Convergence to steady states of solutions of semilinear evolutionary integral equations

We study the convergence and decay rate to a steady state of bounded solutions of the nonlinear evolutionary integral equation and we apply our abstract results to the viscoelastic Euler-Bernoulli beam and to Kelvin-Voigt solids.Received: 25 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004This work was partially supported by the DFG project Regularität und Asymptotik für elliptische und parabolische Probleme and by the grants GAR 201/01/D094, MSM 113200007.     

Convergence to diffusion waves for nonlinear evolution equations with different end states

In this paper, we consider the global existence and the asymptotic decay of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects:

(E)     

Anti-periodic solutions to nonlinear evolution equations

We deal with anti-periodic problems for nonlinear evolution equations with nonmonotone perturbations. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.     

Convergence to steady state and attractors for doubly nonlinear equations

This paper is concerned with the asymptotic behaviour of a class of doubly nonlinear parabolic-type equations. For degenerate equations, we prove thanks to the Lojasiewicz inequality that the solutions to some nonautonomous equations converge to a steady state. In the nondegenerate case, the existence of exponential attractors is shown by using the ?-trajectories method.     

Periodic solutions of nonlinear integral equations

The existence of a continuous periodic solution of the system

The first integral method for constructing exact and explicit solutions to nonlinear evolution equations

Problems that are modeled by nonlinear evolution equations occur in many areas of applied sciences. In the present study, we deal with the negative order KdV equation and the generalized Zakharov system and derive some further results using the so‐called first integral method. By means of the established first integrals, some exact traveling wave solutions are obtained in a concise manner. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions to nonlinear Volterra integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.     

Convergence of solutions of nonlinear differential equations

Summary Solutions of are said to converge if every pair of solutions x(t), y(t) satisfy x(t) − y(t) →0 as t → ∞. An invariance principle of LaSalle is used to determine conditions under which the solutions of converge. In certain cases the approach used does not require boundedness of solutions as has been required in most previous results on convergence of solutions. The results of this investigation are applied to a number of nonlinear second order differential equations. Sufficient conditions are also found for the convergence of solutions of certain functional differential equations. Entrata in Redazione il 10 febbraio 1976.     

Travelling wave solutions of nonlinear evolution equations by using the first integral method

In this paper, we established travelling wave solutions for some (2 + 1)-dimensional nonlinear evolution equations. The first integral method was used to construct travelling wave solutions of nonlinear evolution equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The first integral method presents a wider applicability for handling nonlinear wave equations.     

Convergence to steady state of solutions of the euler equations,I

In this paper we analyze the convergence to steady state of solutions of the compressible and the incompressible isentropic Euler equations in two space dimensions. In the compressible case, the original equations do not converge. We replace the equation of continuity with an elliptic equation for the density, obtaining a new set of equations, which have the same steady solution. In the incompressible case, the equation of continuity is replaced by a Poisson equation for the pressure. In both cases, we linearize the equations around a steady solution and show that the unsteady solution of the linearized equations converges to the steady solution, if the steady solution is sufficiently smooth. In the proof we consider how the energy of the time dependent part developes with time, and find that it decrease exponentially.     

Convergence of solutions to nonlinear dispersive equations without convexity conditions

This paper gets a series of results about the convergence of solutions {u_δ ^c} for partial differential equations of the form u_t + f_x(u) + δu_χχχ ≡ εu_χχ and u_t + f_χ(u) + δu_χχχ ≡ εu_χχ as ε and δ approach zero. Where the flux functions need no convexity conditions     

Existence of solutions to nonlinear Hammerstein integral equations and applications

In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu=u in the real Hilbert space L²(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.     

Stability for solutions to nonlinear nonautonomous evolution equations

In this paper, we find an estimate on d(u(t), K(t)), where u is a mild solution to the nonautonomous Cauchy problem \({\dot{u}(t) + A(t)u(t) \ni 0,\, t \geq s, u(s) = u_0}\) . Here, A(t) is a family of nonlinear multivalued, ω-accretive operators in a Banach space X, with D(A(t)) possibly depending on t, and K(t) a family of closed subsets in X.     

Convergence of solutions of von Karman evolution equations to equilibria

We consider von Karman evolution equations with nonlinear interior dissipation and with clamped boundary conditions. Under some conditions we prove that every energy solution converges to a stationary solution and establish a rate of convergence. Earlier this result was known in the case when the set of equilibria was finite and hyperbolic. In our argument we use the fact that the von Karman nonlinearity is analytic on an appropriate space and apply the Lojasiewicz–Simon method in the form suggested by A. Haraux and M. Jendoubi.     

Convergence of solutions of perturbed nonlinear differential equations

Summary We use the nonlinear variation of parameters formula to investigate the convergence of the solutions of nonlinear perturbed systems of differential equations. This research was supported in part by the National Science Foundation under grant GP-11543. Entrata in Redazione il 9 ottobre 1971.     

Blow-up time for solutions to some nonlinear Volterra integral equations

The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.     

Existence of solutions for some nonlinear integral equations

In this study, we present an existence of solutions for some nonlinear functional- integral equations which include many key integral and functional equations that appear in nonlinear analysis and its applications. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo’s theorem to obtain the mentioned aims in Banach algebra.     

一类非线性演化方程新的行波解

利用双函数法和吴消元法,得到了一类非线性演化方程在不同情况下的一系列显示精确解.Sinh-Gordon方程及Klein-Gordon方程作为该方程的特例也得到了相应的行波解.