Higher-dimensional scalar trichotomy and stabilization cost

Determine whether sufficiently large diffusion for the higher-dimensional Neumann problem with reaction f(u)=(u_+)^{(N+2)/(N-2)}-au, N≥3 and a>0, recovers the scalar trichotomy uniformly on bounded H^1(Ω) families, and identify the sharp cost of uniform subthreshold stabilization.

Background

The authors propose a higher-dimensional analogue involving the Sobolev-critical reaction f(u)=(u_+)^{(N+2)/(N-2)}-au. This reaction retains the convexity structure used in the planar analysis, but the critical Sobolev growth introduces a different concentration mechanism.

The unresolved issue is whether large diffusion can suppress concentration rapidly enough to reproduce the scalar decay–threshold–blow-up classification uniformly over bounded H^1 data, together with determining the asymptotic diffusion threshold required for uniform stabilization below the scalar threshold.