Details
| Original language | English |
|---|---|
| Pages (from-to) | 954-983 |
| Number of pages | 30 |
| Journal | SIAM Journal on Mathematical Analysis |
| Volume | 49 |
| Issue number | 2 |
| Publication status | Published - 2017 |
| Externally published | Yes |
Abstract
We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for ut = uΔu + u ∫Ω |∇u|2 in bounded domains Ω ⊂ ℝn which arises in game theory. We prove that solutions converge to 0 if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with Ω; i.e., the finite-time blow-up is global.
Keywords
- Blow-up, Degenerate diffusion, Evolutionary games, Infinite dimensional replicator dynamics, Nonlocal nonlinearity
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Applied Mathematics
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In: SIAM Journal on Mathematical Analysis, Vol. 49, No. 2, 2017, p. 954-983.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - On a degenerate nonlocal parabolic problem describing infinite dimensional replicator dynamics
AU - Kavallaris, Nikos I.
AU - Lankeit, Johannes
AU - Winkler, Michael
PY - 2017
Y1 - 2017
N2 - We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for ut = uΔu + u ∫Ω |∇u|2 in bounded domains Ω ⊂ ℝn which arises in game theory. We prove that solutions converge to 0 if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with Ω; i.e., the finite-time blow-up is global.
AB - We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for ut = uΔu + u ∫Ω |∇u|2 in bounded domains Ω ⊂ ℝn which arises in game theory. We prove that solutions converge to 0 if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with Ω; i.e., the finite-time blow-up is global.
KW - Blow-up
KW - Degenerate diffusion
KW - Evolutionary games
KW - Infinite dimensional replicator dynamics
KW - Nonlocal nonlinearity
UR - http://www.scopus.com/inward/record.url?scp=85018724882&partnerID=8YFLogxK
U2 - 10.1137/15M1053840
DO - 10.1137/15M1053840
M3 - Article
AN - SCOPUS:85018724882
VL - 49
SP - 954
EP - 983
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
SN - 0036-1410
IS - 2
ER -