Uniform stabilization for Dirichlet families

Determine whether every bounded family of initial data in H_0^1(Ω) for the corresponding homogeneous Dirichlet exponential reaction–diffusion problem is uniformly stabilized to zero by sufficiently large diffusion.

Background

The paper studies uniform stabilization for the planar Neumann problem with exponential reaction on bounded H^1(Ω) families and identifies an exponential diffusion threshold governed by the sharp mean-zero Moser–Trudinger inequality. The authors ask whether an analogous stabilization result holds under homogeneous Dirichlet boundary conditions.

The Dirichlet setting differs substantially because the Dirichlet heat flow tends toward zero and the spatial mean no longer satisfies a diffusion-independent scalar comparison. The authors note that concentration is expected to occur in the interior and that the relevant sharp Moser–Trudinger coefficient is 4π.