Abstract
Asymptotic behavior of solutions of a Fisher equation with free boundaries and nonlocal term
Highlights
Consider the following free boundary problem with nonlocal term ut = uxx + (1 − u) k(x − y)u(t, y)dy, g(t) < x < h(t), t > 0, R u(t, x) = 0, x ∈ R \ (g(t), h(t)), t > 0,g (t) = −μux(t, g(t)), t > 0, (1.1) h (t) =−μux(t, h(t)), t > 0,−g(0) = h(0) = h0, u(0, x) = u0(x), −h0 ≤ x ≤ h0, where x = g(t) and x = h(t) are moving boundaries to be determined together with u(t, x), μ > 0 is a constant, h0 > 0
The main purpose of this paper is to study the asymptotic behavior of bounded solutions of (1.1)
We mainly consider the asymptotic behavior of solutions of the problem (1.1) by constructing some suitable upper and lower solutions, so the comparison principle is essential here
Summary
Problem (1.1) with Fisher–KPP nonlinearity, i.e., ut = uxx + u(1 − u) was studied by [9, 10], etc They used this model to describe the spreading of a new or invasion species, with the free boundaries h(t) and g(t) representing the expanding fronts of the species whose density is represented by u(t, x). We added the nonlocal term into the equation Such nonlocal interactions are used in epidemic reaction-diffusion models, such as [7] studied the following system ut = d∆u − au + Ω k(x, y)v(t, y)dy, t > 0, x ∈ Ω,. Considered the nonlocal SIS epidemic model with free boundaries and obtained some sufficient conditions for spreading (limt→∞ u(t, ·) C([g(t),h(t)]) > 0) and vanishing.
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