TY - JOUR

T1 - Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory

AU - Lankeit, Johannes

AU - Winkler, Michael

PY - 2019/7/9

Y1 - 2019/7/9

N2 - We consider the degenerate parabolic equation with nonlocal source given by ut=uΔu+u∫ℝn|∇u|2, which has been proposed as a model for the evolution of the density distribution of frequencies with which different strategies are pursued in a population obeying the rules of replicator dynamics in a continuous infinite-dimensional setting. Firstly, for all positive initial data u0 ∈ C0(ℝn) satisfying u0 ∈ Lp(ℝn) for some p ∈ (0, 1) as well as ∫ℝnu0=1, the corresponding Cauchy problem in ℝn is seen to possess a global positive classical solution with the property that ∫ℝnu(⋅,t)=1 for all t > 0. Thereafter, the main purpose of this work consists in revealing a dependence of the large time behavior of these solutions on the spatial decay of the initial data in a direction that seems unexpected when viewed against the background of known behavior in large classes of scalar parabolic problems. In fact, it is shown that all considered solutions asymptotically decay with respect to their spatial H1 norm, so that ℰ(t):=∫0t∫ℝn|∇u(⋅,t)|2,t>0, always grows in a significantly sublinear manner in that (0.1) ℰ(t)t→0ast→∞; the precise growth rate of ℰ, however, depends on the initial data in such a way that fast decay rates of u0 enforce rapid growth of ℰ. To this end, examples of algebraic and certain exponential types of initial decay are detailed, inter alia generating logarithmic and arbitrary sublinear algebraic growth rates of ℰ, and moreover indicating that (0.1) is essentially optimal.

AB - We consider the degenerate parabolic equation with nonlocal source given by ut=uΔu+u∫ℝn|∇u|2, which has been proposed as a model for the evolution of the density distribution of frequencies with which different strategies are pursued in a population obeying the rules of replicator dynamics in a continuous infinite-dimensional setting. Firstly, for all positive initial data u0 ∈ C0(ℝn) satisfying u0 ∈ Lp(ℝn) for some p ∈ (0, 1) as well as ∫ℝnu0=1, the corresponding Cauchy problem in ℝn is seen to possess a global positive classical solution with the property that ∫ℝnu(⋅,t)=1 for all t > 0. Thereafter, the main purpose of this work consists in revealing a dependence of the large time behavior of these solutions on the spatial decay of the initial data in a direction that seems unexpected when viewed against the background of known behavior in large classes of scalar parabolic problems. In fact, it is shown that all considered solutions asymptotically decay with respect to their spatial H1 norm, so that ℰ(t):=∫0t∫ℝn|∇u(⋅,t)|2,t>0, always grows in a significantly sublinear manner in that (0.1) ℰ(t)t→0ast→∞; the precise growth rate of ℰ, however, depends on the initial data in such a way that fast decay rates of u0 enforce rapid growth of ℰ. To this end, examples of algebraic and certain exponential types of initial decay are detailed, inter alia generating logarithmic and arbitrary sublinear algebraic growth rates of ℰ, and moreover indicating that (0.1) is essentially optimal.

UR - http://www.scopus.com/inward/record.url?scp=85068870703&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1805.11492

DO - 10.48550/arXiv.1805.11492

M3 - Article

AN - SCOPUS:85068870703

VL - 233

SP - 249

EP - 296

JO - Israel journal of mathematics

JF - Israel journal of mathematics

SN - 0021-2172

IS - 1

ER -