Z. Angew. Math. Phys. (2018) 69:12
2018 Springer International Publishing AG,
part of Springer Nature
published online January 10, 2018
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Multidimensional stability of traveling fronts in combustion and non-KPP monostable
Zhen-Hui Bu and Zhi-Cheng Wang
Abstract. This paper is concerned with the multidimensional stability of traveling fronts for the combustion and non-KPP
monostable equations. Our study contains two parts: in the ﬁrst part, we ﬁrst show that the two-dimensional V-shaped
traveling fronts are asymptotically stable in R
with n ≥ 1 under any (possibly large) initial perturbations that decay at
space inﬁnity, and then, we prove that there exists a solution that oscillates permanently between two V-shaped traveling
fronts, which implies that even very small perturbations to the V-shaped traveling front can lead to permanent oscillation.
In the second part, we establish the multidimensional stability of planar traveling front in R
with n ≥ 1.
Mathematics Subject Classiﬁcation. 35K57, 35B10, 35B35, 35C07.
Keywords. Planar traveling front, V-shaped traveling front, Combustion nonlinearity, Non-KPP monostable nonlinearity,
Multidimensional asymptotic stability.
1. Introduction and main results
In this paper, we investigate the Cauchy problem for the following equation
=Δu + f(u), x ∈ R
(x), x ∈ R
where m ∈ N. Here a given initial function u
is continuous and bounded. The following is the standing
assumption on the nonlinearity f:
(A1) f(u) is of class C
([−, 1+], R) for some constants ϑ ∈ (0, 1] and ∈ (0, 1) and such that
f(0) = f (1) = 0,f
(0) ≥ 0,f
(1) < 0,f(u) ≥ 0foru ∈ (0, 1).
In one-dimensional space, we write x = z. A traveling front of (1.1) is a special translation invariant
solution of the form u(t, z)=φ(z− bt). φ is the wave proﬁle that propagates through the one-dimensional
spatial domain at a constant velocity b. Of interest are traveling fronts connecting an equilibrium state 0
and an asymptotically stable equilibrium state 1. Set P = z − bt, then the function φ(P) satisﬁes
(P)+f(φ(P)) = 0,φ
(P) < 0, P∈R,
The function φ is called the planar traveling front of (1.1)onR. The existence, uniqueness and stability
of such planar traveling fronts for (1.2) with various types of nonlinearities f in one-dimensional space
have been studied by many works. We can refer to [1,2,9,13,28,30,33,42–44,53,60,61] and the references
therein for more details. Throughout this paper, we further assume that the following condition holds.
(A2) There exists a traveling front φ(P) ∈ C
(R) with speed c
> 0 satisfying (1.2)and