TY - JOUR

T1 - Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity

AU - Lankeit, Johannes

PY - 2016/4

Y1 - 2016/4

N2 - We prove convergence of positive solutions to ut = uΔu+u ∫Ω|∇u|2, u|∂ω=0 u(.,0)=u0 in a bounded domain Ω⊂Rn, n≥1, with smooth boundary in the case of ∫Ω u0=1 and identify the W01,2 (Ω)-limit of u(t) as t→∞ as the solution of the corresponding stationary problem. This behaviour is different from the cases of ∫Ω u0<1 and ∫Ω u0>1 which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of ∫Ω |∇u|2 along trajectories and the analysis of an associated constrained minimization problem.

AB - We prove convergence of positive solutions to ut = uΔu+u ∫Ω|∇u|2, u|∂ω=0 u(.,0)=u0 in a bounded domain Ω⊂Rn, n≥1, with smooth boundary in the case of ∫Ω u0=1 and identify the W01,2 (Ω)-limit of u(t) as t→∞ as the solution of the corresponding stationary problem. This behaviour is different from the cases of ∫Ω u0<1 and ∫Ω u0>1 which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of ∫Ω |∇u|2 along trajectories and the analysis of an associated constrained minimization problem.

KW - 35K55

KW - 35K65

KW - Degenerate diffusion

KW - Long-term behaviour

KW - MSC 35B40

KW - Nonlocal nonlinearity

UR - http://www.scopus.com/inward/record.url?scp=84959526792&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1511.01885

DO - 10.48550/arXiv.1511.01885

M3 - Article

AN - SCOPUS:84959526792

VL - 135

SP - 236

EP - 248

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -