Layered stable equilibria of a reaction–diffusion equation with nonlinear Neumann boundary condition

Abstract

In this work we investigate the existence and asymptotic profile of a family of layered stable stationary solutions to the scalar equation u t = ε 2 Δ u + f ( u ) in a smooth bounded domain Ω ⊂ R 3 under the boundary condition ε ∂ ν u = δ ε g ( u ) . It is assumed that Ω has a cross-section which locally minimizes area and lim ε → 0 ε ln δ ε = κ , with 0 ⩽ κ < ∞ and δ ε > 1 when κ = 0 . The functions f and g are of bistable type and do not necessarily have the same zeros what makes the asymptotic geometric profile of the solutions on the boundary to be different from the one in the interior.