V -shaped fronts around an obstacle
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Mathematische Annalen
V -shaped fronts around an obstacle Hongjun Guo1 · Harunori Monobe2 Received: 14 January 2019 / Revised: 27 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, we investigate V -shaped fronts around an obstacle K . We first prove that there exist solutions emanating from any homogeneous transition front including V shaped front for exterior domains Ω = R N \K . By providing the complete propagation of the V -shaped front, we prove that the V -shaped front can recover after passing the obstacle. Mathematics Subject Classification 35A18 · 35B08 · 35B30 · 35C07 · 35K57
1 Introduction This paper is concerned with the following reaction-diffusion equation in exterior domains u t = Δu + f (u), t ∈ R, x ∈ Ω = R N \K ⊂ R N , (1.1) on x ∈ ∂Ω, u ν = 0, where the obstacle K is a compact set of R N which is the closure of an open set with smooth boundary and Ω is an exterior domain. Here, ν = ν(x) is the outward unit normal on the boundary ∂Ω and u ν = ∂u ∂ν . On the boundary ∂Ω, the homogeneous Neumann boundary condition is imposed. Throughout of this paper, the reaction term f is assumed to be of bistable type, namely u = 0 and u = 1 are both stable stationary states. More precisely, we assume
Communicated by Y. Giga.
B
Hongjun Guo [email protected] Harunori Monobe [email protected]
1
Department of Mathematics, University of Miami, Coral Gables, FL, USA
2
Research Institute for Interdisciplinary Science, Okayama University, 3-1-1 Tsushima-naka, Kita-ku, Okayama 700-8530, Japan
123
H. Guo, H. Monobe
that f is of C 1 ([0, 1], R) and satisfies f (0) = f (1) = 0, f (0) < 0
and
f (1) < 0.
(1.2)
For mathematical purposes, the function f is extended in R as a C 1 (R) function such that f (s) = f (0)s > 0 for s ∈ (−∞, 0) and f (s) = f (1)(s − 1) < 0 for s ∈ (1, +∞). A typical example is the cubic nonlinearity f (s) = s(1 − s)(s − θ ) with 0 < θ < 1. Notice that the existence of V -shaped front requires the bistable reaction term f being unbalanced. Thus, we assume additionally that
1
f (s)ds > 0,
(1.3)
0
1 (for 0 f (s)ds < 0, one can only reverse the roles of 0 and 1). For the balanced case 1 0 f (s)ds = 0, no more V -shaped fronts exist, see [14]. Instead, some fronts with their level sets being exponential shape (N = 2) or parabolic shape (N ≥ 3) may exist, see [8]. Since we consider the propagation of homogeneous transition fronts and V -shaped fronts, we assume throughout this paper that if Ω = R, it admits a unique traveling front u(t, x) = φ(x − c f t) such that
φ + c f φ + f (φ) = 0 in R, φ(−∞) = 1 and φ(+∞) = 0.
(1.4)
It follows from [9] that the propagation speed c f is only determined by f and has 1 the sign of 0 f (s)ds. As we consider in this paper, c f > 0 by (1.3). We also point out that the existence and nonexistence of traveling fronts relies on conditions of the bistable nonlinearity, see [9]. The first aim of this paper is to prove the existence of entire solutions emanating from any homogeneous
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