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 ? ?ⁿ under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g ₀(x, t)+ ε^?α g ₁ (x, t/ε), x ∈ Ω, t ∈ ?, where α > 0, 0 < ε ≤ 1. In E = H ₀ ¹ × L ₂, this equation has an absorbing set B ^ε estimated as ‖B ^ε‖ _E ≤C ₁+C ₂ε^?α and, therefore, can increase without bound in the norm of E as ε → 0+. Under certain additional constraints on the function g ₁(x, z), x ∈ Ω, z ∈ ?, 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 < ε ≤ 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.     

Averaging of a 3D primitive equations with oscillating external forces

In this article, we consider a non-autonomous three-dimensional primitive equations of the ocean with a singularly oscillating external force g ^ε?=?g ₀(t)?+?ε^?ρ g ₁(t/ε) depending on a small parameter ε?>?0 and ρ?∈?[0,?1) together with the averaged system with the external force g ₀(t), formally corresponding to the case ε?=?0. Under suitable assumptions on the external force, we prove as in [V.V. Chepyzhov, V. Pata, and M.M.I. Vishik, Averaging of 2D Navier–Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), pp. 351–370] the boundness of the uniform global attractor 𝒜^ε as well as the convergence of the attractors 𝒜^ε of the singular systems to the attractor 𝒜⁰ of the averaged system as ε?→?0⁺. When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by Kε^(1?ρ). Let us note that the main difference between this work and that of Chepyzhov et al. (2009) is that the non-linearity involved in the three-dimensional primitive equation is stronger than the one in the two-dimensional Navier–Stokes equations considered in Chepyzhov et al. (2009), which makes the analysis of the problem studied in this article more involved.     

Kolmogorov ε-entropy of attractor for a non-autonomous strongly damped wave equation

In this paper, we study the long-time behavior of solutions for a non-autonomous strongly damped wave equation. We first prove the existence of a uniform attractor for the equation with a translation compact driving force and then obtain an upper estimate for the Kolmogorov ε-entropy of the uniform attractor. Finally we obtain an upper bound of the fractal dimension of the uniform attractor with quasiperiodic force.     

Convergence of the wave equation damped on the interior to the one damped on the boundary

In this paper, we study the convergence of the wave equation with variable internal damping term γn(x)ut to the wave equation with boundary damping γ(x)⊗δx∈∂Ωut when (γn(x)) converges to γ(x)⊗δx∈∂Ω in the sense of distributions. When the domain Ω in which these equations are defined is an interval in R, we show that, under natural hypotheses, the compact global attractor of the wave equation damped on the interior converges in X=H¹(Ω)×L²(Ω) to the one of the wave equation damped on the boundary, and that the dynamics on these attractors are equivalent. We also prove, in the higher-dimensional case, that the attractors are lower-semicontinuous in X and upper-semicontinuous in H1−ε(Ω)×H−ε(Ω).     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

On the almost automorphy of bounded solutions of differential equations with piecewise constant argument

In this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=Ax([t])+f(t), t∈R, where A is a bounded linear operator in X and f is an X-valued almost automorphic function.     

A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

In this paper, we propose a new high accuracy numerical method of O(k² + k²h² + h⁴) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u_tt = A(x,y,z,t)u_xx + B(x,y,z,t)u_yy + C(x,y,z,t)u_zz + g(x,y,z,t,u,u_x,u_y,u_z,u_t), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in  and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method.     

Functionally separable solutions to nonlinear wave equations by group foliation method

We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: utt=A(x)(Dx(u)ux)+B(x)Q(u), Ax≠0. A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.     

Polynomially bounded solutions to the Loewner differential equation in several complex variables

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z,t)=etAz+?, where A∈L(Cn,Cn) has the property m(A)>0. Here m(A)=min{R〈A(z),z〉:‖z‖=1}. We also give sufficient conditions for g(z,t)=L(f(z,t)) to be polynomially bounded, where f(z,t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.     

Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation

The paper studies the existence of the finite-dimensional global attractors and exponential attractors for the dynamical system associated with the Kirchhoff type equation with a strong dissipation utt−M(‖∇u^‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor. For application, the fact shows that for the concerned viscoelastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.     

Existence of the mild solution for some fractional differential equations with nonlocal conditions

We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D ^q x(t)=Ax(t)+t ⁿ f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ⁺, x(0)+g(x)=x ₀, where 0<q<1, A is the infinitesimal generator of a C ₀-semigroup of bounded linear operators on a Banach space X.     

On the dynamics of a class of nonclassical parabolic equations

We consider the first initial and boundary value problem of nonclassical parabolic equations ut−μΔut−Δu+g(u)=f(x) on a bounded domain Ω, where μ∈[0,1]. First, we establish some uniform decay estimates for the solutions of the problem which are independent of the parameter μ. Then we prove the continuity of solutions as μ→0. Finally we show that the problem has a unique global attractor Aμ in in the topology of H²(Ω); moreover, Aμ→A₀ in the sense of Hausdorff semidistance in as μ goes to 0.     

The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen-Cahn type equation of the form ut=Δu+ε⁻²fε(x,t,u), where ε is a small parameter and fε(x,t,u)=f(u)−εgε(x,t,u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u₀ that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order ε²|lnε|, and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where gε≡0.Next we consider systems of reaction-diffusion equations of the form

     

Nonlinear Geometric Optics for Short Pulses

This paper studies the propagation of pulse-like solutions of semilinear hyperbolic equations in the limit of short wavelength. The pulses are located at a wavefront Σ?{φ=0} where φ satisfies the eikonal equation and dφ lies on a regular sheet of the characteristic variety. The approximate solutions are u^ε_approx=U (t, x, φ(t, x)/ε) where U(t, x, r) is a smooth function with compact support in r. When U satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions u^ε_exact such that u^ε_approx has relative error O(ε) as ε→0. While the transport equation is familiar, the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope.     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

On bounded solutions of nonlinear first- and second-order equations with a Carathéodory function

In the paper we study the existence and uniqueness of bounded solutions for differential equations of the form: x^′−Ax=f(t,x), x^″−Ax=f(t,x), where A∈L(Rm), is a Carathéodory function and the homogeneous equations x^′−Ax=0, x^″−Ax=0 have nontrivial solutions bounded on R. Using a perturbation of the equations, the Leray-Schauder Topological Degree and Fixed Point Theory, we overcome the difficulty that the linear problems are non-Fredholm in any reasonable Banach space.     

An abstract doubly nonlinear equation with a measure as initial value

The solvability of the abstract implicit nonlinear nonautonomous differential equation (A(t)u^′(t))+B(t)u(t)+C(t)u(t)∋f(t) will be investigated in the case of a measure as an initial value. It will be shown that this problem has a solution if the inner product of A(t)x and B(t)x+C(t)x is bounded below.     

Existence of asymptotically almost periodic solutions for damped wave equations

In this paper, a class of nonlinear damped wave equations of the form αu^?(t)+u^″(t)=βAu(t)+γAu^′(t)+f(t,u(t)), t?0, satisfying αβ<γ with prescribed initial conditions are studied. Some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution. These results have significance in the study of vibrations of flexible structures possessing internal material damping. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.     

Stability of the Absolutely Continuous Spectrum of Random Schrödinger Operators on Tree Graphs

According to the Smolukowski-Kramers approximation, we show that the solution of the semi-linear stochastic damped wave equations μ u _tt (t,x)=Δu(t,x)?u _t (t,x)+b(x,u(t,x))+Q (t),u(0)=u ₀, u _t (0)=v ₀, endowed with Dirichlet boundary conditions, converges as μ goes to zero to the solution of the semi-linear stochastic heat equation u _t (t,x)=Δ u(t,x)+b(x,u(t,x))+Q (t),u(0)=u ₀, endowed with Dirichlet boundary conditions. Moreover we consider relations between asymptotics for the heat and for the wave equation. More precisely we show that in the gradient case the invariant measure of the heat equation coincides with the stationary distributions of the wave equation, for any μ>0.     

Existence and stability of solutions with periodically moving weak internal layers

We consider the periodic parabolic differential equation under the assumption that ε is a small positive parameter and that the degenerate equation f(u,x,t,0)=0 has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution up(x,t,ε) which is T-periodic in t, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as ε→0.