DCDS-B

This paper is concerned with a geodesic curvature flow on the unit
sphere. In each zone between the equator and the circle with
latitude $\theta_0 \in (0, \frac{\pi}{2} ]$, we give the existence
and uniqueness of a spiral rotating wave of the geodesic curvature
flow.

PROC

We study a generalized curvature flow equation in the plane: $V =F(k,$ **n**, $x)$, where for a simple plane curve $\Gamma$ and for any $P \in \Gamma, k$ denotes the curvature of $\Gamma$ at $P$, **n** denotes the unit normal vector at $P$ and $V$ denotes the velocity in direction **n**, $F$ is a smooth function which is 1-periodic in $x$. For any given $\alpha \in ( - \pi/2, \pi/2)$, we prove the existence and uniqueness of a planar-like traveling wave solution of $V = F(k,$**n**,$x)$, that is, a curve: $y = v$*$(x) + c$*$t$ traveling in $y$-direction in speed $c$*, the graph of $v$*$(x)$ is in a
bounded neighborhood of the line $x$tan$\alpha$. Also, we show that the graph of $v$*$(x)$ is periodic in the direction (cos$\alpha$, sin$\alpha$).

DCDS

We consider a curvature flow in heterogeneous media in the plane: $
V= a(x,y) \kappa + b$, where for a plane curve, $V$ denotes its
normal velocity, $\kappa$ denotes its curvature, $b$ is a constant
and $a(x,y)$ is a positive function, periodic in $y$. We study
periodic traveling waves which travel in $y$-direction with given
average speed $c \geq 0$. Four different types of traveling waves
are given, whose profiles are straight lines, ''V"-like curves,
cup-like curves and cap-like curves, respectively. We also show
that, as $(b,c)\rightarrow (0,0)$, the profiles of the traveling
waves converge to straight lines. These results are connected with
spatially heterogeneous version of Bernshteĭn's Problem and De
Giorgi's Conjecture, which are proposed at last.

NHM

We study a two-point free boundary problem in a sector for a
quasilinear parabolic equation. The boundary
conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and
asymptotic stability of an expanding solution which is self-similar at discrete
times. We also study the existence and uniqueness of a shrinking solution which is
self-similar at discrete times.

DCDS-B

We study two systems of reaction diffusion equations with monostable or bistable type of nonlinearities
and with free boundaries. These systems are used as multi-species competitive model.
For two-species models, we prove the existence of traveling wave solutions, each of which consists
of two semi-waves intersecting at the free boundary.
For three-species models, we also obtain some traveling wave solutions. In this case, however, every
traveling wave solution consists of
two semi-waves and one compactly supported wave in between, each intersecting with its neighbors at the free boundaries.

DCDS-B

Recently, Gu et al. [7,8] studied a reaction-diffusion-advection equation $u_t =u_{xx} -β u_x + f(u)$ in $(g(t), h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions, $f(u)$ is a Fisher-KPP type of nonlinearity. When $β ∈ [0,c_0)$, where $c_0 := 2\sqrt{f'(0)}$, they found that for a spreading solution $(u,g,h)$, $h(t)/t \to c^*_r (β)$ and $g(t)/t \to c^*_l (β)$ as $t \to ∞$, and $c^*_r (β) > c^*_r(0) = - c^*_l (0) > - c^*_l (β) >0$. In this paper we study the expanding speed $C^*(β) :=c^*_r(β) - c^*_l (β)$ of the habitat $(g(t), h(t))$, and show that $C^*(β)$ is strictly increasing in $β ∈ [0,c_0)$. When $β ∈ [c_0, β^*)$ for some $β^*>c_0$, [8] also found a virtual spreading phenomena: $h(t)/t \to c^*_r(β)$ as $t\to∞$, and a back forms in the solution which moves rightward with a speed $β - c_0$. Hence the expanding speed of the main habitat for such a solution is $C^*(β) := c^*_r(β) -[β -c_0]$. In this paper we show that $C^*(β)$ is strictly decreasing in $β∈ [c_0, β^*)$ with $C^*(β^* -0)=0$, and so there exists a unique $β_0∈ (c_0, β^*)$ such that the advection is favorable to the expanding speed of the habitat if and only if $β∈ (0,β_0)$.

NHM

We study a curvature-dependent motion of plane curves in a
two-dimensional cylinder with periodically undulating
boundary. The law of motion is given by $V=\kappa + A$, where
$V$ is the normal velocity of the curve, $\kappa$ is the curvature,
and $A$ is a positive constant. We first establish a necessary
and sufficient condition for the existence of periodic traveling
waves, then we study how the average speed of the periodic
traveling wave depends on the geometry of the domain boundary.
More specifically, we consider the homogenization problem as the
period of the boundary undulation, denoted by $\epsilon$, tends to
zero, and determine the homogenization limit of the average
speed of periodic traveling waves. Quite surprisingly,
this homogenized speed depends only on the maximum opening angle
of the domain boundary and no other geometrical features are
relevant. Our analysis also shows that, for any
small $\epsilon>0$, the average speed of the traveling wave is
smaller than $A$, the speed of the planar front.
This implies that boundary undulation always lowers the speed
of traveling waves, at least when the bumps are small enough.