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.

     

A Stable Rapidly Oscillating Periodic Solution for an Equation with State-Dependent Delay

We exhibit a differential delay equation with state-dependent delay

$$\begin{aligned} x'(t) = f(x(t - h(x_t))) \end{aligned}$$

for which the familiar non-increasing “oscillation speed” is defined and for which there exists an asymptotically stable rapidly oscillating periodic solution.

     

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.

     

Large Time Behavior for a Quasilinear Diffusion Equation with Critical Gradient Absorption

We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption

$$\begin{aligned} \partial _t u-\Delta _{p}u+|\nabla u|^{q_*}=0 \quad \hbox {in}\, (0,\infty )\times \mathbb {R}^N, \end{aligned}$$

for \(p\in (2,\infty )\) and \(q_*:=p-N/(N+1)\). We show that the asymptotic profile of compactly supported solutions is given by a source-type self-similar solution of the p-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results.

     

Propagation of Regularity and Persistence of Decay for Fifth Order Dispersive Models

This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equation

$$\begin{aligned} \partial _tu - \partial _x^5u -30u^2\partial _xu + 20\partial _xu\partial _x^2u + 10u\partial _x^3u = 0. \end{aligned}$$

The main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times.

     

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.

     

Multiple Solutions to Neumann Problems with Indefinite Weight and Bounded Nonlinearities

We study the Neumann boundary value problem for the second order ODE

$$\begin{aligned} u^{\prime \prime } + (a^+(t)-\mu a^-(t))g(u) = 0, \qquad t \in [0,T], \end{aligned}$$

(1)

where \(g \in {\mathcal {C}}^1({\mathbb {R}})\) is a bounded function of constant sign, \(a^+,a^-: [0,T] \rightarrow {\mathbb {R}}^+\) are the positive/negative part of a sign-changing weight \(a(t)\) and \(\mu > 0\) is a real parameter. Depending on the sign of \(g^{\prime }(u)\) at infinity, we find existence/multiplicity of solutions for \(\mu \) in a “small” interval near the value

$$\begin{aligned} \mu _c = \frac{\int _0^T a^+(t) \, dt}{\int _0^T a^-(t) \, dt}\,. \end{aligned}$$

The proof exploits a change of variables, transforming the sign-indefinite Eq. (1) into a forced perturbation of an autonomous planar system, and a shooting argument. Nonexistence results for \(\mu \rightarrow 0^+\) and \(\mu \rightarrow +\infty \) are given, as well.

     

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.

     

Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable,Hyperbolic Case

In this paper, we use the parameterization method to construct quasi-periodic solutions of state-dependent delay differential equations. For example

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}(t)&=f(\theta ,x(t),\epsilon x(t-\tau (x(t))))\\ \dot{\theta }(t)&=\omega . \end{aligned} \right. \end{aligned}$$

Under the assumption of exponential dichotomies for the \(\epsilon =0\) case, we use a contraction mapping argument to prove the existence and smoothness of the quasi-periodic solution. Furthermore, the result is given in an a posteriori format. The method is very general and applies also to equations with several delays, distributed delays etc.

     

Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water

In this paper, the Lie symmetry analysis and the dynamical system method are performed on an integrable evolution equation for surface waves in deep water

$$\begin{aligned} 2\sqrt{\frac{k}{g}}u_{xxt}=k^2u_x-\frac{3}{2}k(uu_x)_{xx}. \end{aligned}$$

All of the geometric vector fields of the equation are presented, as well as some exact similarity solutions with an arbitrary function of t are obtained by using a special symmetry reduction and the dynamical system method. Different kinds of traveling wave solutions also be found by selecting the function appropriately.

     

Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an indefinite weight. In particular we are interested in the case where such a weight is positive in a ball and negative outside, or in the reversed situation. We extend the approach to elliptic equations in presence of Hardy potentials, i.e. to

$$\begin{aligned} \varDelta u +\frac{h(|\text {x}|)}{|\text {x}|^2} u+ f(u, |\text {x}|)=0 \end{aligned}$$

where h is not necessarily constant. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.

     

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.

     

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 \([0,\pi ]\).

     

Centers and Uniform Isochronous Centers of Planar Polynomial Differential Systems

For planar polynomial vector fields of the form

$$\begin{aligned} (-y+X(x,y))\dfrac{\partial }{\partial x}+(x+Y(x,y))\dfrac{\partial }{\partial y}, \end{aligned}$$

where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers.

     

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.

     

Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in $$\mathbb {R}^{N}$$

This paper is concerned with the following fractional Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$

where \(s\in (0,1),N> 2s, (-\Delta )^{s}\) is the fractional Laplacian, k is a bounded positive function, \(h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0\) is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that \(|h|_{2}\) is sufficiently small.

     

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 \({\mathbb {R}}\).

     

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.

     

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.