Dissipative Hyperbolic Equations Research Articles

Finding the exact decay rate of all solutions to some second order evolution equations with dissipation

We consider an abstract second order evolution equation with damping. The “elastic” term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions.We prove the coexistence of slow solutions and fast solutions. Slow solutions live close to the kernel of A, and decay as negative powers of t as solutions of the first order equation obtained by neglecting the operator A and the second order time-derivatives in the original equation. Fast solutions live close to the range of A and decay exponentially as solutions of the linear homogeneous equation obtained by neglecting the nonlinear terms in the original equation.The abstract results apply to semilinear dissipative hyperbolic equations.

Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations

We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as t \to \infty , at least as fast as a suitable negative power of t . Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of t . Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.

Asymptotically free property of the solutions of an abstract linear hyperbolic equation with time-dependent coefficients

This paper is concerned with an abstract dissipative hyperbolic equation with time-dependent coefficient. Under an assumption which ensures that the energy does not decay, this paper provides a condition on the coefficient, which is necessary and sufficient so that the solutions tend to the solutions of the free wave equation.

The Generalized Diffusion Phenomenon and Applications

We study the asymptotic behavior of solutions to dissipative wave equations involving two noncommuting self-adjoint operators in a Hilbert space. The main result is that the abstract diffusion phenomenon takes place. Thus solutions of such equations approach solutions of diffusion equations at large times. When the diffusion semigroup has the Markov property and satisfies a Nash-type inequality, we obtain precise estimates for the consecutive diffusion approximations and remainders. We present several important applications including sharp decay estimates for dissipative hyperbolic equations with variable coefficients on an exterior domain. In the nonlocal case we obtain the first decay estimates for nonlocal wave equations with damping; the decay rates are sharp.

Inertial Manifolds for Weakly and Strongly Dissipative Hyperbolic Equations

One considers mixed boundary value problems for a quasilinear hyperbolic equation with a weak, as well as strong, dissipation. The nonlinear function in the equation is assumed Lipschitz continuous. For each of these problems one obtains the conditions on the Lipschitz constant that ensure the existence of inertial manifolds.

An attractor for the singularly perturbed Kirchhoff equation with supercritical nonlinearity

We consider a quasilinear wave equation of Kirchhoff typeϵutt−(1+‖∇u‖2)Δu+ut+f(u)=g(x), where ϵ>0 is a small parameter. Without any growth restrictions on the nonlinearity f(u), we prove the existence of a finite-dimensional global attractor on an appropriate (bounded) phase space. The key step is the estimate of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation.

Approximation of periodic solutions for a dissipative hyperbolic equation

This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force \(f\). These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on \(f\), the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term \(f\) is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.

Global solvability of the Cauchy problem for semilinear dissipative hyperbolic equations with anisotropic elliptic part

Global solvability of the Cauchy problem for semilinear dissipative hyperbolic equations with anisotropic elliptic part

Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $u_{tt} + 2u_t - a_{ij} (u_t ,\nabla u)\partial _i \partial _j u = f$ corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation $ - a_{ij} (0,\nabla v)\partial _i \partial _j v = h$ . We then give conditions for the convergence, as t → ∞, of the solution of the evolution equation to its stationary state.

Diffusion phenomenon in Hilbert spaces and applications

We prove an abstract version of the striking diffusion phenomenon that offers a strong connection between the asymptotic behavior of abstract parabolic and dissipative hyperbolic equations. An important aspect of our approach is that we use in a natural way spectral analysis without involving complicated resolvent estimates. Our proof of the diffusion phenomenon does not use the individual behavior of solutions; instead we show that only their difference matters. We estimate the Hilbert norm of the difference in terms of the Hilbert norm of solutions to the parabolic problems, which allows us to transfer the decay from the parabolic to the hyperbolic problem. The application of these estimates to operators with Markov property combined with a weighted Nash inequality yields explicit and sharp decay rates for hyperbolic problems with variable (x-dependent) coefficients in exterior domains. Our method provides new insight in this area of extensive research which was not well understood until now.

Explicit construction of attracting integral manifolds for a dissipative hyperbolic equation

A dissipative sine-Gordon-type equation in a bounded domain is considered. A simple explicit construction of finite-dimensional integral manifolds with exponential tracking is proposed.

Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic–parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic–parabolic singular perturbation problem for linear equations with a time-dependent coefficient.

Attractors of dissipative hyperbolic equations with singularly oscillating external forces

We study a uniform attractor $$\mathcal{A}^\varepsilon $$ for a dissipative wave equation in a bounded domain Ω ⋐ ℝn under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g 0(x, t)+ e−α g 1 (x, t/e), x ∈ Ω, t ∈ ℝ, where α > 0, 0 < e ≤ 1. In E = H 0 1 × L 2, this equation has an absorbing set B e estimated as ‖B e‖ E ≤C 1+C 2e−α and, therefore, can increase without bound in the norm of E as e → 0+. Under certain additional constraints on the function g 1(x, z), x ∈ Ω, z ∈ ℝ, we prove that, for 0 < α ≤ α 0, the global attractors $$\mathcal{A}^\varepsilon $$ of such an equation are bounded in E, i.e., $$\parallel \mathcal{A}^\varepsilon \parallel _E \leqslant C_3 $$ , 0 < e ≤ 1. Along with the original equation, we consider a “limiting” wave equation with external force g 0(x, t) that also has a global attractor $$\mathcal{A}^0 $$ . For the case in which g 0(x, t) = g 0(x) and the global attractor $$\mathcal{A}^0 $$ of the limiting equation is exponential, it is established that, for 0 < α ≤ α 0, the Hausdorff distance satisfies the estimate $$dist_E (\mathcal{A}^\varepsilon ,\mathcal{A}^0 ) \leqslant C\varepsilon ^{\eta (\alpha )} $$ , where η(α) > 0. For η(α) and α 0, explicit formulas are given. We also study the nonautonomous case in which g 0 = g 0(x, t). It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors $$\mathcal{A}^\varepsilon $$ from $$\mathcal{A}^0 $$ , similar to those given above.

Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds

Exponential dichotomy properties are studied for non-autonomous quasilinear partial differential equations that can be written as an ordinary differential equation in a Hilbert space . It is assumed that the non-linear function is essentially subordinated to the linear operator ; namely, the gap property from the theory of inertial manifolds must hold. Integral manifolds and attracting at an exponential rate an arbitrary solution of this equation as and , respectively, are constructed. The general results established are applied to the study of the dichotomy properties of solutions of a one-dimensional reaction-diffusion system and of a dissipative hyperbolic equation of sine-Gordon type.

A MEAN CONDITION ON FORCING TERM FOR MULTIPLICITY OF PERIODIC SOLUTIONS FOR NONLINEAR DISSIPATIVE HYPERBOLIC EQUATIONS

A condition on forcing term insuring the multiplicity of Dirichlet-periodic solutions of nonlinear dissipative hyperbolic equations is discussed. The nonlinear term is assumed to have co- ercive growth.

Trajectory and global attractors of dissipative hyperbolic equations with memory

We consider in this article a general construction of trajectory attractorsand global attractors of evolution equations with memory. In our approach,the corresponding dynamical system acts in the space of initial data of theCauchy problem under study; we can note that, in previous studies, theso-called history space setting was introduced and the study of globalattractors was made in an extended phase space.As an application, we construct trajectory and global attractors fordissipative hyperbolic equations with linear memory. We also prove theexistence of a global Lyapunov function for the dissipative hyperbolicequation with memory. The existence of such a Lyapunov function implies aregular structure for the trajectory and global attractors of the equationunder consideration.

On the Decay of Solutions for Some Nonlinear Dissipative Hyperbolic Equations

In this paper we show the decay of solutions to the initial-boundary value problem for some nonlinear hyperbolic equation with a nonlinear dissipative term, by using a difference inequality.

Averaging of trajectory attractors of evolution equations with rapidly oscillating terms

Evolution equations containing rapidly oscillating terms with respect to the spatial variables or the time variable are considered. The trajectory attractors of these equations are proved to approach the trajectory attractors of the equations whose terms are the averages of the corresponding terms of the original equations. The corresponding Cauchy problems are not assumed here to be uniquely soluble. At the same time if the Cauchy problems for the equations under consideration are uniquely soluble, then they generate semigroups having global attractors. These global attractors also converge to the global attractors of the averaged equations in the corresponding spaces. These results are applied to the following equations and systems of mathematical physics: the 3D and 2D Navier-Stokes systems with rapidly oscillating external forces, reaction-diffusion systems, the complex Ginzburg-Landau equation, the generalized Chafee-Infante equation, and dissipative hyperbolic equations with rapidly oscillating terms and coefficients.

Double-periodic boundary value problem for non-linear dissipative hyperbolic equations

Let Z and R be the set of all integers and real numbers, respectively, and let Q = [0,27c] x [0, 2711. Let L’(Q) be the space of measurable real-valued functions U: Sz + R which are Lebesgue integrable over 52 with usual norm II.jI L1. Let L2(Q) be the space of measurable real-valued functions U: Sz + R which are Lebesgue square integrable over Q with usual inner product ( , ) and usual norm I/ . ljL2 and let L”(Q) be the space of measurable real-valued functions U: Q + R which are essentially bounded with the norm 1) u 11 Lm = ess sup 1 u( t, x) I . (I. x) E R Let C“(Q) be the space of all continuous functions u.: 52 --f R such that the partial derivatives up to order k with respect to both variables are continuous on 52, while C(Q) is used for Co(Q) with the usual norm 11. Iloo.

Global existence for quasi-linear dissipative hyperbolic equations with large data and small parameter

Global existence for quasi-linear dissipative hyperbolic equations with large data and small parameter