On the decay and blow‐up of solution for coupled nonlinear wave equations with nonlinear damping and source terms

In this work, we consider a nonlinear coupled wave equations with initial‐boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on the damping terms and source terms and initial data in the stable set, we obtain that the decay estimates of the energy function is exponential or polynomial by using Nakao's method. By using the energy method, we obtain the blow‐up result of solution with some positive or nonpositive initial energy. Copyright © 2016 John Wiley & Sons, Ltd.     

On decay and blow-up of solutions for a system of nonlinear wave equations

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with the nonlinear damping and the source terms in a bounded domain is considered. We prove that, under suitable conditions on the nonlinearity of the damping and the source terms and certain initial data in the stable set and for a wider class of relaxation functions, the decay estimates of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s method. Conversely, for certain initial data in the unstable set, we obtain the blow-up of solutions in finite time when the initial energy is nonnegative. This improves earlier results in the literature.     

Global existence and blow-up of solutions for a system of Petrovsky equations

The initial-boundary value problem for a system of Petrovsky equations with memory and nonlinear source terms in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the exponential decay estimate of global solutions. Meanwhile, under suitable conditions on relaxation functions and the positive initial energy as well as non-positive initial energy, it is proved that the solutions blow up in the finite time and the lifespan estimates of solutions are also given.     

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

In this work, we consider a nonlinear system of viscoelastic equations of Kirchhoff type with degenerate damping and source terms in a bounded domain. Under suitable assumptions on the initial data, the relaxation functions g_i(i = 1,2) and degenerate damping terms, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow‐up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.     

Optimal time decay of the Boltzmann equation with frictional force

In this paper, we use the combination of energy method and Fourier analysis to obtain the optimal time decay of the Boltzmann equation with frictional force towards equilibrium. Precisely speaking, we decompose the equation into macroscopic and microscopic partitions and perform the energy estimation. Then, we construct a special solution operator to a linearized equation without source term and use Fourier analysis to obtain the optimal decay rate to this solution operator. Finally, combining the decay rate with the energy estimation for nonlinear terms, the optimal decay rate to the Boltzmann equation with frictional force is established.     

Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions

A variable coefficient viscoelastic wave equation with acoustic boundary conditions and nonlinear source term is considered. Under suitable conditions on the initial data and the relaxation function g, we show the polynomial decay of the energy solution and the blow up of solutions by energy methods. The estimates for the lifespan of solutions are also given.     

Global existence and decay of solutions of a nonlinear system of wave equations

This work is concerned with a system of two wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we show that our problem has a unique local solution. Also, we prove that, for some restrictions on the initial data, the rate of decay of the total energy is exponential or polynomial depending on the exponents of the damping terms in both equations.     

Global Existence and Uniform Decay of Solutions for a Coupled System of Nonlinear Viscoelastic Wave Equations with Not Necessarily Differentiable Relaxation Functions

In this paper, we consider the initial boundary value problem for a coupled system of nonlinear viscoelastic wave equations with source terms. By using the Faedo–Galerkin method, potential well theory and perturbed energy technique, we establish the global existence and exponential decay of solutions under weaker conditions on the relaxation functions that are not necessarily differentiable.     

Stabilization and control for the subcritical semilinear wave equation

In this paper, we analyze the exponential decay property of solutions of the semilinear wave equation in with a damping term which is effective on the exterior of a ball. Under suitable and natural assumptions on the nonlinearity we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p<5. The method of proof combines classical energy estimates for the linear wave equation allowing to estimate the total energy of solutions in terms of the energy localized in the exterior of a ball, Strichartz's estimates and results by P. Gérard on microlocal defect measures and linearizable sequences. We also give an application to the stabilization and controllability of the semilinear wave equation in a bounded domain under the same growth condition on the nonlinearity but provided the nonlinearity has been cut-off away from the boundary.     

Energy decay of variable‐coefficient wave equation with nonlinear acoustic boundary conditions and source term

In this paper, we consider a variable‐coefficient wave equation with nonlinear acoustic boundary conditions and source term. Using the Riemannian geometry method, we prove the general energy decay of the system corresponds to the ordinary differential equation (ODE), which certainly is stable under some suitable assumptions.     

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.     

General decay of solutions of a nonlinear system of viscoelastic wave equations

This work is concerned with a system of two viscoelastic wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we prove that, for certain class of relaxation functions and for some restrictions on the initial data, the rate of decay of the total energy depends on those of the relaxation functions. This result improves many results in the literature, such as the ones in Messaoudi and Tatar (Appl. Anal. 87(3):247–263, 2008) and Liu (Nonlinear Anal. 71:2257–2267, 2009) in which only the exponential and polynomial decay rates are considered.     

Global existence,asymptotic stability and blow-up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.     

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.     

Energy decay rate for a coupled hyperbolic system with nonlinear damping

In this paper we investigate the energy decay rate for the solution of a coupled hyperbolic system. The explicit energy decay rate is established by using multiplier techniques and constructing a suitable energy functional.     

On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution

In this paper, we consider a nonlinear viscoelastic wave equation with nonlinear boundary damping and source terms. Under some appropriate assumptions on the relaxation function $g$ and with certain initial data, the global existence of solutions and a general decay for the energy have been established.     

GENERAL DECAY FOR A VISCOELASTIC EQUATION OF VARIABLE COEFFICIENTS WITH A TIME-VARYING DELAY IN THE BOUNDARY FEEDBACK AND ACOUSTIC BOUNDARY CONDITIONS

A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay results of the energy are established via suitable Lyapunov functionals and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and the elements of the matrix A and the kernel function g.     

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems(coupled parabolic systems)with nonlinear coupled source terms is considered in order to classify the initial data for the global existence,finite time blowup and long time decay of the solution.The whole study is conducted by considering three cases according to initial energy:the low initial energy case,critical initial energy case and high initial energy case.For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence,long time decay and finite time blowup are given to show a sharp-like condition.In addition,for the high initial energy case the possibility of both global existence and finite time blowup is proved first,and then some sufficient initial conditions of finite time blowup and global existence are obtained,respectively.     

On the spatial blow-up and decay for some nonlinear parabolic equations with nonlinear boundary conditions

In this note we investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions. Under suitable conditions on the nonlinear terms we prove that the solutions either cease to exist for a finite value of the spatial variable or else they decay algebraically. The main tool used is the weighted energy method. Our results can be applied to several situations concerning heat conduction. Received: April 4, 2004; revised: September 20, 2004     

Decay estimates for second order evolution equations with memory

This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t→+∞. Our approach is based on integral inequalities and multiplier techniques.These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system.