General decay of solutions of a wave equation with a boundary control of memory type

In this paper we consider a semilinear wave equation, in a bounded domain, where the memory-type damping is acting on a part of the boundary. We establish a general decay result, from which the usual exponential and polynomial decay rates are only special cases. Our work allows certain relaxation functions which are not necessarily of exponential or polynomial decay and, therefore, generalizes and improves earlier results in the literature.     

General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type

In this paper we consider linear porous-thermoelasticity systems, in a bounded domain, where the memory-type damping is acting on a part of the boundary. We establish a general decay result, for which the usual exponential and polynomial decay rates are just special cases. Our work allows certain relaxation functions which are not necessarily of exponential or polynomial decay and, therefore, generalizes and improves on earlier results from the literature.     

Mathematical modeling of time fractional reaction–diffusion systems

We study a fractional reaction–diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction–diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.     

Mathematical models for the non‐isothermal Johnson–Segalman viscoelasticity in porous media: stability and wave propagation

We present a nonlinear model for Johnson–Segalman type polymeric fluids in porous media, accounting for thermal effects of Oldroyd‐B type. We provide a thermodynamic development of the Darcy's theory, which is consistent with the interlacement between thermal and viscoelastic relaxation effects and diffusion phenomena. The appropriate invariant convected time derivative for the flux vector and the stress tensor is discussed. This is performed by investigating the local balance laws and entropy inequality in the spatial configuration, within the single‐fluid approach. For constant parameters, our thermomechanical setting is of Jeffreys type with two delay time parameters, and hence, in the linear/linearized version, it is strictly related to phase‐lag theories within first‐order Taylor approximations. A detailed spectral analysis is carried out for the linearized version of the model, with a scrutiny to some significant limit situations, enhancing the stabilizing effects of the dissipative and elastic mechanisms, also for retardation responses. For polymeric liquids, rheological aspects, wave propagation properties and analogies with other theories with lagging are pointed out. Copyright © 2014 John Wiley & Sons, Ltd.     

Time-invariant and time-varying discontinuity structures for models described by the generalized Korteweg–Burgers equation

Taking the generalized Korteweg–Burgers equation as the example, it is established by numerical analysis that three types of discontinuity structures are encountered for weakly dissipative media with dispersion and non-linearity: time-invariant structures, time-periodic structures and stochastic structures. Time-invariant weakly dissipative structures contain internal non-dissipative discontinuity structures of the type of transitions between homogeneous or wave states. The structure of a discontinuity can be non-unique. Hystereses arise on account of this, that is, the type of discontinuity depends on the evolutionary path of the system. The dependence of the type of discontinuity on its amplitude and the dissipation parameter has been investigated. The time-invariant solutions of the generalized Korteweg – de Vries equation: the periodic solutions, soliton solutions and the structures of the discontinuities were studied in order to explain the observed phenomena and to predict the type of discontinuity. A technique is developed for analysing the branches of the biperiodic solutions. A correspondence between the structural types of weakly dissipative discontinuity and the branch arrangement pattern is revealed.     

On the Long-Time Behavior of the Quantum Fokker-Planck Equation

We analyze the long-time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type diffusion operator, subject to confining potentials of harmonic oscillator type. We establish the existence and uniqueness of a non-equilibrium steady state for the corresponding dynamics. Further, using a (classical) convex Sobolev inequality, we prove an optimal exponential rate of decay towards this state and additionally give precise dispersion estimates in those cases, where no stationary state exists.     

带松弛项的单个守恒律方程解的时态渐近性质

本文研究一维空间中带松弛项的单个守恒律方程解的大时间状态估计.在松弛项满足耗散条件下通过对线性化方程Green函数的逐点估计得到方程解在时间充分大时的衰减估计,并由此反映出“弱”惠更斯原理.     

Nonlinear Transmission Problem for Wave Equation with?Boundary Condition of Memory Type

In this paper we consider a transmission problem with a boundary damping condition of memory type, that is, the wave propagation over bodies consisting of two physically different types of materials. One component is clamped, while the other is in a viscoelastic fluid producing a dissipative mechanism on the boundary. We will study the global existence of solutions for the transmission problem, and moreover we show that if the relaxation function decays exponentially or polynomially, then the solutions for the problem have the same decay rates.     

Singular limit problem of the Camassa-Holm type equation

We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove the strong convergence and establish the condition under which, as diffusion and dispersion parameters tend to zero, smooth solutions of the shallow water equation converge to the entropy solution of a scalar conservation law using methodology developed by Hwang and Tzavaras [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254]. The proof relies on the kinetic formulation of conservation laws and the averaging lemma.     

Diffusion phenomena for partially dissipative hyperbolic systems

We consider a partially dissipative hyperbolic system with time-dependent coefficients and show that under natural assumptions its solutions behave like solutions to a parabolic problem modulo terms of faster decay. This generalises the well-known diffusion phenomenon for damped waves and gives some further insights into the structure of dissipative hyperbolic systems.     

On three types of dynamics and the notion of attractor

We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor–repeller merger. We identify regimes observed in dynamical systems with attractors as defined in a paper by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior, conservative and dissipative, while the attractors of the third type, reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal; i.e., it displays dynamics of maximum possible richness and complexity.     

Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate

We consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal BGK model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic Maxwellian. We observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part, leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter θ reaches zero. Accordingly, we employ two different sets of micro–macro system to derive the full coercivity and close the energy estimate.     

Weak solutions for equations defined by accretive operators II: relaxation limits

In the first part of this paper we define solutions for certain nonlinear equations defined by accretive operators, “dissipative solution”. This kind of solution is equivalent to the viscosity solutions for Hamilton-Jacobi equations and to the entropy solutions for conservation laws.In this paper we use dissipative solutions to obtain several relaxation limits for systems of semilinear transport equations and quasilinear conservation laws. These converge to diffusion second-order equations and in one case to a single conservation law. The relaxation limit is obtained using a version of the perturbed test function method to pass to the limit. This guarantees existence for the considered equations.     

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.     

Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow

We establish global solutions of nonconcave hyperbolic equations with relaxation arising from traffic flow. One of the characteristic fields of the system is neither linearly degenerate nor genuinely nonlinear. Furthermore, there is no dissipative mechanism in the relaxation system. Characteristics travel no faster than traffic. The global existence and uniqueness of the solution to the Cauchy problem are established by means of a finite difference approximation. To deal with the nonconcavity, we use a modified argument of Oleinik (Amer. Math. Soc. Translations 26 (1963) 95). It is also shown that the zero relaxation limit of the solutions exists and is the unique entropy solution of the equilibrium equation.     

Relaxation self-oscillations in neuron systems: II

A singularly perturbed system of nonlinear delay differential equations that models the diffusion interaction of two neurons is considered. We study the existence and stability of relaxation periodic motions in this system.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

Dissipative solutions for equations of viscoelastic diffusion in polymers

We study the system of differential equations which describes the diffusion of a penetrant liquid in a polymer. We construct dissipative solutions to the Cauchy problem for this system in the space.     

Multiparameter Schemes for Evolutionary Equations

Multiparameter extensions (MP) of (linear and nonlinear) descent methods have been proposed for the solution of finite dimensional time independent problems; these new methods are based on a different treatment of several blocks of components of the solution, basically via the substitution of a scalar relaxation by a (suitable) matricial relaxation. Similarly, the Nonlinear Galerkin Method (NLG), that stems from the dynamical system theory, propose to apply distinct temporal integration schemes to different sets of data scales when solving dissipative PDEs. In this paper, the algebraic similarity of Richardson iteration and Forward-Euler time integration is extended to new grounds through the expansion of the realm of MP methods to the field of the numerical integration of dissipative PDEs. The separation of the structures is realized by the utilization of hierarchical preconditioners in finite differences, which are conjugated to a MP temporal integration steeming from NLG theory. Numerical examples of fluid dynamics problems show the improved temporal stability of these new methods as compared to the classical ones.     

Optimal decay estimates of a regularity-loss type system with constraint condition

In the paper [14], the authors formulated a new structural condition which includes the Kawashima–Shizuta condition, and analyzed the weak dissipative structure called the regularity-loss type for general systems which contain the Timoshenko system and the Euler–Maxwell system. However, this new structural condition can not cover all of dissipative systems. Indeed we introduce a dissipative system which does not satisfy the new condition and analyze the weaker dissipative structure in this paper. Precisely we first derive the $L^2$ decay estimate of solutions and discuss the type of the corresponding regularity-loss structure. Moreover, in order to show the optimality of the decay estimate, we analyze the expansion for the corresponding eigenvalue of our problem and derive that the solution approaches the diffusion wave as time tends to infinity.