ミニワークショップ “Recent Trends in Traveling Waves”
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世話人代表

俣野博記
日時： 2015 年 1 月 30 日 (金) 10:00〜17:20
場所： 東京大学大学院数理科学研究科棟 002 教室
京王井の頭線駒場東大前駅よりすぐ――Program――
10:00〜10:40 Danielle Hilhorst (CNRS / Univ. Paris-Sud)Front propagation in nonlinear parabolic equations
11:00〜11:40 Thomas Giletti (University of Lorraine)
Speed-up of propagation by a road – the periodically heterogeneous framework
11:50〜12:30 Masaharu Taniguchi (Okayama University)
An (N − 1)-dimensional convex compact set gives an N -dimensional travelingfront in the Allen-Cahn equation
12:30〜14:00 Lunch
14:00〜14:40 Cyrill Muratov (New Jersey Institute of Technology)
A variational approach to supercritical fronts for reaction-diﬀusion equations
in cylinders
14:50〜15:30 Xing Liang (Univ. Science and Technology of China)
The spreading speeds of monostable parabolic equations with free boundary
in the spatially and temporally periodic domain
15:50〜16:30 Matthieu Alfaro (University of Montpellier)
Acceleration or not in some nonlocal reaction diﬀusion equations
16:40〜17:20 Hirokazu Ninomiya (Meiji University)
Traveling waves in two dimensional excitable media
本ミニワークショップは，以下の助成によるものです．

• 東京大学数物フロンティア・リーディング大学院 (FMSP) 事業• 科学研究費補助金 基盤研究 A 課題番号 23244017「非線形偏微分方程式の定性的理論と特異性の解析」（研究代表者： 俣野博）

Danielle Hilhorst (CNRS / Univ. Paris-Sud)“Front propagation in nonlinear parabolic equations”
We study the existence and the stability of travelling waves for nonlinear convectiondiﬀusion equations in one-space dimension. The diﬀusion coeﬃcient depends on thegradient as in the case of the p-Laplacian and it may be degenerate. Unconditional
stability is established with respect to initial data perturbations in L1 . We also provethat the solution converges to ±1 outside of an interface which moves with constantvelocity; our results include both generation and propagation of interface properties.
This is joint work with Eduard Feireisl, Hana Petzeltová and Peter Takáč.

Thomas Giletti (University of Lorraine)“Speed-up of propagation by a road – the periodically heterogeneousframework”
In this talk, I will discuss a new model which was initially proposed by H. Berestycki,
J.-M. Roquejoﬀre and Luca Rossi in order to account for the eﬀects of a road on
the propagation of a biological species. Under Fisher-KPP type assumptions on thereproduction rate of the species, they showed that the asymptotic speed of spreadingis accelerated when the diﬀusion on the road exceeds some threshold. I will give a
brief overview of some recent developments for this type of models, and introduce
some work in progress (in collaboration with L. Monsaingeon and M. Zhou) on theextension of the above result to the heterogeneous framework.

Masaharu Taniguchi (Okayama University)“An (N −1)-dimensional convex compact set gives an N -dimensional traveling front in the Allen-Cahn equation”
We study traveling fronts to the Allen-Cahn equation in RN for N ≥ 3. Let (N − 2)dimensional smooth surfaces be the boundaries of compact sets in RN −1 , and assume
that all principal curvatures are positive everywhere. We deﬁne an equivalence relation between them, and prove that there exists a traveling front associated with agiven surface and that it is asymptotically stable for given initial perturbation. The
associated traveling fronts coincide up to phase transition if and only if the given
surfaces satisfy the equivalence relation.

Cyrill Muratov (New Jersey Institute of Technology)“A variational approach to supercritical fronts for reaction-diﬀusion equations in cylinders”
In this talk, I will present a variational treatment of the supercritical traveling front
solutions for scalar reaction-diﬀusion equations in inﬁnite cylinders which invade alinearly unstable equilibrium. These equations are known to possess traveling wave
solutions connecting an unstable equilibrium to the closest stable equilibrium for all
speeds exceeding a critical value. I will show how to construct these fronts by minimizing a suitable functional and show that these are, in fact, the only traveling front
solutions and are long-time attractors in the considered problems for suﬃciently
large speeds and suitable initial data. In addition, I will show that other travelingfronts connecting to the unstable equilibrium may exist in a certain range of thewave speed.

Xing Liang (Univ. Science and Technology of China)“The spreading speeds of monostable parabolic equations with freeboundary in the spatially and temporally periodic domain”
In this talk, we investigate the spreading speed of the following free boundary
problem:ut − duxx = f (t, x, u),t > 0, g(t) < x < h(t), u(t, h(t)) = u(t, g(t)) = 0, t > 0,h′ (t) = −µux (t, h(t)),
t > 0,(∗)
′g (t) = −µux (t, g(t)),
t > 0,

u(0, x) = u0 (x) > 0,g(0) ≤ x ≤ h(0),where we assume that the media is space-time periodic in the sense that f is periodic
in t, x. We also assume that f is smooth and f (t, x, 0) = 0. We say that the equationis ‘monostable’ if the equation ut − duxx = f (t, x, u), x ∈ R, has precisely two timespace periodic solutions, namely 0 and a positive one ū(x, t), the latter being stablein the space of periodic continuous functions. Then we can prove that there exist
constants c∗ > c∗ ≥ 0 such that for any initial data (g0 , h0 , u0 ) with g0 ≪ h0 andu0 ∈ H01 ((g0 , h0 )) with u0 ≥ 0, u0 ̸≡ 0,lim−c′ t<x<ct,t→∞lim

x<−c′ t,t→∞|u(x, t) − ū(x, t)| = 0u(x, t) =lim

x>ct,t→∞u(x, t) = 0

if c < c∗ and c′ < c∗ ,if c > c∗ , c′ > c∗ .

In proving this result, we extend the general theory of Weinberger [2002 JMB] andthat of Liang and Zhao [2010 JFA] on the existence of the spreading speed.

Matthieu Alfaro (University of Montpellier)“Acceleration or not in some nonlocal reaction diﬀusion equations”
We are concerned with the speed of propagation of solutions to nonlocal reactiondiﬀusion equations.
First, we consider a class of problems, referred to as replicator-mutator equations
in evolutionary genetics. By some tricky change of unknown function, we compute
their solution explicitly. Depending on the tails of the initial data, they can accelerate or become extinct (Joint work with R. Carles).
Then, we consider dipersion kernels with heavy tails and degenerate monostablenonlinearities (weak Allee eﬀect). We investigate the balance between these twoeﬀects on the propagation of solutions (traveling waves versus acceleration) (Ongoingwork with J. Coville).

Hirokazu Ninomiya (Meiji University)“Traveling waves in two dimensional excitable media”
In this talk we consider the singular limit problems arising from FitzHugh-Nagumo
type model, which is a free boundary problem. The singular limit problems of FHN
system have been studied by many authors. However, the dynamics of the freeboundary problems have not been well known yet. To understand the dynamics, weﬁrst introduce the singular limit problem of the modiﬁed FitzHugh-Nagumo modeland study the dynamics in one dimensional space. Using this observation, I explain
the existence of two types of the two-dimensional traveling waves of this model whichinclude the front and the back.