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

2015/11/05 by Lankeit, Johannes
#35B40 #35K55 #35K65 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1511.01885

Abstract

We prove convergence of positive solutions to u_t = 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 ∫_Ωu₀=1 and identify the W₀^1,2(Ω)-limit of u(t) as t→ ∞ as the solution of the corresponding stationary problem. This behaviour is different from the cases of ∫_Ωu₀<1 and ∫_Ωu₀>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. Keywords: degenerate diffusion, nonlocal nonlinearity, long-term behaviour

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