Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Some remarks on global existence to the Cauchy problem of the wave equation with nonlinear dissipation

In this paper we prove the existence of global decaying H² solutions to the Cauchy problem for a wave equation with a nonlinear dissipative term by constructing a stable set in H¹(?ⁿ ). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H² solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.     

Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system

In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure, and are given as the union of the unstable manifolds of a set of hyperbolic global solutions which are the non-autonomous analogues of the hyperbolic equilibria. We also prove, again parallel to the autonomous case, that all solutions converge as t→+∞ to one of these hyperbolic global solutions. We then show how to apply these results to systems that are asymptotically autonomous as t→−∞ and as t→+∞, and use these relatively simple test cases to illustrate a discussion of possible definitions of a forwards attractor in the non-autonomous case.     

Existence and regularity of solution of mixed boundary value problem of a Busemann-equation with nonlinear absorb term

The Busemann-equation is a classical equation coming from fluid dynamics. The well-posed problem and regularity of solution of Busemann-equation with nonlinear term are interesting and important. The Busemann-equation is elliptic in y>0 and is degenerate at the line y=0 in R². With a special nonlinear absorb term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain. By means of elliptic regularization technique, a delicate prior estimate and compact argument, we show that the solution of mixed boundary value problem of Busemann-equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary on some conditions. The result is better than the result of classical boundary degenerate elliptic equation.     

Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system

We consider a structural acoustic wave equation with nonlinear acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with boundary conditions on the interface. We prove wellposedness in the Hadamard sense for strong and weak solutions. The main tool used in the proof is the theory of nonlinear semigroups. We present the system of partial differential equations as a suitable Cauchy problem . Though the operator A is not maximally dissipative we are able to show that it is a translate of a maximally dissipative operator. The obtained semigroup solution is shown to satisfy a suitable variational equality, thus giving weak solutions to the system of PDEs. The results obtained (i) dispel the notion that the model does not generate semigroup solutions, (ii) provide treatment of nonlinear models, and (iii) provide existence of a correct state space which is invariant under the flow-thus showing that physical model under consideration is a dynamical system. The latter is obtained by eliminating compatibility conditions which have been assumed in previous work (on the linear case).     

双摆内共振分岔分析

应用normal form理论,首先分析了复摆自治系统在1:1内共振临界点附近的Hopf分岔解及其在参数平面上的分岔转迁集的解析表达式,并与数值解进行了比较;然后,应用数值方法,得到了复摆非自治系统通向混沌的过程。     

Global attractors for von Karman evolutions with a nonlinear boundary dissipation

Dynamic von Karman equations with a nonlinear boundary dissipation are considered. Questions related to long time behaviour, existence and structure of global attractors are studied. It is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global (compact) attractor for all weak (finite energy) solutions. This result has been known in the case of full interior dissipation, but it is new in the case when the boundary damping is the main dissipative mechanism in the system. In addition, we prove that fractal dimension of the attractor is finite. The proofs depend critically on the infinite speed of propagation associated with the von Karman model considered.     

A non-autonomous second order boundary value problem on the half-line

By variational arguments, the existence of a solution to a non-autonomous second-order boundary problem on the half-line is proven. The corresponding autonomous problem has no solution, revealing significant differences between the autonomous and the non-autonomous case.     

Critical exponents for nonlinear diffusion equations with nonlinear boundary sources

This paper is concerned with the large time behavior of solutions to two types of nonlinear diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q₀ and the critical Fujita exponent qc for the problems considered, and show that q₀=qc for the multi-dimensional porous medium equation and non-Newtonian filtration equation with nonlinear boundary sources. This is quite different from the known results that q₀<qc for the one-dimensional case.     

Adaptive synchronization to a general non-autonomous chaotic system and its applications

In this paper, new adaptive synchronous criteria for a general class of n-dimensional non-autonomous chaotic systems with linear and nonlinear feedback controllers are derived. By suitable separation between linear and nonlinear terms of the chaotic system, the phenomenon of stable chaotic synchronization can be achieved using an appropriate adaptive controller of feedback signals. This method can also be generalized to a form for chaotic synchronization or hyper-chaotic synchronization. Based on stability theory on non-autonomous chaotic systems, some simple yet less conservative criteria for global asymptotic synchronization of the autonomous and non-autonomous chaotic systems are derived analytically. Furthermore, the results are applied to some typical chaotic systems such as the Duffing oscillators and the unified chaotic systems, and the numerical simulations are given to verify and also visualize the theoretical results.     

A nonlinear singular diffusion equation with source

In this paper, the existence, uniqueness and dependence on initial value of solution for a singular diffusion equation with nonlinear boundary condition are discussed. It is proved that there exists a unique global smooth solution which depends on initial data in L¹ continuously.     

Error estimates for discontinuous Galerkin method for nonlinear parabolic equations

We consider the nonlinear parabolic partial differential equations. We construct a discontinuous Galerkin approximation using a penalty term and obtain an optimal L^∞(L²) error estimate.     

On an exact control problem for a semilinear wave equation with an unknown source

The exact controllability of a semilinear wave equation, with Dirichlet boundary control on a part of the boundary and an unknown source, is shown. The nonlinear term has at most a linear growth, the initial and target spaces are L²(Ω)×H⁻¹(Ω).     

Long time stability and convergence for fully discrete nonlinear galerkin methods

The aim of this paper is to analyze the fully discrete nonlinear Galerkin methods, which are well suited to the long time integration of dissipative partial differential equations.

With the help of several time discrete Gronwall lemmas, we are able to prove L ^∞(IR⁺,H ^α ) (α=0,1) stabilities of the fully discrete nonlinear Galerkin methods under a less restrictive time step constraint than that of the classical Galerkin methods.     

Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity

We study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ|u|^p−1u, where 1<p<1+2/n, n is the space dimension and λ is a complex constant satisfying Imλ<0. We show the time decay estimates and the large-time asymptotics of the solution, when the space dimension n?3, p is sufficiently close to 1+2/n and the initial data is sufficiently small.     

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H¹ and L² error estimates for the velocity and the L² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.