Focusing Inhomogeneous Hartree Equation

• The focusing inhomogeneous Hartree equation is a nonlinear model characterized by a spatially singular weight and an energy-critical nonlocal convolution nonlinearity.
• The analysis employs Strichartz estimates, Hardy–Littlewood–Sobolev inequalities, and weighted Sobolev embeddings to establish local well-posedness and small-data global solutions.
• The model highlights threshold phenomena with ground state dynamics, using virial identities to demarcate the global versus blow-up solution regimes.

The focusing generalized inhomogeneous Hartree equation provides a nonlocal model for nonlinear dispersive dynamics, incorporating both a spatial inhomogeneity and an energy-critical convolution nonlinearity. In recent years, the rigorous analysis of its local and global behavior has drawn considerable attention, especially regarding the subtle interplay between singular weights, nonlocal potentials, and scaling-critical nonlinearities. The core equation is

$$i\partial_t u + \Delta u = -\,|x|^{-b}\;\bigl(|x|^{-\gamma}\ast |x|^{-b}|u|^p\bigr)\;|u|^{p-2}\,u, \qquad u(0,x)=u_0(x)\in H^1(\R^n)$$

with dimensional parameter $n\geq3$, spatial inhomogeneity exponent $b > 0$, Riesz kernel exponent $\gamma \in (0,n)$, and nonlinear power $p\geq2$. The focusing nature is signified by the choice of the negative sign. This model unifies, extends, and challenges classical results on local and global well-posedness, threshold phenomena, and critical dynamics.

1. Equation Structure and Energy-Criticality

The canonical form as treated in Kim–Saanouni–Seo (Kim et al., 25 May 2026) introduces both a spatially inhomogeneous weight $|x|^{-b}$ and a nonlocal interaction via a Riesz convolution operator $|x|^{-\gamma} * (\cdot)$. The energy-critical exponent is determined by requiring the invariance of the homogeneous $\dot H^1$ norm under the scaling

$$u(t,x)\mapsto u_\lambda(t,x)=\lambda^{\frac{2-2b+\gamma}{2(p-1)}}u(\lambda^2 t,\lambda x)$$

so that

$p = 2 + \frac{\gamma - n + 4 - 2b}{n-2}.$

This ensures that the equation is critical for the Sobolev space $n\geq3$0, i.e., the scaling leaves both the energy and the $n\geq3$1 norm invariant.

2. Local and Small-Data Global Well-Posedness

Kim–Saanouni–Seo prove that for all dimensions $n\geq3$2 and for parameter ranges

$n\geq3$3

there exists $n\geq3$4 such that the Cauchy problem has a unique solution in

$n\geq3$5

for all Schrödinger-admissible pairs $n\geq3$6. The solution map is continuous with respect to initial data. In particular, for sufficiently small data in $n\geq3$7, the solution exists globally and scatters. This result fills the remaining parameter gaps, notably for small $n\geq3$8 and extreme $n\geq3$9, and, in dimensions $b > 0$0, it closes the last previously unresolved regime (Kim et al., 25 May 2026).

3. Analytical Techniques and Fundamental Estimates

The well-posedness analysis rests on three principal analytical tools:

• Strichartz Estimates: Dispersive and smoothing space-time inequalities for the linear Schrödinger propagator, extended to all admissible pairs $b > 0$1.
• Hardy–Littlewood–Sobolev Inequality: Controls the nonlocal convolution, with $b > 0$2 estimated in Lebesgue spaces.
• Weighted Sobolev Embedding (Caffarelli–Kohn–Nirenberg): Deals with the singular weight $b > 0$3, embedding weighted $b > 0$4 norms of $b > 0$5 into homogeneous Sobolev spaces. Specifically,

$b > 0$6

for suitable exponents, provided $b > 0$7 and $b > 0$8.

The nonlinearity is controlled in dual Strichartz norms: $b > 0$9 where $\gamma \in (0,n)$0 denotes the Hartree nonlinearity and $\gamma \in (0,n)$1 are the relevant Strichartz spaces.

A fixed-point argument in the intersection space $\gamma \in (0,n)$2 completes the local theory. This approach accommodates the full range of admissible data below the mass/energy threshold.

4. Threshold Phenomena and Ground States

The critical threshold for global versus blow-up dynamics is organized by the ground state $\gamma \in (0,n)$3, i.e., a radial positive solution to the nonlinear elliptic problem

$\gamma \in (0,n)$4

which minimizes an associated variational Gagliardo–Nirenberg functional. Small data global existence and scattering hold for data beneath explicit thresholds in energy and gradient norm—see, for instance, the full Kenig–Merle classification and concentration–compactness schemes in related models (Guzmán et al., 2023, Arora et al., 2019).

The sharp dichotomy is: for data with energy and critical norm below those of $\gamma \in (0,n)$5, solutions are global and scatter; otherwise, finite-time blow-up or norm divergence is possible, particularly in the focusing case.

5. Blow-Up and Virial/Morawetz Identities

Finite-time blow-up is demonstrated via localized virial identities, adapted for inhomogeneous and nonlocal nonlinearities: $\gamma \in (0,n)$6 for carefully chosen radial weights $\gamma \in (0,n)$7. These functionals encode the convexity or concavity properties leading to blow-up, and are crucially modified to control the error terms from the weights and the convolution.

In the energy-critical setting, conservation laws for mass and energy robustly complement these arguments. When $\gamma \in (0,n)$8 and the critical gradient norm is exceeded, the virial action forces finite-time blow-up.

6. Extensions, Open Problems, and Significance

Kim–Saanouni–Seo highlight that the weighted Sobolev embedding/Strichartz approach is versatile and extends beyond energy-criticality to both super- and sub-critical exponents, as well as to other classes of inhomogeneous convolution-type problems (Kim et al., 25 May 2026). It relaxes previous reliance on Lorentz-space or Hardy inequality techniques (Kim et al., 2021, Kim, 2022).

Global theory for large data, particularly for the focusing energy-critical regime, remains open except under mass/energy thresholds. The use of concentration–compactness and rigidity methodology (Kenig–Merle) is expected to further clarify the full classification of dynamics in the critical setting, potentially leading to a complete global theory. The inhomogeneous Hartree model thus serves as a testbed for understanding threshold dynamics, dispersive blow-up, and deep harmonic analysis in nonlocal, singular settings.