appropriate Sobolev space. This is established by showing coercivity of the functional, precluding norm blow-up.

• Above or at the threshold: If the initial data exceeds the ground-state kinetic threshold (or equivalent) and an associated sign condition holds (often a negative virial), then finite-time blow-up or "grow-up" occurs:

$$\delta(u_0) < \delta(Q), \quad K(u_0) > K(Q), \quad V_t(0) \leq 0 \implies \text{finite-time blow-up}$$

• In many frameworks, the equality $\delta(u_0) = \delta(Q)$ or $K(u_0) = K(Q)$ corresponds to stationary or traveling ground states, or marks a bifurcation between scattering and blow-up (Akahori et al., 2010).

3. Quantitative Criteria and Scaling-Invariance

Beyond qualitative existence assertions, recent advances provide quantitative bounds and exact regimes for scattering versus blow-up, extending well beyond the classical Kenig–Merle threshold theory:

• For finite variance data, Duyckaerts–Roudenko-type criteria combine mass, energy, variance, and the variance time derivative in a scaling-invariant algebraic constraint:

$$\mathcal{ME}[u_0] \left(1 - \frac{V_t(0)^2}{32 E[u_0] V(0)} \right) \leq 1$$

and similar criteria for inhomogeneous or nonlocal models (Duyckaerts et al., 2014, Campos et al., 2020, Miller, 9 Dec 2024).

• In the energy-supercritical or fractional settings, scattering is ensured under scale-invariant norm control or even slow growth of critical norms—substantially refining the dichotomy mechanism (Bulut, 2020, Guo et al., 2017).
• Such results are robust to the introduction of potentials, inhomogeneity, or lack of scaling invariance, provided the threshold and functionals are adjusted appropriately (Gou et al., 2023, Gou et al., 26 Aug 2024, Ji et al., 30 Nov 2024).

4. Analytical and Virial-Morawetz Toolkits

The central technical apparatus spans:

• Virial and localized virial identities: These yield sign-definite second-order differential inequalities for quantities such as the variance $V(t) = \int |x|^2 |u(x,t)|^2$, often leading to concavity arguments that force blow-up, or monotonicity that underpins scattering.
• Sharp Gagliardo–Nirenberg inequalities and variational coercivity: Exploited to control nonlinear terms and maintain invariant sets below threshold, crucial for both scattering and blow-up regimes.
• Concentration–compactness and rigidity: Particularly in the borderline and critical cases, these methods rule out the existence of non-scattering "critical elements," often via profile decomposition and Morawetz estimates (Hamano et al., 2023).
• Strichartz and Morawetz estimates: These control spacetime norms and enable interpolation between local bounds and global behavior (Georgiev et al., 2022, Guo et al., 2018).

5. Extensions: Models, Geometries, and Quantitative Refinement

The dichotomy paradigm is now pervasive across a broad range of dispersive PDEs:

• Nonlinear wave and Klein–Gordon equations: The Strauss and Fujita exponents delineate blow-up versus scattering in semilinear wave (and damped wave) equations, with critical exponents shifting based on damping regime, space dimension, and nonlinearity (Bernhardt et al., 19 Apr 2024, Lai et al., 2017, Cheng et al., 2020).
• Nonlinear systems and coupled models: The Zakharov system and other coupled dispersive-wave systems show analogous dichotomies with thresholds tied to the wave component's mass or energy (Guo et al., 2018).
• Metric and geometric effects: The structure of the dichotomy is retained in hyperbolic space or other non-Euclidean geometries with threshold functionals adapted to sharp Sobolev or Poincaré inequalities (Banica et al., 2014).
• Equations without scaling or translation invariance: The dichotomy survives with significant technical modification, with thresholds and virial functionals keyed to the specific competition of nonlinearities and their weights (Gou et al., 2023, Gou et al., 26 Aug 2024).

6. Applications and Examples

Quantitative dichotomy theorems yield constructive regimes and explicit initial data scenarios:

• Quadratic-phase modulation: Data far above the ground-state energy but satisfying the mass-variance-kinetic constraints can lead to scattering or blow-up, depending on the sign and size of modulation parameters (Duyckaerts et al., 2014).
• Energy-supercritical and fractional models: For models where classical theory gives no conclusion, the explicit criteria extend the dichotomy and give uniform scattering bounds under critical norm control (Bulut, 2020, Guo et al., 2017).
• Noncompact or nonradial data: The dichotomy extends to nonradial and even noncompact data for scattering, provided the mass escape criteria and localized Morawetz identities are met (Gou et al., 2023, Bernhardt et al., 19 Apr 2024).

7. Significance and Open Challenges

The quantitative blow-up versus scattering dichotomy unifies disparate results and provides explicit, checkable criteria for the global behavior of solutions to nonlinear dispersive PDEs across a vast model spectrum. It underpins the analysis of soliton resolution, instability, and singularity formation, while remaining robust under perturbations including nonlocality, nonlinearity, external fields, and lack of symmetries. Future developments center on extending these techniques to broader classes, including higher-dimensional, non-integrable, random, or stochastic dispersive systems, with an eye toward the explicit computation of sharp thresholds in even more complex regimes (Georgiev et al., 2022, Miller, 9 Dec 2024, Duyckaerts et al., 2014, Gou et al., 2023).