Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions

The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

The lifespan for 3D quasilinear wave equations with nonlinear damping terms

In this paper, we study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain Ω. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results in [K. Agre, M.A. Rammaha, System of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006) 1235-1270], especially, the blow-up of weak solutions in the case of non-negative energy.     

Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping

This paper is concerned with the p-system of hyperbolic conservation laws with nonlinear damping. When the constant states are small, the solutions of the Cauchy problem for the damped p-system globally exist and converge to their corresponding nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation given by the Darcy's law. The optimal convergence rates are also obtained. In order to overcome the difficulty caused by the nonlinear damping, a couple of correction functions have been technically constructed. The approach adopted is the elementary energy method together with the technique of approximating Green function. On the other hand, when the constant states are large, the solutions of the Cauchy problem for the p-system will blow up at a finite time.     

Optimal convergence rates to nonlinear diffusion waves for the compressible Euler-Poisson system with damping

In this work, we study the 1-D isentropic bipolar hydrodynamic model. This model takes the form of compressible Euler-Poisson system with nonlinear damping added to the momentum equations. Under some smallness conditions, the solutions to the Cauchy problem of the system globally exist and convergence to the nonlinear diffusion waves, which are the corresponding solutions of nonlinear parabolic equations given by the Darcy's law with a specified initial data. The optimal convergence rates are obtained by Green function method when the initial perturbation is in L¹-space.     

Asymptotic stability for anisotropic Kirchhoff systems

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the p(x)-Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms.     

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.     

On shape changing solutions of a generalized inhomogeneous Hirota equation

In a recent series of papers, Kavitha et al. [2,3,4] solved three inhomogeneous nonlinear Schrödinger (INLS) integro-differential equation under the influence of a variety of nonlinear inhomogeneities and nonlocal damping by the modified extended tangent hyperbolic function method. They obtained several kinds of exact solitary solutions accompanied by the shape changing property. In this paper, we demonstrate that most of exact solutions derived by them do not satisfy the nonlinear equations and consequently are wrong. Furthermore, we study a generalized Hirota equation with spatially-inhomogenetiy and nonlocal nonlinearity. Its integrability is explored through Painlevé analysis and N-soliton solutions are obtained based on the Hirota bilinear method. Effects of linear inhomogeneity on the profiles and dynamics of solitons are also investigated graphically.     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping

In this paper we consider the so-called p-system with linear damping, and we will prove an optimal decay estimates without any smallness conditions on the initial error. More precisely, if we restrict the initial data (V₀,U₀) in the space H³(R⁺)∩L1,γ(R⁺)×H²(R⁺)∩L1,γ(R⁺), then we can derive faster decay estimates than those given in [P. Marcati, M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech. 7 (2) (2005) 224-240; H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differential Equations 174 (1) (2001) 200-236] and [M. Jian, C. Zhu, Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant, J. Differential Equations 246 (1) (2009) 50-77].     

Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schrödinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

DECAY RATES AND CONVERGENCE OF SOLUTIONS TO SYSTEM OF ONE-DIMENSIONAL VISCOELASTIC MODEL WITH DAMPING

This paper considers the asymptotic behavior of solutions to the system of onedimensional viscoelastic model with damping and prove that the corresponding solutions time-asymptotically behave like nonlinear diffusion wave as in [4,11]. In addition, It is also shown that the system of one-dimensional viscoelastic model with damping is a viscosity approximation of a hyperbolic conservation laws with damping.     

DECAY RATES AND CONVERGENCE OF SOLUTIONS TO SYSTEM OP ONE-DIMENSIONAL VISCOELASTIC MODEL WITH DAMPING

This paper considers the asymptotic behavior of solutions to the system of one-dimensional viscoelastic model with damping and prove that the corresponding solutions time-asymptotically behave like nonlinear diffusion wave as in [4,11]. In addition, It is also shown that the system of one-dimensional viscoelastic model with damping is a viscosity approximation of a hyperbolic conservation laws with damping.     

Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the rate of decay of solutions of the wave equation with localized nonlinear damping without any growth restriction and without any assumption on the dynamics. Providing regular initial data, the asymptotic decay rates of the energy functional are obtained by solving nonlinear ODE. Moreover, we give explicit uniform decay rates of the energy. More precisely, we find that the energy decays uniformly at last, as fast as 1/(ln(t+2))^2−δ,∀δ>0, when the damping has a polynomial growth or sublinear, and for an exponential damping at the origin the energy decays at last, as fast as 1/(ln(ln(t+e²)))^2−δ,∀δ>0.     

Global nonexistence for logarithmic wave equations with nonlinear damping and distributed delay terms

This paper is concerned with global nonexistence of solutions for a logarithmic wave equation with nonlinear damping and distributed delay terms. Due to the simultaneous presence of nonlinear damping and logarithmic source terms, we have difficulty in use of the concavity method. Applying the energy estimates, we show the global nonexistence of solutions with not only non-positive initial energy but also positive initial energy.     

Global existence and nonexistence for a nonlinear wave equation with damping and source terms

In this paper we consider a nonlinear wave equation with damping and source term on the whole space. For linear damping case, we show that the solution blows up in finite time even for vanishing initial energy. The criteria to guarantee blowup of solutions with positive initial energy are established both for linear and nonlinear damping cases. Global existence and large time behavior also are discussed in this work. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when α?max{m,p}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension N=1, the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if 0?α<1, and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown.     

Oscillation of second-order nonlinear differential equations with damping

We study oscillatory properties of solutions to a class of nonlinear second-order differential equations with a nonlinear damping. New oscillation criteria extend those reported in [ROGOVCHENKO, Yu. V.—TUNCAY, F.: Oscillation criteria for second-order nonlinear differential equations with damping, Nonlinear Anal. 69 (2008), 208–221] and improve a number of related results.     

Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms

This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.     

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.     

具阻尼的Gross-Pitaevskii(GP)方程在二维空间中的稳定性

本文讨论出现在吸引玻色-爱因斯坦凝聚中的一类带调和势的阻尼非线性Schr dinger方程.对照玻色-爱因斯坦凝聚的物理性质,证明了阻尼参数存在一个门槛值,即当阻尼参数大于该门槛值时,初值问题的解整体存在;当阻尼参数小于该门槛值时,其初值问题的解将在有限时间内坍塌.