Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations

This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equations

$$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$

where N ≥ 3, \(g(s): \mathbb {R}\rightarrow \mathbb {R}^{+}\) is C ¹ nondecreasing function with respect to |s|, V is a positive potential bounded away from zero and h(u) is a nonlinear term of subcritical type. By introducing a variable replacement and using minimax methods, we show the existence of a nontrivial solution in \(C^{\alpha }_{loc}(\mathbb {R}^{N})\).

     

Lower and Upper Bounds for the Blow-Up Time of a Class of Damped Fourth-Order Nonlinear Evolution Equations

This paper is concerned with the study of the nonlinear damped wave equation

$$u_{tt}+{\Delta}^{2}u-{\Delta} u-\omega{\Delta} u_{t}+\alpha(t)u_{t}=\left\vert u\right\vert^{p-2}u, $$

in a bounded domain with smooth boundary. The blow-up of solutions are investigated under some conditions. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.

     

Uniform Stability of the Solution for a Memory-Type Elasticity System with Nonhomogeneous Boundary Control Condition

In this paper, we consider the memory-type elasticity system \(\boldsymbol {u}_{tt}-\upmu {\Delta }{\boldsymbol {u}}-(\upmu +\lambda )\nabla (\text {div}\boldsymbol {u})+{{\int }^{t}_{0}}g(t-\tau ){\Delta }{\boldsymbol {u}}(s)ds=0,\) with nonhomogeneous boundary control condition and establish the uniform stability result of the solution. The exponential decay result and polynomial decay result in some literature are the special cases of this paper.     

Study on Existence of Solutions for p-Kirchhoff Elliptic Equation in ℝN with Vanishing Potential

In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem \(\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ &u(x)>0, \;\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} \) ?????(0.1) where a, μ > 0, τ > 0, and f(x, u) = h ₁(x)|u| ^m?2 u + λ h ₂(x)|u| ^r?2 u with the parameter λ ∈ ?, 1 < p < N, 1 < r < m < \(p^{*}=\frac {pN}{N-p}\) , and the functions h ₁ (x), h ₂(x) ∈ C(?^N) satisfy some conditions. The potential V(x) > 0 is continuous in ? ^N and V(x)→0 as |x|→+∞. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.     

Analytic Solutions of Systems

In this paper, functional series solutions of the nonlinear analytic system for the unknown state variable x(t), and functional series solutions of the analytic infinite-dimension

Multifractal Spectrum for Barycentric Averages

Let \(\left (X,\nu \right ) \) and Y be a measured space and a C A T(0) space, respectively. If \(\mathcal {M}_{2}(Y)\) is the set of measures on Y with finite second moment then a map \(bar:\mathcal {M}_{2}(Y)\rightarrow Y\) can be defined. Also, for any x∈X and for a map \(\varphi :X\rightarrow Y\), a sequence \(\left \{\mathcal {E}_{N,\varphi }(x)\right \} \) of empirical measures on Y can be introduced. The sequence \(\left \{ bar\left (\mathcal {E}_{N,\varphi }(x)\right ) \right \} \) replaces in C A T(0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map \(\overline {\varphi }\left (x\right )\)) almost surely for any x∈X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X:

$$K_{y,\varphi}=\left\{ x:\lim\limits_{N\rightarrow\infty}bar\left(\mathcal{E}_{N,\varphi}(x)\right) =y\right\} , $$

and we will obtain a variational formula for this multifractal spectrum.

     

Controllability of the Semilinear Beam Equation

In this paper, we prove the approximate controllability of the following semilinear beam equation: $$ \left\{ \begin{array}{lll} \displaystyle{\partial^{2} y(t,x) \over \partial t^{2}} & = & 2\beta\Delta\displaystyle\frac{\partial y(t,x)}{\partial t}- \Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),\; \mbox{in}\; (0,\tau)\times\Omega, \\ y(t,x) & = & \Delta y(t,x)= 0 , \ \ \mbox{on}\; (0,\tau)\times\partial\Omega, \\ y(0,x) & = & y_{0}(x), \ \ y_{t}(x)=v_{0}(x), x \in \Omega, \end{array} \right. $$ in the states space $Z_{1}=D(\Delta)\times L^{2}(\Omega)$ with the graph norm, where β?>?1, Ω is a sufficiently regular bounded domain in IR ^N , the distributed control u belongs to L ²([0,τ];U) (U?=?L ²(Ω)), and the nonlinear function $f:[0,\tau]\times I\!\!R\times I\!\!R\times I\!\!R\longrightarrow I\!\!R$ is smooth enough and there are a,c?∈?IR such that $a<\lambda_{1}^{2}$ and $$ \displaystyle\sup\limits_{(t,y,v,u)\in Q_{\tau}}\mid f(t,y,v,u) - ay -cu\mid<\infty, $$ where Q _τ ?=?[0,τ]×IR×IR×IR. We prove that for all τ?>?0, this system is approximately controllable on [0,τ].     

General energy decay rates for a weakly damped Timoshenko system

In this paper, we consider the following Timoshenko-type system:

Rolling of manifolds without spinning

The control model of rolling of a Riemannian manifold (M; g) onto another one $ \left( {\hat{M},\hat{g}} \right) $ consists of a state space Q of relative orientations (isometric linear maps) between their tangent spaces equipped with a so-called rolling distribution $ {\mathcal D} $ _R, which models the natural constraints of no-spinning and no-slipping of the rolling motion. It turns out that the distribution $ {\mathcal D} $ _R can be built as a sub-distribution of a so-called no-spinning distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ on Q that models only the no-spinning constraint of the rolling motion. One is thus motivated to study the control problem associated to $ {{\mathcal{D}}_{\overline{\nabla}}} $ and, in particular, the geometry of $ {{\mathcal{D}}_{\overline{\nabla}}} $ -orbits. Moreover, the definition of $ {{\mathcal{D}}_{\overline{\nabla}}} $ (contrary to the definition of $ {\mathcal D} $ _R) makes sense in the general context of vector bundles equipped with linear connections. The purpose of this paper is to study the distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ determined by the product connection $ \nabla \times \hat{\nabla} $ on a tensor bundle $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ induced by linear connections ?, $ \hat{\nabla} $ on vector bundles $ E\to M,\,\,\,\hat{E}\to \hat{M} $ . We describe completely the orbit structure of $ {{\mathcal{D}}_{\overline{\nabla}}} $ in terms of the holonomy groups of ?, $ \hat{\nabla} $ and characterize the integral manifolds of it. Moreover, we describe the general formulas for the Lie brackets of vector elds in $ {E^{*}}\otimes \hat{E} $ in terms of $ {{\mathcal{D}}_{\overline{\nabla}}} $ and the vertical tangent distribution of $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ . In the particular case of tangent bundles $ TM\to M,\,\,\,T\hat{M}\to \hat{M} $ and Levi-Civita connections, we describe in more detail how $ {{\mathcal{D}}_{\overline{\nabla}}} $ is related to the above mentioned rolling model, where these Lie brackets formulas provide an important tool for the study of controllability of the related control system.     

On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials

In this paper, we consider the high dimensional Schrödinger equation \( -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, \) where u(t) is a real analytic quasi-periodic symmetric matrix, \(E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})\) is a diagonal matrix with λ _j >0,j=1,…,n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann’s non-resonant condition, then for most of sufficiently large λ _j ,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions.     

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n?1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem.     

An Estimate for the Entropy of Hamiltonian Flows

In this paper, we present a generalization to Hamiltonian flows on symplectic manifolds of the estimate proved by Ballmann and Wojtkovski in [4] for the dynamical entropy of the geodesic flow on a compact Riemannian manifold of nonpositive sectional curvature. Given such a Riemannian manifold M, Ballmann and Wojtkovski proved that the dynamical entropy h _μ of the geodesic flow on M satisfies the inequalitywhere v is a unit vector in T _p M if p is a point in M, SM is the unit tangent bundle on M, K(v) is defined as , where is the Riemannian curvature of M, and μ is the normalized Liouville measure on SM. We consider a symplectic manifold M of dimension 2n, and a compact submanifold N of M, given by the regular level set of a Hamiltonian function on M; moreover, we consider a smooth Lagrangian distribution on N, and we assume that the reduced curvature of the Hamiltonian vector field with respect to the distribution is non-positive. Then we prove that under these assumptions, the dynamical entropy h _μ of the Hamiltonian flow with respect to the normalized Liouville measure on N satisfies      

Second-Order Viability Problems with Perturbation in Hilbert Spaces

In this paper, we study the existence of viable solutions to the differential inclusion

$ \ddot{x}(t) \in f\left( {t,x(t),\dot{x}(t)} \right) + F\left( {x(t),\dot{x}(t)} \right), $

where f is a Carathéodory single-valued map and F is an upper semi-continuous multifunction with compact values contained in the Clarke subdifferential ? _c V of an uniformly regular function V.

     

Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, { _s , s S} be the semigroup of translations in the function space X = V ^S , A = {A _n } be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X _f,,Adefined by the following formula:

On the Singularities of Families of Curve Congruences on Lorentzian Surfaces

In Nabarro (2011), we define and study the families of ??-conjugate curve congruences \({\mathcal {L}\mathcal {C}}^{i}_{\alpha }\) and the families of reflected ??-conjugate curve congruences \({\mathcal {L}\mathcal {R}}^{i}_{\alpha }\), i = 1, 2, associated to a self-adjoint operator ?? on a smooth and oriented surface M endowed with a Lorentzian metric. These families parametrize parts of the pencils of forms that link the equation of the ??-asymptotic (resp. ??-characteristic) curves and that of the ??-principal curves. There is a crucial difference with the Riemannian case due to the existence of lightlike curves. In this paper, we study the generic local singularities in the members of these families and describe the way they bifurcate within the families.     

Optimal Control on Nilpotent Lie Groups

Let G be a nilpotent Lie group and let = {X ₁,X ₂} be a bracket generating left invariant distribution on G. In this paper we study the left invariant optimal control problem on G defined by the differential equation

Genericity for Non-Wandering Surface Flows

Consider the set \(\chi ^{0}_{\text {nw}}\) of non-wandering continuous flows on a closed surface M. Then we show that such a flow can be approximated by a non-wandering flow v such that the complement M?Per(v) of the set of periodic points is the union of finitely many centers and finitely many homoclinic saddle connections. Using the approximation, the following are equivalent for a continuous non-wandering flow v on a closed connected surface M: (1) the non-wandering flow v is topologically stable in \(\chi ^{0}_{\text {nw}}\); (2) the orbit space M/v is homeomorphic to a closed interval; (3) the closed connected surface M is not homeomorphic to a torus but consists of periodic orbits and at most two centers. Moreover, we show that a closed connected surface has a topologically stable continuous non-wandering flow in \(\chi ^{0}_{\text {nw}}\) if and only if the surface is homeomorphic to either the sphere \(\mathbb {S}^{2}\), the projective plane \(\mathbb {P}^{2}\), or the Klein bottle \(\mathbb {K}^{2}\).     

The Deligne–Simpson Problem and the Connectedness of some Varieties

The Deligne–Simpson problem is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes or so that there exist irreducible (p+1)-tuples of matrices M _j ∈ C _j or A _j ∈ c _j satisfying the equality M ₁ ⋯ M _p+1 = I or A ₁ + ⋯ + A _p+1 = 0. The matrices M _j and A _j are interpreted as monodromy operators of regular linear systems and as matrices-residues of Fuchsian ones on the Riemann sphere. We prove that in the so-called simple case the subset or of the variety or consisting of all irreducible (p+1)-tuples (if nonempty) is connected. “Simple” means that the greatest common divisor of all quantities of Jordan blocks of a given size, of a given matrix M _j or A _j , and with a given eigenvalue is 1. To the memory of my mother     

Horizontal Holonomy for Affine Manifolds

In this paper, we consider a smooth connected finite-dimensional manifold M, an affine connection ? with holonomy group H ^? and Δ a smooth completely non integrable distribution. We define the Δ-horizontal holonomy group \({H^{\;\nabla }_{\Delta }}\) as the subgroup of H ^? obtained by ?-parallel transporting frames only along loops tangent to Δ. We first set elementary properties of \({H^{\;\nabla }_{\Delta }}\) and show how to study it using the rolling formalism Chitour and Kokkonen (2011). In particular, it is shown that \({H^{\;\nabla }_{\Delta }}\) is a Lie group. Moreover, we study an explicit example where M is a free step-two homogeneous Carnot group with m ≥ 2 generators, and ? is the Levi-Civita connection associated to a Riemannian metric on M, and show in this particular case that \({H^{\;\nabla }_{\Delta }}\) is compact and strictly included in H ^? as soon as m ≥ 3.