Fourth order wave equations with nonlinear strain and source terms

In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition I(u₀)?0, E(0)=d. So the Esquivel-Avila's results are generalized and improved.     

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic- parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the Hs ∩ Ll-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L2 decay rate of the solution and the almost optimal L2 decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.     

The Birman-Schwinger principle on the essential spectrum

Let H₀ and H be self-adjoint operators in a Hilbert space. We consider the spectral projections of H₀ and H corresponding to a semi-infinite interval of the real line. We discuss the index of this pair of spectral projections and prove an identity which extends the Birman-Schwinger principle onto the essential spectrum. We also relate this index to the spectrum of the scattering matrix for the pair H₀, H.     

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H ¹(Ω) × L ²(?Ω) with boundary data from L ²(?Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H ¹(Ω) into L ²(Ω) or L ²(?Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.     

Long range scattering for the modified Schrödinger map in two space dimensions

We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schrödinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a by product, we obtain global existence for small data in Hk∩FHk with k>1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.     

Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors AV in space V (has H²-regularity, see notation in Section 2) and AH in space H (has L²-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.     

Differential equations in Schatten classes of operators

We study a system of differential equations in Schatten classes of operators, ${\mathcal{C}_p(\mathcal{H})\,(1 \leq p < \infty}$ ), with ${\mathcal{H}}$ a separable complex Hilbert space. The systems considered are infinite dimensional generalizations of mathematical models of unsupervised learning. In this new setting, we address the usual questions of existence and uniqueness of solutions. Under some restrictions on the spectral properties of the initial conditions, we explicitly solve the system. We also discuss the long-term behavior of solutions.     

A dissipative singular Sturm-Liouville problem with a spectral parameter in the boundary condition

In the Hilbert space , we consider nonselfadjoint singular Sturm-Liouville boundary value problem (with two singular end points a and b) in limit-circle cases at a and b, and with a spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Sturm-Liouville equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and Sturm-Liouville boundary value problem.     

Perturbation of invariant subspaces

We investigate by how much the invariant subspaces of a bounded linear operator on a Banach space change when the operator is slightly perturbed. If E and F are the spectral projector frames associated with A and A + H respectively, we answer the natural question about how far the two frames are in terms of the perturbation H and the separation of parts of the spectrum of the operator A. These results depend on how to measure the difference between the two frames and how to measure the separation between parts of the spectrum. These two measures are introduced and analysed.     

Solvability of a Model Oblique Derivative Problem for the Heat Equation in the Zygmund Space <Emphasis Type="Italic">H</Emphasis><Subscript>1</Subscript>

We consider the oblique derivative problem for the heat equation in a model statement. We introduce a difference matching condition for the initial and boundary functions, under which we establish conditions on the data of the problem sufficient for the solution to belong to the parabolic Zygmund space H₁, which is an analog of the parabolic Hölder space for the case of an integer smoothness exponent. We present an example showing that if the above-mentioned matching condition is not satisfied, then the solution may fail to belong to the space H₁.     

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

GLOBAL EXISTENCE, UNIFORM DECAY AND EXPONENTIAL GROWTH FOR A CLASS OF SEMI-LINEAR WAVE EQUATION WITH STRONG DAMPING

In this paper, we consider the nonlinearly damped semi-linear wave equation associated with initial and Dirichlet boundary conditions. We prove the existence of a local weak solution and introduce a family of potential wells and discuss the invariants and vacuum isolating behavior of solutions. Furthermore, we prove the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively, and the asymptotic behavior of solutions for the cases of potential well family with 0相似文献   

Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms

This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.     

Walsh-Marcinkiewicz means and Hardy spaces

The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H _p , when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space H _p , and prove a strong convergence theorem for the Walsh-Marcinkiewicz means.     

Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system

We derive some new results concerning the Cauchy problem and the existence of bound states for a class of coupled nonlinear Schrödinger-gKdV systems. In particular, we obtain the existence of strong global solutions for initial data in the energy space H¹(R)×H¹(R), generalizing previous results obtained in Tsutsumi (1993) [11], Corcho and Linares (2007) [13] and Dias et al. (submitted for publication) [14] for the nonlinear Schrödinger-KdV system.     

Regularity of variational solutions to linear boundary value problems in Lipschitz domains

In a bounded Lipschitz domain in ?ⁿ, we consider a second-order strongly elliptic system with symmetric principal part written in divergent form. We study the Neumann boundary value problem in a generalized variational (or weak) setting using the Lebesgue spaces H _p ^σ (Ω) for solutions, where p can differ from 2 and σ can differ from 1. Using the tools of interpolation theory, we generalize the known theorem on the regularity of solutions, in which p = 2 and {σ ? 1} < 1/2, and the corresponding theorem on the unique solvability of the problem (Savaré, 1998) to p close to 2. We compare this approach with the nonvariational approach accepted in numerous papers of the modern theory of boundary value problems in Lipschitz domains. We discuss the regularity of eigenfunctions of the Dirichlet, Neumann, and Poincaré-Steklov spectral problems.     

Porous medium equation and fast diffusion equation as gradient systems

We show that the Porous Medium Equation and the Fast Diffusion Equation, \(\dot u - \Delta {u^m} = f\), with m ∈ (0, ∞), can be modeled as a gradient system in the Hilbert space H ^?1(Ω), and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets Ω ? ? ⁿ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.     

Almost periodicity for the nonlinear cubic Schrödinger equation

For initial data inH ² we prove that the periodic solutions in space to the nonlinear cubic Schrödinger equation are almost periodic in time.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

Global strong solutions for a class of compressible non‐Newtonian fluids with vacuum

We study a class of compressible non‐Newtonian fluids in one space dimension. We prove, by using iterative method, the global time existence and uniqueness of strong solutions provided that the initial data satisfy a compatibility condition and the initial density is small in its H¹‐norm. The main difficulty is due to the strong nonlinearity of the system and the initial vacuum. Copyright © 2010 John Wiley & Sons, Ltd.