A Variational Analysis for the Moving Finite Element Method for Gradient Flows

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.     

Well-posedness for an extended Penrose–Fife phase-field model with energy balance supplied by Dirichlet boundary conditions

In this study we consider the non-isothermal phase-field model proposed by Penrose and Fife [Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62]. The system consists of the energy balance law (a nonlinear heat equation) and an equation that describes space-time changes in the order parameter (the Ginzburg–Landau equation). For the energy balance law, we consider the general nonlinear heat flux arising in non-equilibrium thermodynamics and impose the Dirichlet boundary condition. For the order parameter, we impose a constraint and thus consider a parabolic variational inequality. We prove the well-posedness of the problem: the system yields a unique solution that depends continuously upon given data.     

Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

For the Cauchy problem for the nonlinear wave equation with nonlinear damping and source terms we define stable and unstable sets for the initial data. We prove that, if during the evolution the solution enters into the stable set, the solution is global and we are able to estimate the decay rate of the energy. If during the evolution the solution enters into the unstable set, the solution blows up in finite time.     

A Multiresolution Method for Numerical Reduction and Homogenization of Nonlinear ODEs

The multiresolution analysis (MRA) strategy for the reduction of a nonlinear differential equation is a procedure for constructing an equation directly for the coarse scale component of the solution. The MRA homogenization process is a method for building a simpler equation whose solution has the same coarse behavior as the solution to a more complex equation. We present two multiresolution reduction methods for nonlinear differential equations: a numerical procedure and an analytic method. We also discuss one possible appproach to the homogenization method.     

On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation

The modified method of simplest equation is powerful tool for obtaining exact and approximate solutions of nonlinear PDEs. These solutions are constructed on the basis of solutions of more simple equations called simplest equations. In this paper we study the role of the simplest equation for the application of the modified method of simplest equation. We follow the idea that each function constructed as polynomial of a solution of a simplest equation is a solution of a class of nonlinear PDEs. We discuss three simplest equations: the equations of Bernoulli and Riccati and the elliptic equation. The applied algorithm is as follows. First a polynomial function is constructed on the basis of a simplest equation. Then we find nonlinear ODEs that have the constructed function as a particular solution. Finally we obtain nonlinear PDEs that by means of the traveling-wave ansatz can be reduced to the above ODEs. By means of this algorithm we make a first step towards identification of the above-mentioned classes of nonlinear PDEs.     

Blow up of Solutions of a Nonlinear Viscoelastic Wave Equation

We consider a nonlinear viscoelastic wave equation with nonlinear source term. Under suitable conditions on g, it is proved that any weak solution with negative initial energy blows up in finite time if p>2.     

Exterior nonlinear Neumann problem

We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical. We establish the existence of least energy solutions. In the subcritical case the coefficient b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions.     

Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrödinger type equation on torus

We consider the time local and global well-posedness for the fourth order nonlinear Schrödinger type equation (4NLS) on the torus. The nonlinear term of (4NLS) contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to (4NLS).     

Energy Decay for the Strongly Damped Nonlinear Beam Equation and Its Applications in Moving Boundary

We study the existence and energy decay of solutions for the strongly damped nonlinear beam equation. We apply a method based on Nakao method to show that the solution decays exponentially, and to obtain precise estimates of the constants in the estimates. Finally, we discuss its applications in moving boundary.     

Highly accurate algorithms endowing with boundary functions for solving a nonlinear beam equation involving an integral term

In this paper, the problem of a nonlinear beam equation involving an integral term of the deformation energy, which is unknown before the solution, under different boundary conditions with simply supported, 2‐end fixed, and cantilevered is investigated. We transform the governing equation into an integral equation and then solve it by using the sinusoidal functions, which are chosen both as the test functions and the bases of numerical solution. Because of the orthogonality of the sinusoidal functions, we can find the expansion coefficients of the numerical solution that are given in closed form by using the Drazin inversion formula. Furthermore, we introduce the concept of fourth‐order and fifth‐order boundary functions in the solution bases, which can greatly raise the accuracy over 4 orders than that using the partial boundary functions. The iterative algorithms converge very fast to find the highly accurate numerical solutions of the nonlinear beam equation, which are confirmed by 6 numerical examples.     

A stable and conservative finite difference scheme for the Cahn-Hilliard equation

Summary. We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is . Numerical examples demonstrate the effectiveness of the proposed scheme. Received July 22, 1997 / Revised version received October 19, 1999 / Published online August 2, 2000     

The Neumann boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution to a nonlocal Cahn-Hilliard equation on a bounded domain. The equation generates a gradient flow for a free energy functional with nonlocal interaction. Also we apply a nonlinear Poincaré inequality to show the existence of an absorbing set in each constant mass affine space.     

Asymptotic Behavior for Petrovsky Equation with Localized Damping

In this paper, we investigate asymptotic behavior for the solution of the Petrovsky equation with locally distributed damping. Without growth condition on the damping at the origin, we extend the energy decay result in Martinez (Rev. Math. Complut. Madr. 12(1):251–283, 1999) for the single wave equation to the Petrovsky equation. The explicit energy decay rate is established by using piecewise multiplier techniques and weighted nonlinear integral inequalities.     

Global well-posedness for strongly damped viscoelastic wave equation

We study the initial boundary value problem for the nonlinear viscoelastic wave equation with strong damping term and dispersive term. By introducing a family of potential wells we not only obtain the invariant sets, but also prove the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, we establish a blow-up result for certain solutions with arbitrary positive initial energy (high energy case)     

Initial Value Problem for a Generalized Korteweg－de Vries Equation with Singular Integral－Differential Terms

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一类非线性高阶Kirchhoff型方程的初边值问题

本文研究了一类具有非线性耗散项的高阶Kirchhoff型方程的初边值问题.通过构造稳定集讨论了此问题整体解的存在性,应用Nakao的差分不等式建立了解能量的衰减估计.在初始能量为正的条件下,证明了解在有限时间内发生blow-up,并且给出了解的生命区间估计.     

Beam evolution equation with variable coefficients

We investigate a initial‐boundary value problem for the nonlinear beam equation with variable coefficients on the action of a linear internal damping. We show the existence of a unique global weak solution and that the energy associated with this solution has a rate decay estimate. Besides, we prove the existence and uniqueness of non‐local strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

Multi-Solitary Waves for the Nonlinear Klein-Gordon Equation

We consider the nonlinear Klein-Gordon equation in ? ^d . We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.