Existence of Traveling Wave Solutions for a Nonlocal Bistable Equation: An Abstract Approach

Abstract

We consider traveling fronts to the nonlocal bistable equationwhere μ is a Borel-measure on R with μ(R) = 1 and f satisfies f (0) = f (1) = 0, f < 0 in (0, α) and f > 0 in (α, 1) for some constant α ∈ (0, 1). We do not assume that μ is absolutely continuous with respect to the Lebesgue measure. We show that there are a constant c and a monotone function φ with φ(−∞) = 0 and φ(+∞) = 1 such that u(t, x) := φ(x+ct) is a solution to the equation, provided f (α) > 0. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.

Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

“…Meanwhile, the study of traveling waves of nonlocal reaction diffusion equations attract much attention, see Bates et al [2], Chen [5], Carr and Chmaj [4], Schumacher [32], Coville and Dupainge [8] and Coville et al [6], Yagisita [42,43] etc., see also [13,20,21,29,30,31,33,34,35,45]. Particularly, for nonlocal dispersal Fisher-KPP equation, Li et al [24] first studied the entire solutions.…”

Bistable traveling waves for monotone semiflows with applications

“…Clearly, the restriction of system (1.1) on X = R is the ordinary differential equation u ′ = u(1 − u)(u − a), which admits a unique unstable equilibrium between two ordered and stable ones. The same property is shared by the nonlocal dispersal equation in [4,16,46] and the lattice equations in [3,49,50]. Chen [13] studied a general nonlocal evolution equation u t = A(u(·, t)), which also possess the above bistability structure.…”

“…In our investigation, we consider seven cases: (I) T = Z + and H = R; (II) T = Z + and H = Z; (III) T = R + and H = R; (IV) T = R + and H = Z; (V) periodic habitat; (VI) weak compactness; (VII) time periodic. For the case (I), we combine the above observations for general bistable semiflows and Yagisita's perturbation idea in [46] to prove the existence of traveling waves. For the case (III), we use the bistable traveling waves φ ± (x+c ±,s ) of discrete-time semiflows {(Q s ) n } n≥0 to approximate the bistable wave of the continuous-time semiflow {Q t } t≥0 .…”

A counterexample to the Liouville property of some nonlocal problems

“…Now, since w = αw 0 + h, since w 0 ≡ 1 on B R 0 and since h is integral free (by (4.13)) we have 15) where C N,J is given by (3.3). We are now left to estimate I.…”

“…Since J ε is radially symmetric (because J is), using the results obtained in [1,8,9,15], we know that, for any 0 < ε < 1, there exists an increasing function φ ε,δ ∈ C 1 (R) and a number c ε,δ > 0 such that the function ϕ ε,δ (x) := φ ε,δ (x · e 1 ) satisfies (5.4) where H e 1 is the hyperplane H e 1 := {x 1 = 0}. Now, for any r 0 > 0, we let ϕ ε,δ,r 0 be the function defined by ϕ ε,δ,r 0 (x) := ϕ ε,δ (x − r 0 ).…”