Fine patterns for a nonlocal periodic-parabolic equation with spatial degeneracy

Abstract

In this paper, we investigate several properties of the positive solution to the nonlocal periodic-parabolic problem{ut=d[∫ΩJ(x−y)u(y,t)dy−u(x,t)]+λu−b(x)q(t)up in Ω×[0,T],u(x,0)=u(x,T) in Ω, where Ω is a bounded smooth domain in RN (N≥1), λ>0 is a parameter, d>0 and p>1 are constants, q(t) is a positive and T-periodic continuous function, b(x) is a nonnegative continuous function, and is strictly positive only outside a domain Ω¯0⊂Ω. So the equation has a degeneracy over Ω0. In such a case, it is known that there is a unique positive solution uλ if and only if λ is between the principle eigenvalues λp(Ω) and λp(Ω0) of the associated nonlocal diffusion operator (over Ω and Ω0, respectively). We show that uλ is stable under perturbations of λ and b(x) in the equation, and when λ increases to λp(Ω0), uλ converges to ∞ uniformly over Ω¯×[0,T] but a clear pattern is exhibited by uλ/‖uλ‖∞:uλ(x,t)‖uλ‖∞→{0 for (x,t)∈(Ω¯∖Ω¯0)×[0,T],ϕ(x) for (x,t)∈Ω¯0×[0,T], where ϕ(x) is the normalised positive eigenfunction associated to λp(Ω0). The proof relies on a general regularity result for the solutions of these types of nonlocal equations, which is of independent interest.