Solutions of a non‐local aggregation equation: Universal bounds, concavity changes, and efficient numerical solutions

• Published in: Mathematical methods in the applied sciences Vol. 43; no. 8; pp. 5398 - 5429
• Main Authors: Fellner, Klemens, Hughes, Barry D.
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30-05-2020
• Subjects: Agglomeration; aggregation models; Boundary value problems; Concavity; Convexity; Differential equations; genetics and population dynamics; initial value problems for non‐linear first‐order systems; integro‐partial differential equations; method of characteristics; numerical solutions; Ordinary differential equations; Symmetry; Upper bounds
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.6281

We consider a one‐dimensional aggregation equation for a non‐negative density
ρ(x,t) associated with a quartic potential
W(x)=βx2+δx4 (
δ>0,
β∈R). We show that for the case of symmetric initial data [
ρ(x,0)≡ρ(−x,0)], the solution of the aggregation equation can be expressed in terms of an explicit function of
x,
L(t), and
Φ(t), where the functions
L(t) and
Φ(t) are determined by an ordinary differential equation initial value problem, the numerical solution of which is relatively straightforward. The function
L(t), which can be interpreted as an upper bound for the radius of the support of
ρ(x,t), is finite for all
t>0, even if the support of
ρ(x,0) is unbounded, while
Φ(t) is a potential related to the time integral of the second spatial moment
ϕ(t) of
ρ(x,t). We develop various bounds on
L(t) and
ϕ(t) and a number of results concerning convexity and monotonicity that require no knowledge of
ρ(x,0) other than its symmetry. We show that many, but not all, of these results persist if the assumption of symmetry is relaxed. Our general results are tested against numerical solutions for two examples of symmetric
ρ(x,0). We also exhibit some counterintuitive behaviour of the model, by finding conditions under which
ρxx(0,t)→∞ as
t→∞, even if we start with
ρxx(0,0)<0 and take
β≥0, which ensures ultimate collapse to a single Dirac component at the origin.