Explosive solutions of stochastic nonlinear beam equations with damping

This study considers a class of damped stochastic nonlinear beam equations driven by multiplicative noise. By an appropriate energy inequality, we provide sufficient conditions such that the local solutions of the stochastic equations blow up with a positive probability or are explosive in an L²$L^2$ sense. We also derive estimates of the upper bound of the blow-up time.     

An existence result for a class of second order evolution equations of mixed type

We give an existence and uniqueness result for a linear abstract evolution equation of second order with some coefficient in front of the second temporal derivative which may degenerate to zero and change sign.     

Boundary feedback control of elastic beam equation with structural damping and stability

In this paper, we consider the partial differential equation of an elastic beam with structural damping by boundary feedback control. First, we prove this closed system is well-posed; then we establish the exponential stability for this elastic system by using a theorem whichbelongs to F. L. Huang^[2]; finally, we discuss the distribution and multiplicity of the spectrum of this system. These results are very important and useful in practical applications.     

New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping

Some new oscillation criteria are established for the nonlinear damped differential equation (r(t)y′)′+p(t)y′+q(t)f(y)=0 that are different from most known ones in the sense that they are based on a class of new functions Φ(t,s,r) defined in the sequel. Our results are sharper than some previous results which can be seen by the examples at the end of this paper.     

On oscillation of nonlinear second-order differential equations with damping term

The paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term. First it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question, and vice versa. Advantages and drawbacks of such translation are carefully examined. Then an oscillation criterion for the new class of equations is established. The principal result of the paper is compared to those reported in the literature, and an illustrative example to which known oscillation criteria fail to apply is provided.     

Some new oscillation theorems for second order nonlinear elliptic equations with damping

In this paper, some new oscillation criteria are obtained for second order elliptic differential equations with damping of the form

     

Stationary distributions of second order stochastic evolution equations with memory in Hilbert spaces

In this paper, we consider stationarity of a class of second-order stochastic evolution equations with memory, driven by Wiener processes or Lévy jump processes, in Hilbert spaces. The strategy is to formulate by reduction some first-order systems in connection with the stochastic equations under investigation. We develop asymptotic behavior of dissipative second-order equations and then apply them to time delay systems through Gearhart–Prüss–Greiner’s theorem. The stationary distribution of the system under consideration is the projection on the first coordinate of the corresponding stationary results of a lift-up stochastic system without delay on some product Hilbert space. Last, two examples of stochastic damped wave equations with memory are presented to illustrate our theory.     

Existence and non-existence of positive solutions of BVPs for fractional order elastic beam equations with a non-Caratheodory nonlinearity

In this article, the existence and non-existence results on positive solutions of two classes of boundary value problems for nonlinear singular fractional order elastic beam equations is established. Here f depends on x  x^′$x^{\prime }$ and x^″,f$x^{″}\text{,}f$ may be singular at t=0$t=0$ and t=1$t=1$ and f may be a non-Caratheodory function. The analysis relies on the well known Schauder’s fixed point theorem. By applying iterative techniques, results on the existence of positive solutions are obtained and the iterative scheme which starts off with zero function for approximating the solution is established. The iterative scheme obtained is very useful and feasible for computational purpose. Examples and their numerical simulation are presented to illustrate the main theorems. A conclusion section is also given at the end of this paper.     

Oscillation criteria for second-order nonlinear differential equations with integrable coefficient

In this paper, we consider the second-order nonlinear differential equation

[a(t)|y′(t)|^σ−1y′(t)|′+q(t)f(y(t))=r(t)

where σ > 0 is a constant, a C(R, (0, ∞)), q C(R, R), f C(R, R), xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples which dwell upon the importance of our results are also included.     

Boundary stabilization for a non‐linear beam on elastic bearings

In this paper, we study the equation under non‐linear boundary conditions which model the vibrations of a beam clamped at x=0 and supported by a non‐linear bearing at x=L. By adding only one damping mechanism at x=L, we prove the existence of a global solution and exponential decay of the energy. Copyright © 2001 John Wiley & Sons, Ltd.     

Existence and uniqueness results for the bending elastic beam equations

This paper deals with the solvability of the fourth-order boundary value problem $\left\{\begin{array}{c}{u}^{\left(4\right)}=f(x,u,u^{\prime \prime }),0\le x\le 1,\hfill \\ u\left(0\right)=u\left(1\right)=u^{\prime \prime }\left(0\right)=u^{\prime \prime }\left(1\right)=0,\hfill \end{array}\right.$ which models a statically bending elastic beam whose two ends are simply supported, where $f:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous. Inequality conditions on $f$ guaranteeing the existence and uniqueness of solution are presented. The inequality conditions allow that $f(x,u,v)$ may be superlinear growth on $u$ and $v$ as $\left|(u,v)\right|\to \infty$.     

Regularity for solutions of nonlinear second order evolution equations

This paper deals with the existence of solutions for the class of nonlinear second order evolution equations. The regularity and a variation of solutions of the given equations are also given. As particular cases of our general formulation, some results for Volterra integrodifferential equations of the hyperbolic type are given.     

Global solution and asymptotic behavior for the variable coefficient beam equation with nonlinear damping

This paper is concerned with the initial‐boundary value problem for a variable coefficient beam equation with nonlinear damping. Such a model arises from the vertical deflections of a damped extensible elastic inhomogeneous beam whose density depends on time and position. By using the Faedo–Galerkin method and energy method, we obtain the existence and uniqueness of global strong solution. Furthermore, the exponential decay estimate for the total energy is also derived. Copyright © 2015 John Wiley & Sons, Ltd.     

广义静态梁方程非负解的存在性

运用Leray—Schauder拓扑理论,证明了广义静态梁方程和静态梁方程非负解的存在性,仅要求非线性项f在原点的某个邻域满足一定的符号条件,突破了以往对非线性项f的增长性限制．所获结果对工程设计具有重要的理论意义和实用价值．     

A monotone iterative technique for solving the bending elastic beam equations

This paper discusses the solvability of the fourth-order boundary value problem

     

An existence theorem of positive solutions for elastic beam equation with both fixed end-points

Abstract. By using the degree theory on cone an existence theorem of positive solution for aclass of fourth-order two-point BVP‘s is obtained. This class of BVP‘s usually describes the de-formation of the elastic beam with both fixed end-points.     

Observability of elastic vibrations of a beam

The observability problem for beam vibrations described by a fourth-order partial differential equation with various boundary conditions is considered. Dynamic observability problems are solved in terms of boundary conditions and observations of the beam state at certain fixed instants of time.     

Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions

We model the roadway of a suspension bridge as a thin rectangular plate and we study in detail its oscillating modes. The plate is assumed to be hinged on its short edges and free on its long edges. Two different kinds of oscillating modes are found: longitudinal modes and torsional modes. Then we analyze a fourth order hyperbolic-like equation describing the dynamics of the bridge. In order to emphasize the structural behavior we consider an isolated equation with no forcing and damping. Due to the nonlinear behavior of the cables and hangers, a structural instability appears. With a finite dimensional approximation we prove that the system remains stable at low energies while numerical results show that for larger energies the system becomes unstable. We analyze the energy thresholds of instability and we show that the model allows to give answers to several questions left open by the Tacoma collapse in 1940.     

Exponential decay for elastic systems with structural damping and infinite delay

ABSTRACT

In this paper, we consider nonlinear evolution equations of second order in Banach spaces involving unbounded delay, which can model an elastic system with structural damping involving infinite delays. By using fixed point for condensing maps, we prove the existence and exponential decay of mild solutions. The obtained results can be applied to the nonlinear vibration equation of elastic beams with structural damping and infinite delay.     

Interval oscillation criteria for a forced second-order nonlinear differential equations with damping

In this paper, we are concerned with the oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using classical variational principle and averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. Example is also given to illustrate the results.