Attractors for strongly damped wave equations with critical exponent

We prove the existence of the global attractor for the semigroup generated by strongly damped wave equations when the nonlinearity has a critical growth exponent.     

Attractor and dimension for strongly damped nonlinear wave equation

1. IntroductionConsider the strongly damped nonlinear wave equationwith the Dirichlet boundary conditionand the initial value conditionswhere u = u(x, t) is a real--valued function on fi x [0, co), fi is an open bounded set of R"with smooth boundary off, or > 0, g e L'(fl), D(--Q) ~ Ha(~~) n H'(fl).We assume for the function f(u) as follows'f(u) E CI (R, R) satisfiesfor any ig al, uZ E R, where k, hi > 0, i ~ 0, 1, 2, 61 > 0 and 0 5 6o < 1'The type of equation (1) can be regarded as the…     

Dimension of the global attractor for damped nonlinear wave equations

An estimate on the Hausdorff dimension of the global attractor for damped nonlinear wave equations, in two cases of nonlinear damping and linear damping, with Dirichlet boundary condition is obtained. The gained Hausdorff dimension is bounded and is independent of the concrete form of nonlinear damping term. In the case of linear damping, the gained Hausdorff dimension remains small for large damping, which conforms to the physical intuition.

     

On non-autonomous strongly damped wave equations with a uniform attractor and some averaging

In this paper we study the existence of a uniform attractor for strongly damped wave equations with a time-dependent driving force. If the time-dependent function is translation compact, then in a certain parameter region, the uniform attractor of the system has a simple structure: it is the closure of all the values of the unique, bounded complete trajectory of the wave equation. And it attracts any bounded set exponentially. At the same time, we consider the strongly damped wave equations with rapidly oscillating external force gε(x,t)=g(x,t,t/ε) having the average g⁰(x,t) as ε→⁰⁺. We prove that the Hausdorff distance between the uniform attractor Aε of the original equation and the uniform attractor A₀ of the averaged equation is less than O(ε1/2). We mention, in particular, that the obtained results can be used to study the usual damped wave equations.     

Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity

This paper deals with the initial boundary value problem for strongly damped semilinear wave equations with logarithmic nonlinearity $u_{tt}-\Delta u-\Delta u_t={\phi }_p\left(u\right)log\left|u\right|$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$. We discuss the existence, uniqueness and polynomial or exponential energy decay estimates of global weak solutions under some appropriate conditions. Moreover, we derive the finite time blow up results of weak solutions, and give the lower and upper bounds for blow-up time by the combination of the concavity method, perturbation energy method and differential–integral inequality technique.     

一类强阻尼波方程解的存在性和爆破性

讨论一类强阻尼波方程解的局部存在性,并利用势井理论研究解的整体存在性和爆破性.     

Qualitative theory for strongly damped wave equations

We investigate the asymptotic periodicity, L^p‐boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications.     

Limiting behavior for strongly damped nonlinear wave equations

In this paper we investigate both the existence and the limiting behavior for the equation $\text{u}{}_{\mathrm{tt}}\text{+ Au}{}_{\mathrm{t}}\text{+ Au = ?(t, u, u}{}_{\mathrm{t}}\text{)}$, where A is a sectorial operator, ? is periodic in t, and ? satisfies certain regularity and growth assumptions. In most results on limiting behavior we will assume A has compact resolvent. We consider the equation as an abstract ODE defined on a paired space X^β × X^α, 0 ? σ ? β < 1. With regard to the limiting behavior, one of our principal results will be to show that if there is a bounded set in one of the spaces considered, for which all points or trajectories enter into and remain, then there is a set J consisting of very “smooth” functions defined on all of the spaces considered, which is the maximum compact invariant set, uniformly asymptotically stable, connected, and having very strong attractivity properties in all these spaces. We will often show it attracts all points in a bounded set uniformly. We will give a few sharper results for the case where A = ?Δ. The work is motivated by recent papers of Webb and Fitzgibbon, and applies techniques found in recent papers by the author.     

One-dimensional global attractor for strongly damped wave equations

We consider the dynamical behavior of the strongly damped wave equations under homogeneous Neumann boundary condition. By the property of limit set of asymptotic autonomous differential equations, we prove that in certain parameter region, the system has a one-dimensional global attractor, which is a periodic horizontal curve.     

A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

In this paper we consider the strongly damped wave equation with time-dependent terms

u_tt−Δu−γ(t)Δu_t+β_ε(t)u_t=f(u),     

Rothe's method to strongly damped wave equations

A strongly damped wave equation is studied as an abstract differential equation in a Banach space. Existence and uniqueness of a strong solution is established with the help of Rothe's method.     

On decay properties of solutions for degenerate strongly damped wave equations of Kirchhoff type

We consider the initial-boundary value problem for the degenerate strongly damped wave equations of Kirchhoff type: . For all t?0, we will give the optimal decay estimate C−1(1+t)^−1/γ?‖A1/2u(t)^‖2?C(1+t)^−1/γ, when either the coefficient ρ is appropriately small or the initial data are appropriately small. And, we will show a decay property of the norm ‖Au(t)^‖2 for t?0.     

Higher order energy decay for damped wave equations with variable coefficients

Under appropriate assumptions the higher order energy decay rates for the damped wave equations with variable coefficients c(x)_utt−div(A(x)∇u)+a(x)_ut=0$c(x)u_{tt}-\mathrm{div}(A(x)\mathrm{\nabla }u)+a(x)u_t=0$ in Rⁿ${\mathbb{R}}^n$ are established. The results concern weighted (in time) and pointwise (in time) energy decay estimates. We also obtain weighted L²$L^2$ estimates for spatial derivatives.     

Analytical treatment and convergence of the Adomian decomposition method for a system of coupled damped wave equations

An abstract result is proved for the convergence of Adomian decomposition method for partial differential equations that model evolutionary systems. Moreover, we prove that this decomposition scheme applied to a system of wave equations coupled in parallel with homogeneous Dirichlet boundary conditions, is convergent in a suitable Hilbert space. Furthermore, this technique is utilized to find closed-form solutions for the problem under consideration.     

Global well-posedness for strongly damped viscoelastic wave equation

We study the initial boundary value problem for the nonlinear viscoelastic wave equation with strong damping term and dispersive term. By introducing a family of potential wells we not only obtain the invariant sets, but also prove the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, we establish a blow-up result for certain solutions with arbitrary positive initial energy (high energy case)     

Asymptotic profile of solutions for strongly damped Klein‐Gordon equations

We consider the Cauchy problem in R ⁿ for strongly damped Klein‐Gordon equations. We derive asymptotic profiles of solutions with weighted L^1,1( R ⁿ) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L²‐norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low‐dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference.