Details
| Original language | English |
|---|---|
| Pages (from-to) | 249-296 |
| Number of pages | 48 |
| Journal | Israel journal of mathematics |
| Volume | 233 |
| Issue number | 1 |
| Publication status | Published - 9 Jul 2019 |
| Externally published | Yes |
Abstract
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.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Israel journal of mathematics, Vol. 233, No. 1, 09.07.2019, p. 249-296.
Research output: Contribution to journal › Article › Research › peer review
}
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 -