Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

Topological Pressure of Generic Points for Amenable Group Actions

Let (X, G) be a G-action topological dynamical system (t.d.s. for short), where G is a countably infinite discrete amenable group. In this paper, we study the topological pressure of the sets of generic points. We show that when the system satisfies the almost specification property, for any G-invariant measure \(\mu \) and any continuous map \(\varphi \),

$$\begin{aligned} P\left( X_{\mu },\varphi ,\{F_n\}\right) = h_{\mu }(X)+\int \varphi d\mu , \end{aligned}$$

where \(\{F_n\}\) is a Følner sequence, \(X_{\mu }\) is the set of generic points of \(\mu \) with respect to (w.r.t. for short) \(\{F_n\}\), \(P(X_{\mu },\varphi ,\{F_n\})\) is the topological pressure of \(X_{\mu }\) for \(\varphi \) w.r.t. \(\{F_n\}\) and \(h_{\mu }(X)\) is the measure-theoretic entropy.

     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity

We consider positive classical solutions of

$$\begin{aligned} v_t=(v^{m-1}v_x)_x, \qquad x\in {\mathbb {R}}, \ t>0, \qquad (\star ) \end{aligned}$$

in the super-fast diffusion range \(m<-1\). Our main interest is in smooth positive initial data \(v_0=v(\cdot ,0)\) which decay as \(x\rightarrow +\infty \), but which are possibly unbounded as \(x\rightarrow -\infty \), having in mind monotonically decreasing data as prototypes. It is firstly proved that if \(v_0\) decays sufficiently fast only in one direction by satisfying

$$\begin{aligned} v_0(x) \le cx^{-\beta } \qquad \text{ for } \text{ all } ~x>0 \quad \hbox { with some }\quad \beta >\frac{2}{1-m} \end{aligned}$$

and some \(c>0\), then the so-called proper solution of (\(\star \)) vanishes identically in \({\mathbb {R}}\times (0,\infty )\), and accordingly no positive classical solution exists in any time interval in this case. Complemented by some sufficient criteria for solutions to remain positive either locally or globally in time, this condition for instantaneous extinction is shown to be optimal at least with respect to algebraic decay of the initial data. This partially extends some known nonexistence results for (\(\star \)) (Daskalopoulos and Del Pino in Arch Rat Mech Anal 137(4):363–380, 1997) in that it does not require any knowledge on the behavior of \(v_0(x)\) for \(x<0\). Next focusing on the phenomenon of extinction in finite time, we show that in this respect a mass influx from \(x=-\infty \) can interact with mass loss at \(x=+\infty \) in a nontrivial manner. Namely, we shall detect examples of monotone initial data, with critical decay as \(x\rightarrow +\infty \) and exponential growth as \(x\rightarrow -\infty \), that lead to solutions of (\(\star \)) which become extinct at a finite positive time, but which have empty extinction sets. This is in sharp contrast to known extinction mechanisms which are such that the corresponding extinction sets coincide with all of \({\mathbb {R}}\).

     

Stationary Solutions and Connecting Orbits for <Emphasis Type="Italic">p</Emphasis>-Laplace Equation

We deal with one dimensional p-Laplace equation of the form

$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$

under Dirichlet boundary condition, where \(p>2\) and \(f:[0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(f(x,0)=0\). We will prove that if there is at least one eigenvalue of the p-Laplace operator between \(\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u\) and \(\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u\), then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.

     

On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on \([0,\pi ]\) with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any \(\epsilon \) small enough, there exist a \(0<m_{\epsilon }\le 1\) and one solution \(u_{\epsilon }(t,x)\) with

$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$

where \(u_{\epsilon }(t,x)\) satisfies 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$

with Dirichlet boundery conditions on \([0,\pi ]\).

     

On the Damped Harmonic Oscillator with Time Dependent Damping Coefficient

Conditions guaranteeing asymptotic stability for the differential equation

$$\begin{aligned} x''+h(t)x'+\omega ^2x=0 \qquad (x\in \mathbb {R}) \end{aligned}$$

are studied, where the damping coefficient \(h:[0,\infty )\rightarrow [0,\infty )\) is a locally integrable function, and the frequency \(\omega >0\) is constant. Our conditions need neither the requirement \(h(t)\le \overline{h}<\infty \) (\(t\in [0,\infty )\); \(\overline{h}\) is constant) (“small damping”), nor \(0< \underline{h}\le h(t)\) (\(t\in [0,\infty )\); \(\underline{h}\) is constant) (“large damping”); in other words, they can be applied to the general case \(0\le h(t)<\infty \) (\(t\in [0,\infty \))). We establish a condition which combines weak integral positivity with Smith’s growth condition

$$\begin{aligned} \int ^\infty _0 \exp [-H(t)]\int _0^t \exp [H(s)]\,\mathrm{{d}}s\,\mathrm{{d}}t=\infty \qquad \left( H(t):=\int _0^t h(\tau )\,\mathrm{{d}}\tau \right) , \end{aligned}$$

so it is able to control both the small and the large values of the damping coefficient simultaneously.

     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

A Shilnikov Phenomenon Due to State-Dependent Delay,by Means of the Fixed Point Index

The first part of this paper is a general approach towards chaotic dynamics for a continuous map \(f:X\supset M\rightarrow X\) which employs the fixed point index and continuation. The second part deals with the differential equation

$$\begin{aligned} x'(t)=-\alpha \,x(t-d_{{\varDelta }}(x_t)). \end{aligned}$$

with state-dependent delay. For a suitable parameter \(\alpha \) close to \(5\pi /2\) we construct a delay functional \(d_{{\varDelta }}\), constant near the origin, so that the previous equation has a homoclinic solution, \(h(t)\rightarrow 0\) as \(t\rightarrow \pm \infty \), with certain regularity properties of the linearization of the semiflow along the flowline \(t\mapsto h_t\). The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion.

     

A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrödinger equations with outer force

$$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$

where \(M_\xi \) is a real Fourier multiplier,\(f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)\) is real analytic and the forced frequencies \(\bar{\omega }\) are fixed Diophantine vectors.

     

Existence and Multiplicity of Solutions for a Class of Elliptic Equations Without Ambrosetti–Rabinowitz Type Conditions

In this paper we establish, using variational methods, the existence and multiplicity of weak solutions for a general class of quasilinear problems involving \(p(\cdot )\)-Laplace type operators, with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz (A-R) type growth conditions, namely

$$\begin{aligned} \left\{ \begin{array}{rcll} -{\text {div}}(a(|\nabla u|^{p(x)})|\nabla u|^{p(x)-2}\nabla u)&{}=&{}\lambda f(x,u) &{} \text{ in } \Omega ,\\ u&{}=&{}0 &{} \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$

By different types of versions of the Mountain Pass Theorem with Cerami condition, as well as, the Fountain and Dual Theorem with Cerami condition, we obtain some existence of weak solutions for the above problem under some considerations. Moreover, we show that the problem treated has at least one nontrivial solution for any parameter \(\lambda >0\) small enough, and also that the solution blows up, in the Sobolev norm, as \(\lambda \rightarrow 0^{+}.\) Finally, by imposing additional hypotheses on the nonlinearity \(f(x,\cdot ),\) we get the existence of infinitely many weak solutions by using the Genus Theory introduced by Krasnoselskii.

     

Periodic and Solitary Travelling Waves in Infinite Lattices without Ambrosetti–Rabinowitz Condition

In this paper, we consider FPU lattices with particles of unit mass. The dynamics of the system is described by the infinite system of second order differential equations

$$\begin{aligned} \ddot{q}_n= U^{\prime }(q_{n+1}-q_n)-U^{\prime }(q_n-q_{n-1}),\quad n\in \mathbb {Z}, \end{aligned}$$

where \(q_n\) denotes the displacement of the \(n\)-th lattice site and \(U\) is the potential of interaction between two adjacent particles. We investigate the existence of two kinds travelling wave solutions: periodic and solitary ones under some growth conditions on \(U\) which is different from the widely used Ambrosetti–Rabinowitz condition.

     

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

     

Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation \(u=(u_{1}, \ldots, u_{N})\):

$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$

where \({\mathbf {X}}\) is a finite, open, symmetric \(N\)-annulus (with \(N \ge2\)), \(\mathscr{P}=\mathscr{P}(x)\) is an unknown hydrostatic pressure field and \(\varphi\) is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when \(N=3\), the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when \(N=2\), the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions \(N \ge4\) and discuss a number of closely related issues.

     

Piecewise Implicit Differential Systems

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form

$$\begin{aligned} \dot{x}=1,\quad \left( \dot{y}\right) ^2=\left\{ \begin{array}{lll} g_1(x,y) \quad \text{ if }\quad \varphi (x,y)\ge 0 \\ g_2(x,y) \quad \text{ if }\quad \varphi (x,y)\le 0 \end{array},\right. \end{aligned}$$

where \(g_1,g_2,\varphi :U\rightarrow \mathbb {R}\) are smooth functions and \(U\subseteq \mathbb {R}^2\) is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form

$$\begin{aligned} \dot{x}= f(x,y,\varepsilon ) ,\quad (\varepsilon \dot{ y})^2=g ( x,y,\varepsilon ) \end{aligned}$$

arise when the Sotomayor–Teixeira regularization is applied with \((x, y) \in U\) , \(\varepsilon \ge 0\), and f, g smooth in all variables.

     

Invariant Tori of Full Dimension for Second KdV Equations with the External Parameters

In this paper, we consider the second KdV equation with the external parameters

$$\begin{aligned} u_{t} =\partial _x^5 u +(M_{\sigma }u+u^3)_{x}, \end{aligned}$$

under zero mean-value periodic boundary conditions

$$\begin{aligned} u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

where \(M_\sigma \) is a real Fourier multiplier. It is proved that the equations admit a Whitney smooth family of small amplitude, real analytic almost periodic solutions with all frequencies. The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property of the perturbation and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property of the perturbation, our normal form part is independent of angle variables in spite of the unbounded perturbation. This is the first attempt to prove the almost periodic solutions for the unbounded perturbation case.

     

Boundedness for Some Doubly Nonlinear Parabolic Equations in Measure Spaces

In the context of measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality, we derive local and global sup bounds of the nonnegative weak subsolutions of

$$\begin{aligned} (u^{q})_t-\nabla \cdot {(|\nabla u|^{p-2}\nabla u)}=0, \quad \mathrm {in} \ U_\tau = U \times (\tau _1, \tau _2] , \quad p>1,\quad q>1 \end{aligned}$$

and of its associated Dirichlet problem, respectively. For particular ranges of the exponents p and q, we show that any locally nonnegative weak subsolution, taken in \(Q (\subset \bar{Q}\subset U_\tau )\), is controlled from above by the \(L^\alpha (\bar{Q}) \)-norm, for \(\alpha = \max \{p, q+1\}\). As for the global setting, under suitable assumptions on the boundary datum g and on the initial datum, we obtain sup bounds for u, in \(U \times \{ t\}\), which depend on the \(\sup g\) and on the \(L^{q+1}(U \times (\tau _1, \tau _1+t])\)-norm of \((u-\sup g)_+\), for all \(t \in (0, \tau _2-\tau _1]\). On the critical ranges of p and q, a priori local and global \(L^\infty \) estimates require extra qualitative information on u.

     

Complex Dynamics in a ODE Model Related to Phase Transition

Motivated by some recent studies on the Allen–Cahn phase transition model with a periodic nonautonomous term, we prove the existence of complex dynamics for the second order equation

$$\begin{aligned} -\ddot{x} + \left( 1 + \varepsilon ^{-1} A(t)\right) G'(x) = 0, \end{aligned}$$

where A(t) is a nonnegative T-periodic function and \(\varepsilon > 0\) is sufficiently small. More precisely, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima \(x_0\) and \(x_1\) of G(x). Such solutions stay close to \(x_0\) or \(x_1\) in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case \(x_0 =0\) and \(x_1 = 1\).

     

Infinitely Many Stability Switches in a Problem with Sublinear Oscillatory Boundary Conditions

We consider the elliptic equation \(-\Delta u +u =0\) with nonlinear boundary condition \(\frac{\partial u}{\partial n}= \lambda u + g(\lambda ,x,u), \) where \(\frac{g(\lambda ,x,s)}{s} \rightarrow 0, \hbox { as }|s|\rightarrow \infty \) and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.     

Time Periodic Traveling Curved Fronts in the Periodic Lotka–Volterra Competition–Diffusion System

This paper is concerned with time periodic traveling curved fronts for periodic Lotka–Volterra competition system with diffusion in two dimensional spatial space

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u_{1}}{\partial t}=\Delta u_{1} +u_{1}(x,y,t)\left( r_{1}(t)-a_{1}(t)u_{1}(x,y,t)-b_{1}(t)u_{2}(x,y,t)\right) ,\\ \dfrac{\partial u_{2}}{\partial t}=d\Delta u_{2} +u_{2}(x,y,t)\left( r_{2}(t)-a_{2}(t)u_{1}(x,y,t)-b_{2}(t)u_{2}(x,y,t)\right) , \end{array}\right. } \end{aligned}$$

where \(\Delta \) denotes \(\frac{\partial ^{2}}{\partial x^{2} }+ \frac{\partial ^{2}}{\partial y^{2} }\), \(x,y\in {\mathbb {R}}\) and \(d>0\) is a constant, the functions \(r_i(t),a_i(t)\) and \(b_i(t)\) are T-periodic and Hölder continuous. Under suitable assumptions that the corresponding kinetic system admits two stable periodic solutions (p(t), 0) and (0, q(t)), the existence, uniqueness and stability of one-dimensional traveling wave solution \(\left( \Phi _{1}(x+ct,t),\Phi _{2}(x+ct,t)\right) \) connecting two periodic solutions (p(t), 0) and (0, q(t)) have been established by Bao and Wang ( J Differ Equ 255:2402–2435, 2013) recently. In this paper we continue to investigate two-dimensional traveling wave solutions of the above system under the same assumptions. First, we establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate super- and subsolutions and prove the existence and non-existence of two-dimensional time periodic traveling curved fronts. Finally, we show that the time periodic traveling curved front is asymptotically stable.