Asymptotic Behavior of Solutions for Some Nonlocal Diffusion Problems
Théodore K. Boni, Firmin K. N’gohissé · 2008
Abstract. In this paper, we address the following initial-value problem ut(x, t) = Ω J(x − y)(u(y, t) − u(x, t))dy − a(x)f(u(x, t)) in Ω × (0,∞), u(x, 0) = u0(x)> 0 in Ω, where Ω is a bounded domain in RN with smooth boundary ∂Ω, J: RN − → R is a kernel which is nonnegative, symmetric, bounded and RN J(z)dz = 1, f: [0,∞) − → [0,∞) is a C1 convex, increasing function, ∫ ∞ ds f(s) 0 for x ∈ Ω, and the potential a ∈ C0(Ω), a(x)> 0 for x ∈ Ω. We reveal that the solution of the above problem exists globally and tends to zero uniformly in x ∈ Ω as t approaches infinity. The description of its asymptotic behavior is also given under some conditions.