A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities in a channel with decreasing cross section

Front blocking in the presence of gradient drift

“…pulsating fronts for periodic media [2] and the very general transition fronts for very general media [3]. In recent years there have been investigations of existence and non existence of transition fronts in outer domains with a compactly supported obstacle [4], in cylinders with varying nonlinearity [11,8] and, with respect to this work especially interesting, in opening or closing cylinders [1,6,9].…”

Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity

“…the construction of the entire solution behaving as the traveling wave pertaining to f 1 coming from infinity is similar. In fact, since regularity of the entire solution is known, the Lyapunov function method can be used to show the approach of the entire solution to the traveling wave pertaining to f 2 , one can refer to [10,11].…”

“…Li et al [16] further obtained the existence of entire solutions for space periodic nonlinearity. Particularly, Eberle [10,11] constructed a heteroclinic orbit connecting two traveling waves for bistable local dispersal equation 1 in cylinders. Meanwhile, Berestycki and Rodríguez [5] considered a bistable nonlocal dispersal equation with a gap in one dimension.…”

“…In this paper, we aim to construct an entire solution connecting traveling waves with the two different nonlinearities, motivated by [10,11]. By constructing suitable sub-and super-solutions, we establish the existence and uniqueness of the entire solution behaving as the traveling wave coming from one side and eventually going to the other side, which is different with the one constructed in [23] for the case L = 0 and f 2 = f 1 .…”

“…The crucial part of this paper is to figure out the long time asymptotic behavior of the entire solution. Since the lack of compactness of the nonlocal operator, we can not use the Lyapunov function argument as [10,11] to show that the entire solution converges to a translation of the other traveling wave as time goes to positive infinity. However, inspired by the idea in [7], we use a "squeezing" technique to address this issue.…”