General Energy Decay of Solutions for A Wave Equation with Nonlocal Nonlinear Damping and Source Terms

We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is how to handle with the nonlocal nonlinear damping term.Our result extends and improves the result in the literature such as the work by Jorge Silva and Narciso(Evolution Equation and Control Theory,2017(6):437-470)and Narciso(Evolution Equations and Control Theory,2020,9(2):487-508).

Symmetry Reductions, Integrability and Solitary Wave Solutions to High-Order Modified Boussinesq Equations with Damping Term

Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.

Energy decay for a class of nonlinear wave equations with a critical potential type of damping

The Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│）-1 and a nonlinearity │u│p-1u is studied.The total energy decay estimates of the global solutions are obtained by using multiplier techniques to establish identity ddtE（t）＋F（t）=0 and skillfully selecting f（t）,g（t）,h（t）when the initial data have a compact support.Using the similar method,the Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│＋t）-1 and a nonlinearity │u│p-1u is studied,similar solutions are obtained when the initial data have a compact support.

Travelling solitary wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order

In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved.

Blow-Up Result for a Semi-Linear Wave Equation with a Nonlinear Memory Term of Derivative Type

In this paper,we study the blow-up of solutions to a semi-linear wave equation with a nonlinear memory term of derivative type.By using methods of an iteration argument and di erential inequalities,we obtain the blow-up result for the semi-linear wave equation when the exponent of p is under certain conditions.Meanwhile,we derive an upper bound of the lifespan of solutions to the Cauchy problem for the semi-linear wave equation.

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

Classification of Single Traveling Wave Solutions to the Generalized Strong Nonlinear Boussinesq Equation without Dissipation Terms in <i>P</i>= 1

By the complete discrimination system for polynomial method, we obtained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq equation without dissipation terms in p=1.

An Improved Numerical Simulation Mode of Nonlinear Wave with Consideration of High Order Terms

SUN Da-peng BAO Wei-bin, WU Hao and LI Yu-cheng （ In this paper the 0-1 combined BEM is adopted to subdivide the computational domain boundary, and to discretize the Green＇s integral expression based on Laplace equation. The FEM is used to subdivide the wave surface and deduce the surface equation which satisfies the nonlinear boundary conditions on the surface. The equations with potential function and wave surface height as an unknown quantity by application of Taylor expansion approach can be solved by iteration within the time step. In m-time iteration within the computational process of time step （n-1）At to nat, the results of the previous iteration are taken as the initial value of the two-order unknown terms in the present iteration. Thus, an improved tracking mode of nonlinear wave surface is estabIished, and numerical results of wave tank test indicate that this mode is improved obviously and is more precise than the previous numerical model which ignored the two-order unknown terms of wave surface location and velocity potential function in comparison with the theoretical values.

GLOBAL EXISTENCE AND L^p ESTIMATES FOR SOLUTIONS OF DAMPED WAVE EQUATION WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL SPACE

In this article, the author studies the Cauchy problem of the damped wave equation with a nonlinear convection term in multi-dimensions. The author shows that a classical solution to the Cauchy problem exists globally in time under smallness condition on the initial perturbation. Furthermore, the author obtains the L^p （2 ≤ p ≤ ∞） decay estimates of the solution.

Oscillation of Second-order Nonlinear Dynamic Equation on Time Scales

： The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many knownresults for second order dynamic equations. Some examples are given to illustrate the main results of this article.

Existence and asymptotic behavior for systems of nonlinear hyperbolic equations

The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.

An Explicit Lower Bound for Blow Up Time in a Class of Nonlinear Wave Equations with Nonlinear Damping and Source Terms

This paper deals with an initial boundary value problem for a class of nonlinear wave equation with nonlinear damping and source terms whose solution may blow up in finite time.An explicit lower bound for blow up time is determined by means of a differential inequality argument if blow up occurs.

Approximate Inertial Manifolds to the Generalized Symmetric Regularized Long Wave Equations with Damping Term

Abstract In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the global attractor are derived.

Nonlinear Internal Damping of Wave Equations with Variable Coefficients

For wave equations with variable coefficients on regions which are not necessarily smooth, we study the energy decay rate when a nonlinear damping is applied on a suitable subrigion.

Stability of Planar Diffusion Wave for the Quasilinear Wave Equation with Nonlinear Damping

In this paper, we will show that under some smallness conditions, the planar diffusion wave v（x1/√＋t）is stable for a quasilinear wave equation with nonlinear damping：vtt-△f（v）＋vt＋g（vt）=0_3 x=（x1,x2…,xn）∈Rn,where v（x1/√1＋t）is the unique similar solution to the one dimensional nonlinear heat equa-tion：vt-f（v）x1 x1=0,f＇（v）〉0,v＋（∞,t）=v±,v＋≠v_.We also obtain the L∞ time decay rate whichreads‖v-v‖L∞=О（1）（1＋t）-r/4,where r=min（3,n）.To get the main result, the energy method and a newinequality have been used.

Shape Analysis and Solution to a Class of NonlinearWave Equation with Cubic Term

In this paper,we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems.Two critical values which can characterize the scale of dissipation effect are obtained.If dissipation effect is not less than a certain critical value,the traveling wave solutions appear as kink profile;while if it is less than this critical value,they appear as damped oscillatory.All expressions of bounded traveling wave solutions are presented,including exact expressions of bell and kink profile solitary wave solutions,as well as approximate expressions of damped oscillatory solutions.For approximate damped oscillatory solution,using homogenization principle,we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions.It can be seen that the error is an infinitesimal decreasing in the exponential form.

Solitons and Other Solutions for the Generalized KdV-mKdV Equation with Higher-order Nonlinear Terms

The generalized sub-ODE method, the rational （G＇/G）-expansion method, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations （PDEs）. Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV （GKdV-mKdV） equation with higherorder nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.

Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant

This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant(x,t)∈R^+×R^+,with the null-Dirichlet boundary condition or the null-Neumann boundary condition on u. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends timeasymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang(1999), and Jiang and Zhu(2009, Discrete Contin Dyn Syst), we obtain a general result when the initial perturbation belongs to the same space. In addition,our main novelty lies in the fact that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.

LOCAL ATTRACTORS FOR WEAKLY DAMPED FORCED KdV EQUATION IN THIN 2D DOMAINS

The existence of local attractors in thin 2D domains far the weakly damped forced KdV equation, whose principal operator is a non-self adjoint and non-sectorial one is given.

具变指数非线性项的强阻尼波动方程的爆破

考虑一类具变指数非线性项的强阻尼波动方程的有限时刻爆破.借助凹方法和适当选取的参数,给出该问题的一个新的爆破准则,并给出爆破时间的上下界估计.结果表明,该爆破准则特别蕴含对任意高的初始能量,该问题存在有限时刻爆破解.