Nonlinear oscillations and boundary value problems for Hamiltonian systems

We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms $$\begin{gathered} \dot x(t) = H'_p (t, x(t), p(t)) + g(t), \hfill \\ \dot p(t) = - H'_x (t, x(t), p(t)) - f(t). \hfill \\ \end{gathered} $$ Among these problems is the existence of T-periodic solutions, namely those satisfying x(t+T)=x(t) and p(t+T)+p(t). A special study is made of the classical case, where H(x, p)=1/2 |p|²+V(x). In the case of parametric oscillations, where (f, g)=(0, 0) and t ? H(t, x, p) is T-periodic, we give a lower bound for the true (minimal) period of the T-periodic solution (x, p) produced by our method, and we prove the existence of an infinite number of subharmonics.     

On exact solutions of a class of fractional Euler–Lagrange equations

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where _a ^c D _t ^α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange

On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument

This paper investigates stability and boundedness of solutions to third order nonlinear differential equation with retarded argument: $$\begin{array}{l}x'''(t)+\varphi\bigl(x(t-r),x'(t-r),x''(t-r)\bigr)x''(t)\\\qquad{}+\psi\bigl(x'(t-r)\bigr)+h\bigl(x(t-r)\bigr)\\\quad=p\bigl(t,x(t),x(t-r),x'(t),x'(t-r),x''(t)\bigr).\end{array}$$ By the use of the Lyapunov functional, sufficient conditions for stability and boundedness of solutions to the considered equations are obtained. Examples are introduced throughout the paper for illustrations.     

Oscillating Solutions and Large ω-limit Sets in a Degenerate Parabolic Equation

The paper deals with positive solutions of the initial-boundary value problem for with zero Dirichlet data in a smoothly bounded domain . Here is positive on (0,∞) with f(0) = 0, and λ₁ is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence of a sufficiently strong degeneracy. More precisely, writing , it is shown that if then there exist global classical solutions of (*) satisfying and . Under the additional structural assumption , s > 0, this result can be sharpened: If then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition ensures that all solutions of (*) will stabilize to a single equilibrium.      

A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains

Archive for Rational Mechanics and Analysis - We investigate quantitative properties of the nonnegative solutions $${u(t,x)\geq 0}$$ to the nonlinear fractional diffusion equation, $${\partial_t u...     

Variational Methods in Semilinear Wave Equation with Nonlinear Boundary Conditions and Stability Questions

We present variational approach to the semilinear equation of the vibrating string \(x_{tt}(t,y)-{\Delta } x(t,y)+l(t,y,x(t,y))=0\) in bounded domain and certain type of nonlinearity on the boundary. To this effect we derive new dual variational method. Next the question of stability of solutions with respect to initial conditions is discussed.     

Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion equation with delay

Delta Shock Waves in the Shallow Water System

We consider a Riemann problem for the shallow water system \(u_{t} +\big (v+\textstyle \frac{1}{2}u^{2}\big )_{x}=0\), \(v_{t}+\big (u+uv\big )_{x}=0\) and evaluate all singular solutions of the form \(u(x,t)=l(t)+b(t)H\big (x-\gamma (t)\big )+a(t)\delta \big (x-\gamma (t)\big )\), \(v(x,t)=k(t)+c(t)H\big (x-\gamma (t)\big )\), where \(l,b,a,k,c,\gamma :\mathbb {R}\rightarrow \mathbb {R}\) are \(C^{1}\)-functions of time t, H is the Heaviside function, and \(\delta \) stands for the Dirac measure with support at the origin. A product of distributions, not constructed by approximation processes, is used to define a solution concept, that is a consistent extension of the classical solution concept. Results showing the advantage of this framework are briefly presented in the introduction.     

Regional Blow-up for a Doubly Nonlinear Parabolic Equation with a Nonlinear Boundary Condition

In this work, positive solutions to a doubly nonlinear parabolic equation with a nonlinear boundary condition are considered. We study the problem where 0 < m, r, α < ∞ are parameters. It is known that for some values of the parameters there are solutions that blow up in finite time. We determine in terms of α ,m, r the blow-up sets for these solutions. We prove that single point blow-up occurs if max{m, r} < α, global blow-up appears for the range of parameters 0 < m < α < r and regional blow-up takes place if 0 < m < α = r and . In this case the blow-up set consists of the interval .     

Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations

We study the solutions of the nonstationary incompressible Navier–Stokes equations in , of self-similar form , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity field to its corresponding initial datum a(x).     

Boundary layer phenomena for differential-delay equations with state-dependent time lags,I.

In this paper we begin a study of the differential-delay equation

Persistence of Semi-Trajectories

We consider diffeomorphisms f of a smooth compact riemannian mainfold M and its suspension flow . Assuming some regularity of the stable (unstable) sets at the points we prove the persistence in the future of {f ⁿ (x), n ≥ 0} or , i.e., that C ⁰ small perturbations g of f have a semi-trajectory that closely shadows {f ⁿ (x), n ≥ 0} and that the suspension of g has also a semi-trajectory that closely shadows . In case x belongs to a minimal set of f we show that the assumptions concerning the regularity of stable and unstable sets could be reduced to a neighbourhood of x.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

The weak damped oscillation in a random medium

The paper gives a solution of the random differential equation with random coefficient, that is , where K (t) is a random process and is a damp coefficient, it is a little parameter.     

Stochastic instability of nonlinear oscillations

Consider a dynamic system whose behavior is described by

$$x'' + \omega ^2 _0 (1 + \alpha x^2 )x + F(t)x = 0$$     

Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type

The paper is concerned with the existence and stability of an almost periodic solution of the system with deviating argument .The Wexler inequality for the Cauchy matrix is used. Conditions for the stability of the solution are given.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 295–301, July–September, 2004.     

Existence Theory by Front Tracking for General Nonlinear Hyperbolic Systems

We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension

2D Stochastic Navier–Stokes Equations with a Time-Periodic Forcing Term

We study the long time behavior of the solution X(t, s, x) of a 2D-Navier–Stokes equation subjected to a periodic time dependent forcing term. We prove in particular that as , approaches a periodic orbit independently of s and x for any continuous and bounded real function .      

Modeling and Analog Realization of the Fundamental Linear Fractional Order Differential Equation

This paper provides a rational function approximation of the irrational transfer function of the fundamental linear fra- ctional order differential equation, namely, whose transfer function is given by for 0<m<2. Simple methods, useful in system and control theory, which consists of approximating, for a given frequency band, the transfer function of this fractional order system by a rational function are presented. The impulse and step responses of this system are derived and simple analog circuit which can serve as fundamental analog fractional order system is also obtained. Illustrative examples are presented to show the exactitude and the usefulness of the approximation methods.