Global existence and general decay for a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density

In this article, we investigate a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density. We prove the global existence of weak solutions and general decay of the energy by using the Faedo–Galerkin method [Z.Y. Zhang and X.J. Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Comput. Math. Appl. 59 (2010), pp. 1003–1018; J.Y. Park and J.R. Kang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Acta Appl. Math. 110 (2010), pp. 1393–1406] and the perturbed energy method [Zhang and Miao (2010); X.S. Han, and M.X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. TMA. 70 (2009), pp. 3090–3098], respectively. Furthermore, for certain initial data and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. This result is an improvement over the earlier ones in the literature.     

Global existence and uniform stability of solutions for a quasilinear viscoelastic problem

In this paper the nonlinear viscoelastic wave equation in canonical form with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Global existence and time decay of the non-cutoff Boltzmann equation with hard potential

This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces $H^N\left({\mathbb{R}}_{x,v}^6\right)$ with $N\ge 2$. But if we want to obtain the optimal temporal decay estimates, we need to add the velocity weight function, in this case the global existence and the optimal temporal decay estimate of the Boltzmann equation are all established. Meanwhile, we further gain a more accurate energy estimate, which can guarantee the validity of the assumption in Chen et al. (0000).     

Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term,nonlocal boundary condition,and localized damping term

The paper deals with the existence of a global solution of a singular one-dimensional viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized frictional damping a(x)u_t using the potential well theory. Furthermore, the general decay result is proved. We construct a suitable Lyapunov functional and make use of the perturbed energy method.     

Global existence and blowup of solutions for a parabolic equation with a gradient term

The author discusses the semilinear parabolic equation with . Under suitable assumptions on and , he proves that, if with , then the solutions are global, while if with 1$">, then the solutions blow up in a finite time, where is a positive solution of , with .

     

Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms

In this paper, we consider a system of two coupled wave equations with dispersive and viscosity dissipative terms under Dirichlet boundary conditions. The global existence of weak solutions as well as uniform decay rates (exponential one) of the solution energy are established.     

Asymptotic Behavior for a Viscoelastic Wave Equation with a Time-varying Delay Term

The following viscoelastic wave equation with a time-varying delay term in internal feedback $|u_t|^ρu_{tt}-Δu-Δu_{tt}+∫^t_0g(t-s)Δu(s)ds+μ_1u_t(x,t)+μ_2u_t(x,t-τ(t))=0$, is considered in a bounded domain. Under appropriate conditions on μ_1, μ_2 and on the kernel g, we establish the general decay result for the energy by suitable Lyapunov functionals.     

Global existence of weak solutions to a diffuse interface model for magnetic fluids

This article is devoted to the derivation and analysis of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier–Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn–Hilliard equations. We show global in time existence of weak solutions to the system using the time discretization method.     

Global existence of solutions for the heat equation with a nonlinear boundary condition

We consider the initial-boundary value problem for the heat equation with a nonlinear boundary condition:

     

Global existence of solutions of the Boltzmann equation for Bose–Einstein particles with anisotropic initial data

In this paper we prove the global in time existence and uniqueness of solutions of the spatially homogeneous Boltzmann equation for Bose–Einstein particles for the hard sphere model for bounded anisotropic initial data. The main idea of our proof is as follows: we first establish an intermediate equation which is closely related to the original equation and is relatively easily proven to have global in time and unique solutions, then we use the multi-step iterations of the collision gain operator to obtain a desired uniform $L^{\infty }$-bound for solutions of the intermediate equation so that if an initial datum is sufficiently small relative to the inverse of the Planck constant (which belongs to the case of very high temperature), then the corresponding solution of the intermediate equation becomes the solution of the original equation.     

Optimal decay rates for the viscoelastic wave equation

In this paper, we consider a viscoelastic equation with minimal conditions on the relaxation function g, namely, , where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of gat infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H(s)=s^p and p covers the full admissible range [1,2). We get the best decay rates expected under this level of generality, and our new results substantially improve several earlier related results in the literature.     

Global existence of classical solutions to the minimal surface equation with slow decay initial value

In this paper, we prove that the global existence of classical solutions to the Cauchy problem for the minimal surface equation with slow decay initial value in Minkowskian space time.