Sharp criteria for a degenerate diffusion-aggregation system with the intermediate exponent
Tiantian Zhou, Li Chen, Yutian Lei
TL;DR
This work analyzes a multidimensional degenerate diffusion-aggregation equation with a nonlocal singular kernel in the intermediate diffusion range, establishing two sharp criteria that separate global existence from finite-time blow-up based on initial data. The first criterion uses the $L^{m_*}$-norm and an initial free-energy bound, while the second uses an $L^m$-norm threshold tied to the extremal function of the Hardy-Littlewood-Sobolev inequality; the authors prove equivalence between these criteria and between the underlying initial-energy conditions. The methodology combines regularized weak solutions, energy dissipation, moment estimates, and Lions-Aubin compactness, enabling global existence results on the whole space without requiring $L^\infty$ initial data and yielding a unified classification of solution behavior from two complementary perspectives. These results advance the understanding of nonlocal aggregation-diffusion dynamics and provide precise thresholds for global behavior versus blow-up with potential applications in biological pattern formation and related PDE models.
Abstract
In this paper, we investigate a multi-dimensional nonlocal degenerate diffusion-aggregation equation with a diffusion exponent $m$ in the intermediate range $\frac{2d}{2d-γ}<m<\frac{d+γ}{d}$, where the nonlocal aggregation term is given by singular potential $|x|^{-γ}$, $0<γ\leq d-2$. Under two different assumptions on the initial data, we establish two sharp criteria (i.e., the critical thresholds in Theorem 1.1 and Theorem 1.2) governing the global existence and finite-time blow-up of solutions. Once the initial free energy is less than a constant that depends on the total mass (or depends on the extremum function of the Hardy-Littlewood-Sobolev inequality), the first criterion depends on the relationship between the $L^{\frac{2d}{2d-γ}}$-norm of initial data and total mass, while the second relies on the relationship between the $L^m$-norm of initial data and extremal function. In the discussion of the second criterion, we do not require $L^\infty(\mathbb{R}^d)$ boundedness of the initial data, which is necessary in reference \cite{B}. Furthermore, with the help of moment estimate, we manage to prove the compactness argument on the whole space by using the Lions-Aubin Lemma. Importantly, we demonstrate that the two initial free energy conditions on which two criteria are based are equivalent. Building on this, we further prove that the two sharp criteria themselves are also equivalent, thereby unifying the classification results obtained from two different approaches.