Sharp criteria for a degenerate diffusion-aggregation system with the intermediate exponent

TL;DR

This paper establishes sharp criteria for the global existence or finite-time blow-up of solutions to a multi-dimensional degenerate diffusion-aggregation equation with a nonlocal singular potential, unifying two different analytical approaches.

Contribution

It introduces two sharp, equivalent criteria for solution behavior, applicable under less restrictive initial data conditions, and unifies different analytical methods for this class of equations.

Findings

Established sharp criteria for global existence and blow-up.

Proved equivalence of two initial free energy conditions.

Unified different analytical approaches for the diffusion-aggregation system.

Abstract

In this paper, we investigate a multi-dimensional nonlocal degenerate diffusion-aggregation equation with a diffusion exponent $m$ in the intermediate range $\frac{2d}{2d-\gamma}<m<\frac{d+\gamma}{d}$, where the nonlocal aggregation term is given by singular potential $|x|^{-\gamma}$, $0<\gamma\leq d-2$. 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 $L^{\frac{2d}{2d-\gamma}}$-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.