We study symmetric hyperbolic systems with memory-type dissipation and investigate their dissipative structures under Craftsmanship condition. We treat two cases: memory-type diffusion and memory-type relaxation, and observe that the dissipative structures of these two cases are essentially different. Namely, we show that the dissipative structure of the system with memory-type diffusion is of the standard type, while that of the system with memory-type relaxation is of the regularity-loss type. Moreover, we investigate the asymptotic profiles of the solutions for t ->infinity. In the diffusion case, it is proved that the systems with memory and without memory have the same asymptotic profile for t ->infinity, which is given by the superposition of linear diffusion waves. We have the same result also in the relaxation case under enough regularity assumption on the initial data.

In this article, we show the global existence of solution for some semi-discrete finite difference schemes of some model system of quasilinear hyperbolic balance laws with the Cattaneo law. In [10] and [11] the first author proposed the energy method for the structure-preserving fully discrete numerical schemes for semilinear partial differential equations. The result here gives its application to quasilinear partial differential equations. (C) 2021 Elsevier Inc. All rights reserved.

We study the decay property for symmetric hyperbolic systems with memory-type diffusion. Under the structural condition (called Craftsmanship condition) we prove that the system is uniformly dissipative and the solutions satisfy the corresponding decay property. Our proof is based on a technical energy method in the Fourier space which makes use of the properties of strongly positive definite kernels. (C) 2020 Elsevier Inc. All rights reserved.

We investigate the initial value problem for the generalized double dispersion equation in any dimensions. Inspired by [28] for the hyperbolic system of first order PDEs, we develop Littlewood-Paley pointwise energy estimates for the dissipative wave equation of high-order. Furthermore, with aid of the frequency-localization Duhamel principle, we establish the global existence and optimal decay estimates of solutions in spatially critical Besov spaces. Our results could hold true for any dimensions (n >= 1). Indeed, the proofs are different in case of high dimensions and low dimensions owing to interpolation tricks. (C) 2019 Elsevier Inc. All rights reserved.

This paper is concerned with the rarefaction waves for a model system of hyperbolic balance laws in the whole space and in the half space. We prove the asymptotic stability of rarefaction waves under smallness assumptions on the initial perturbation and on the amplitude of the waves. The proof is based on the L-2 energy method.

This paper is concerned with the weak dissipative structure for linear symmetric hyperbolic systems with relaxation. The authors of this paper had already analyzed the new dissipative structure called the regularity-loss type in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239-266]. Compared with the dissipative structure of the standard type in [T. Umeda, S. Kawashima and Y. Shizuta, On the devay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math. 1 (1984) 435-457
Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249-275], the regularity-loss type possesses a weaker structure in the high-frequency region in the Fourier space. Furthermore, there are some physical models which have more complicated structure, which we discussed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure of two hyperbolic relaxation models with regularity loss, Kyoto J. Math. 57(2) (2017) 235-292]. Under this situation, we introduce new concepts and extend our previous results developed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239-266] to cover those complicated models.

In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a 2 × 2 system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.

We consider the initial-history value problem for the one-dimensional equation of thermoelasticity with fading memory. It is proved that if the data are smooth and small, then a unique smooth solution exists globally in time and converges to the constant equilibrium state as time goes to infinity. Our proof is based on a technical energy method which makes use of the strict convexity of the entropy function and the properties of strongly positive definite kernels. (C) 2017 Elsevier Inc. All rights reserved.

This article investigates two types of decay structures for linear symmetric hyperbolic systems with nonsymmetric relaxation. Previously, the same authors introduced a new structural condition which is a generalization of the classical Kawashima-Shizuta condition and also analyzed the weak dissipative structure called the regularity loss type for general systems with nonsymraetric relaxation, which includes the Timoshenko system and the Euler-Maxwell system as two concrete examples. Inspired by the previous work, we further construct in this article two more complex models which satisfy some new decay structure of regularity-loss type. The proof is based on the elementary Fourier energy method as well as the suitable linear combination of different energy inequalities. The results show that the model of type I has a decay structure similar to that of the Timoshenko system with heat conduction via the Cattaneo law, and the model of type II is a direct extension of two models considered previously to the case of higher phase dimensions.

Dissipative hyperbolic systems of regularity-loss have been recently received increasing attention. Extra higher regularity is usually assumed to obtain the optimal decay estimates, in comparison with the global-in-time existence of solutions. In this paper, we develop a new frequency-localization time-decay property, which enables us to overcome the technical difficulty and improve the minimal decay-regularity for dissipative systems. As an application, it is shown that the optimal decay rate of L-1(R-3)-L-2(R-3) is available for Euler-Maxwell equations with the critical regularity s(c) = 5/2, that is, the extra higher regularity is not necessary. (C) 2016 Elsevier Inc. All rights reserved.

The compressible Euler-Maxwell two-fluid system arises in the modeling of magnetized plasmas. We first design crucial energy functionals to capture its dissipative structure, which is relatively weaker in comparison with the one-fluid case in the whole space R-3, due to the nonlinear coupling and cancelation between electrons and ions. Furthermore, with the aid of L-p(R-n)-L-q(Rn)-L-r(R-n) time-decay estimates, we obtain the L-1(R-3)-L-2(R-3) decay rate with the critical regularity (s(c) = 3) for the global-in-time existence of smooth solutions, which solves the decay problem left open in [Y. J. Peng, Global existence and long-time behavior of smooth solutions of two-fluid Euler-Maxwell equations, Ann. IHP Anal. Non Lineaire 29 (2012) 737-759].

Recently, a time-decay framework L-2(R-n) boolean AND (B)over dot(2,infinity)(-s) (s > 0) has been given by [49] for linearized dissipative hyperbolic systems, which allows to pay, less attention to the traditional spectral analysis. However, owing to interpolation techniques, those decay results for nonlinear hyperbolic systems hold true only in higher dimensions (n >= 3), and the analysis in low dimensions (say, n = 1,2) was left open. We try to give a satisfactory Answer in the current work. First of all, we develop new time-decay properties on the frequency-localization Duhamel principle, and then it is-shown that the classical solution and its derivatives of fractional order decay at the optimal algebraic rate in dimensions n = 1,2, by using a new technique which is the so-called "piecewise Duhamel principle" in localized time-weighted energy approaches compared to [49]. Finally, as direct applications, explicit decay statements are worked out for some relevant examples subjected to the same dissipative structure, for instance, damped compressible Euler equations, the thermoelasticity with second sound, and Timoshenko systems with equal wave speeds. (C) 2016 Elsevier Inc. All rights reserved.

In this paper we discuss the asymptotic stability of stationary solutions for the non-isentropic Euler-Maxwell system in R-3. It is known in the authors' previous works [17, 18, 19] that the Euler-Maxwell system verifies the decay property of the regularity-loss type. In this paper we first prove the existence and uniqueness of a small stationary solution. Then we show that the non-stationary problemhas a global solution in a neighborhood of the stationary solution under smallness condition on the initial perturbation. Moreover, we show the asymptotic convergence of the solution toward the stationary solution as time tends to infinity. The crucial point of the proof is to derive a priori estimates by using the energy method.

We study the Timoshenko system with Cattaneo's type heat conduction in the one-dimensional whole space. We investigate the dissipative structure of the system and derive the optimal L-2 decay estimate of the solution in a general situation. Our decay estimate is based on the detailed pointwise estimate of the solution in the Fourier space. We observe that the decay property of our Timoshenko-Cattaneo system is of the regularity-loss type. This decay property is a little different from that of the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647-667]) in the low frequency region. However, in the high frequency region, it is just the same as that of the Timoshenko-Fourier system (see [N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier's type heat conduction, J. Hyperbolic Differential Equations 11 (2014) 135-157]) or the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647-667]), although the stability number is different. Finally, we study the decay property of the Timoshenko system with the thermal effect of memory-type by reducing it to the Timoshenko-Cattaneo system.

In this paper, we introduce a notion of the mathematical entropy for hyperbolic systems of balance laws with (not necessarily symmetric) relaxation. As applications, we deal with the Timoshenko system, the Euler-Maxwell system and the Euler-Cattaneo-Maxwell system. Especially, for the Euler-Cattaneo-Maxwell system, we observe that its dissipative structure is of the regularity-loss type and investigate the corresponding decay property. Furthermore, we prove the global existence and asymptotic stability of solutions to the Euler-Cattaneo-Maxwell system for small initial data.

As a continued work of 1181, we are concerned with the Timoshenko system in the case of non-equal wave speeds, which admits the dissipative structure of regularity-loss. Firstly, with the modification of a priori estimates in 1181, we construct global solutions to the Timoshenko system pertaining to data in the Besov space with the regularity s = 3/2. Owing to the weaker dissipative mechanism, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions, so it is almost impossible to obtain the optimal decay rates in the critical space. To overcome the outstanding difficulty, we develop a new frequency-localization time-decay inequality, which captures the information related to the integrability at the high-frequency part. Furthermore, by the energy approach in terms of high-frequency and low-frequency decomposition, we show the optimal decay rate for Timoshenko system in critical Besov spaces, which improves previous works greatly. (C) 2015 Elsevier Inc. All rights reserved.

Due to the dissipative structure of regularity-loss, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions to dissipative systems. The aim of this paper is to seek the lowest regularity index for the optimal decay rate of L-1(R-n)-L-2(R-n). Consequently, a notion of minimal decay regularity for dissipative systems of regularity-loss is firstly proposed. To do this, we develop a new time-decay estimate of L-p(R-n)-L-q(R-n)-L-r(R-n) type by using the low-frequency and high-frequency analysis in Fourier spaces. As an application, for compressible Euler-Maxwell equations with the weaker dissipative mechanism, it is shown that the minimal decay regularity coincides with the critical regularity for global classical solutions. Moreover, the recent decay property for symmetric hyperbolic systems with non-symmetric dissipation is also extended to be the L-p-version. (C) 2015 Elsevier Masson SAS. All rights reserved.

We give a new decay framework for the general dissipative hyperbolic system and the hyperbolic-parabolic composite system, which allows us to pay less attention to the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood-Paley pointwise energy estimates and new time-weighted energy functionals to establish optimal decay estimates on the framework of spatially critical Besov spaces for the degenerately dissipative hyperbolic system of balance laws. Based on the embedding and the improved Gagliardo-Nirenberg inequality, the optimal decay rates and decay rates are further shown. Finally, as a direct application, the optimal decay rates for three dimensional damped compressible Euler equations are also obtained.

In this paper, we are concerned with the optimal decay estimates for the Euler-Poisson two-fluid system. It is first revealed that the irrotationality of the coupled electronic field plays a key role such that the two-fluid system has the same dissipative structure as generally hyperbolic systems satisfying the Shizuta-Kawashima condition. This fact inspires us to obtain decay properties for linearized systems in the framework of Besov spaces. Furthermore, various decay estimates of solution and its derivatives of fractional order are deduced by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As the direct consequence, the optimal decay rates of L-p(R-3)-L-2 (R-3) (1 <= p < 2) type for the Euler-Poisson two-fluid system are also shown. Compared with previous works in Sobolev spaces, a new observation is that the difference of variables exactly consists of a one-fluid Euler-Poisson equations, which leads to the sharp decay estimates for velocities.

In this paper, we study the initial value problem for the generalized cubic double dispersion equation in n-dimensional space. Under a small condition on the initial data, we prove the global existence and asymptotic decay of solutions for all space dimensions n >= 1. Moreover, when n >= 2, we show that the solution can be approximated by the linear solution as time tends to infinity.

We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L-1 (R)-L-2 (R) type is also shown. (C) 2014 Elsevier Inc. All rights reserved.

This article is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima Shizuta stability condition formulated in [8], [4]. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (cf. the Timoshenko system and the Euler-Maxwell system). Therefore our purpose of this paper is to formulate a new structural condition and to analyze the weak dissipative structure for general systems with non-symmetric relaxation. This article is a survey of the paper [5].

In this paper, we consider a class of second order hyperbolic systems of viscoelasticity in the whole space and show the global existence and sharp decay estimates of solutions under the smallness condition on the initial data. The tools that we use here are the time-weighted energy method and the semigroup argument.

The non-isentropic compressible Euler-Maxwell system is investigated in R-3 in this paper, and the L-q time decay rate for the global smooth solution is established. It is shown that the density and temperature of electron converge to the equilibrium states at the same rate (1 + t)(-11/4) in L-q norm. This phenomenon on the charge transport shows the essential relation of the equations with the non-isentropic Euler-Maxwell and the isentropic Euler-Maxwell equations.

We study the Timoshenko system with Fourier's type heat conduction in the one-dimensional (whole) space. We observe that the dissipative structure of the system is of the regularity-loss type, which is somewhat different from that of the dissipative Timoshenko system studied earlier by Ide-Haramoto-Kawashima. Moreover, we establish optimal L-2 decay estimates for general solutions. The proof is based on detailed pointwise estimates of solutions in the Fourier space. Also, we introuce here a refinement of the energy method employed by Ide-Haramoto-Kawashima for the dissipative Timoshenko system, which leads us to an improvement on their energy method.

The basic existence theory of Kato and Majda enables us to obtain local-in-time classical solutions to generally quasilinear hyperbolic systems in the framework of Sobolev spaces (in x) with higher regularity. However, it remains a challenging open problem whether classical solutions still preserve well-posedness in the case of critical regularity. This paper is concerned with partially dissipative hyperbolic system of balance laws. Under the entropy dissipative assumption, we establish the local well-posedness and blow-up criterion of classical solutions in the framework of Besov spaces with critical regularity with the aid of the standard iteration argument and Friedrichs' regularization method. Then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner spaces (mixed space-time Besov spaces). This fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well-posedness and stability by assuming at all times the Shizuta-Kawashima algebraic condition. As a direct application, the corresponding well-posedness and stability of classical solutions to the compressible Euler equations with damping are also obtained.

We construct (uniform) global classical solutions to the damped compressible Euler equations on the framework of general Besov spaces which includes both the usual Sobolev spaces H-s (R-d) (s > 1 + d/2) and the critical Besov space B-2,1(1+d/2) (R-d). Such extension heavily depends on a revision of commutator estimates and an elementary fact that indicates the connection between homogeneous and inhomogeneous Chemin-Lerner spaces. Furthermore, we obtain the diffusive relaxation limit of Euler equations towards the porous medium equation, by means of Aubin-Lions compactness argument. (C) 2013 Elsevier Inc. All rights reserved.

We study the initial value problem for the generalized cubic double dispersion equation in one space dimension. We establish a nonlinear approximation result to our global solutions that was obtained in [6]. Moreover, we show that as time tends to infinity, the solution approaches the superposition of nonlinear diffusion waves which are given explicitly in terms of the self-similar solution of the viscous Burgers equation. The proof is based on the semigroup argument combined with the analysis of wave decomposition.

In this paper we consider the initial-value problem for the nonlinear Timoshenko system with a memory term. Due to the regularity-loss property and weak dissipation, we have to assume stronger nonlinearity than usual. By virtue of the semi-group arguments, we obtain the global existence and optimal decay of solutions to the nonlinear problem under smallness and enough regularity assumptions on the initial data, where we employ a time-weighted L2 energy method combined with the optimal L2 decay of lower-order derivatives of solutions. © 2013 Elsevier Ltd. All rights reserved.

In this paper, we study two-fluid compressible Euler-Maxwell equations in the whole space or periodic space. In comparison with the one-fluid case, we need to deal with the difficulty mainly caused by the nonlinear coupling and cancelation between electrons and ions. Precisely, the expected dissipation rates of densities for two carriers are no longer available. To capture the weaker dissipation, we develop a continuity for compositions, which is a natural generalization from Besov spaces to Chemin-Lerner spaces (space-time Besov spaces). An elementary fact that indicates the relation between homogeneous Chemin-Lerner spaces and inhomogeneous Chemin-Lerner spaces will been also used. © 2013 Society for Industrial and Applied Mathematics.

We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space R-n admits a unique stationary solution in a general situation. Moreover, it is proved that when n >= 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted L-p energy method.

This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima-Shizuta stability condition formulated in Umeda et al. (Jpn J Appl Math 1:435-457, 1984) and Shizuta and Kawashima (Hokkaido Math J 14:249-275, 1985) and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (for example, the Timoshenko system and the Euler-Maxwell system). Moreover, it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity-loss type (see Duan in J Hyperbolic Differ Equ 8:375-413, 2011; Ide et al. in Math Models Meth Appl Sci 18:647-667, 2008; Ide and Kawashima in Math Models Meth Appl Sci 18:1001-1025, 2008; Ueda et al. in SIAM J Math Anal 2012; Ueda and Kawashima in Methods Appl Anal 2012). Therefore our purpose in this paper is to formulate a new structural condition which includes the Kawashima-Shizuta condition, and to analyze the weak dissipative structure for general systems with non-symmetric relaxation.

In this paper we consider the initial value problem for the Timoshenko system with a memory term. We construct the fundamental solution by using the Fourier-Laplace transform and obtain the solution formula of the problem. Moreover, applying the energy method in the Fourier space, we derive the pointwise estimate of solutions in the Fourier space, which gives a sharp decay estimate of solutions. It is shown that the decay property of the system is of the regularity-loss type and is weaker than that of the Timoshenko system with a frictional dissipation.

We consider the initial value problem for some nonlinear hyperbolic-dispersive systems in one space dimension. Combining the classical energy method and the smoothing estimates for the Airy equation, we guarantee the time local well-posedness for this system. We also discuss the extension of our results to more general hyperbolic-dispersive system. Copyright (C) 2011 John Wiley & Sons, Ltd.

This paper is devoted to the study of the sharp decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity in the whole space. We develop the time-weighted energy method for our system, which can yield the decay estimate of solutions for small initial data in L-2, provided that n >= 2. Also, we discuss the fundamental solutions to the linearized system and study the decay properties for the corresponding solution operators. Then, by employing the same time-weighted energy method together with the semigroup argument, we show the optimal decay estimate of solutions for small initial data in L-2 boolean AND L-1 and for all n >= 1.

We consider the large-time behavior of solutions to the initial value problem for the Euler-Maxwell system in R-3. This system verifies the decay property of the regularity-loss type. Under smallness condition on the initial perturbation, we show that the solution to the problem exists globally in time and converges to the equilibrium state as time tends to infinity. The crucial point of the proof is to derive a priori estimates of solutions by using the energy method.

In this paper we focus on the initial value problem of a quasi-linear dissipative plate equation with arbitrary spatial dimensions (n >= 1). This equation verifies the decay property of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted (with negative exponent) L(2) energy method combined with the optimal L(2) decay estimates of lower-order derivatives of solutions. We obtain the global existence and optimal decay estimates of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is the fundamental solution of the corresponding fourth-order linear parabolic equation.

In the present paper, we study a large-time behavior of solutions to a quasilinear second-order hyperbolic system which describes a motion of viscoelastic materials. The system has dissipative properties consisting of a memory term and a damping term. It is proved that the solution exists globally in time in the Sobolev space, provided that the initial data are sufficiently small. Moreover, we show that the solution converges to zero as time tends to infinity. The crucial point of the proof is to derive uniform a priori estimates of solutions by using an energy method. © 2011 by Kyoto University.

In this paper we focus on the initial value problem of the semi-linear plate equation with memory in multi-dimensions (n >= 1), the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operators, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.

The aim in this paper is to develop the time-weighted energy method for quasilinear hyperbolic systems of viscoelasticity. As a consequence, we prove the global existence and decay estimate of solutions for the space dimension n >= 2; provided that the initial data are small in the L-2-Sobolev space.

In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space (n >= 2). This equation verifies the decay property of the regularity-loss type, which causes the difficulty in deriving the global a priori estimates of solutions. We overcome this difficulty by employing a time-weighted L(2) energy method which makes use of the integrability of parallel to(partial derivative(2)(x)u(t); partial derivative(3)(x)u)(t)parallel to L(infinity). This L(infinity) norm can be controlled by showing the optimal L(2) decay estimates for lower-order derivatives of solutions. Thus we obtain the desired a priori estimate which enables us to prove the global existence and asymptotic decay of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is given explicitly in terms of the fundamental solution of a fourth-order linear parabolic equation.

This article is concerned with the existence of traveling wave solutions, including standing waves, to some models based on configurational forces, describing respectively the diffusionless phase transitions of solid materials, e.g., Steel, and phase transitions due to interface motion by interface diffusion, e.g., Sintering. These models were proposed by Alber and Zhu in [3]. We consider both the order-parameter-conserved case and the non-conserved one, under suitable assumptions. Also we compare our results with the corresponding ones for the Allen-Cahn and the Cahn-Hilliard equations coupled with linear elasticity, which are models for diffusion-dominated phase transitions in elastic solids.

In this paper we focus on the initial value problem for a hyperbolic-elliptic coupled system of a radiating gas in multi-dimensional space. By using a time-weighted energy method, we obtain the global existence and optimal decay estimates of solutions. Moreover, we show that the solution is asymptotic to the linear diffusion wave which is given in terms of the heat kernel.

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space R-n. In this paper, we study half space problems in R-+(n) = R+ x Rn-1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x' is an element of Rn-1. For the variable x(i) is an element of R+ in the normal direction, we use L-2 space or weighted L-2 space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t -> infinity. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13]. (C) 2010 Elsevier Inc. All rights reserved.

The main concern of this paper is to study large-time behavior of solutions to an ideal polytropic model of compressible viscous gases in one-dimensional half-space. We consider an outflow problem and obtain a convergence rate of solutions toward a corresponding stationary solution. Here the existence of the stationary solution is proved under a smallness condition on the boundary data with the aid of center manifold theory. We also show the time asymptotic stability of the stationary solution under smallness assumptions on the boundary data and the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. The proof is based on deriving a priori estimates by using a time and space weighted energy method. © 2010 World Scientific Publishing Company.

We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

We study a class of second order hyperbolic systems with dissipation which describes viscoelastic materials. The considered dissipation is given by the sum of the memory term and the damping term. When the dissipation is effective over the whole system, we show that the solution decays in L(2) at the rate t(-n/4) as t ->infinity provided that the corresponding initial data are in L(2) boolean AND L(1). where n is the space dimension. The proof is based on the energy method in the Fourier space. Also, we discuss similar systems with weaker dissipation by introducing the operator (1 - Delta)(-theta/2) with theta > 0 in front of the dissipation terms and observe that the decay structure of these systems is of the regularity-loss type. (C) 2009 Elsevier Inc. All rights reserved.

In this short paper, we report global existence and temporal decay properties for the solution of theNavier-Stokes equations with free boundary, describing the motion of infinite mass of viscous, incompressible fluid without surface tension. (C) 2009 Elsevier Ltd. All rights reserved.

This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states, according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove the stability of the rarefaction wave, we use the solution to Burgers' equation to construct a suitably smooth approximation of the rarefaction wave and establish some time-decay estimates in L (p) -norm for the smoothed rarefaction wave. We then employ the L (2)-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity.

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t(-alpha/4) as t -> infinity, provided that the initial perturbation is in the weighted space L(alpha)(2) = L(2)(R(+); (1 + x)(alpha)) for alpha < alpha(c)(q) := 3 + 2/q, where q is the degeneracy exponent. This restriction on a is best possible in the sense that the corresponding linearized operator cannot be dissipative in L(alpha)(2) for alpha > alpha(c)(q). Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant. (c) 2009 Elsevier Inc. All rights reserved.

We study the stationary problem in the whole space R(n) for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We prove the existence and uniqueness of stationary solutions in the weighted LP spaces. The proof is based on a fixed point theorem of the Leray-Schauder type. Copyright (C) 2008 John Wiley & Sons, Ltd.

We study the decay property of a certain nonlinear hyperbolic-elliptic system with 2mth-order elliptic part, which is a modified version of the simplest radiating gas model It is proved that, for m >= 2, the system verifies a decay property of the regularity-loss type that is characterized by the parameter m This dissipative property is very weak in the high-frequency region and causes a difficulty in showing the global existence of solutions to the nonlinear problem By employing the time-weighted energy method together with the optimal decay for lower-order derivatives of solutions, we overcome this difficulty and establish a global existence and asymptotic decay result Furthermore, we show that the solution approaches the nonlinear diffusion wave described by the self-similar solution of the Burgers equation as time tends to infinity

This work is concerned with time-decay properties of small-amplitude global smooth solutions to the initial value problem for hyperbolic systems of balance laws admitting an entropy and satisfying the stability condition. By using energy methods in both the physical space and the Fourier space, we obtain the optimal decay estimates of solutions and their derivatives in the L(2)-norm up to order s - 1, provided that the initial data are in H(s). A key ingredient in our analysis is a time-weighted energy estimate, leading to a decay estimate for multi-dimensional problems without assuming the L(1) property on initial data.

We Study the drift-diffusion system arising in the semiconductor device simulation and the plasma physics. We consider the initial value problem for this system and derive the optimal L(P) decay estimate of solutions by applying the time weighted L(P) energy method. Furthermore, we show that the solutions approach the corresponding heat kernels as time tends to infinity. This asymptotic result is based on the optimal L(P) - L(q) decay estimate for the linearized problem.

We consider the initial value problem for a nonlinear version of the dissipative Timoshenko system. This syetem verifies the decay property of regularity-loss type. To overcome this difficulty caused by the regularity-loss property, we employ the time weighed L-2 energy method which is combined with the optimal L-2 decay estimates for lower order derivatives of solutions. Then we show the global existence and asymptotic decay of solutions under smallness and enough regularity conditions on the initial data. Moreover, we show that the solution approaches the linear diffusion wave expressed in terms of the superposition of the heat kernels as time tends to infinity.

In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L-2-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are "good" and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave. (c) 2008 Elsevier Inc. All rights reserved.

We study the decay property of the dissipative Timoshenko system in the one-dimensional whole space. We derive the L-2 decay estimates of solutions in a general situation and observe that this decay structure is of the regularity-loss type. Also, we give a refinement of these decay estimates for some special initial data. Moreover, under enough regularity assumption on the initial data, we show that the solution approaches the linear diffusion wave expressed in terms of the heat kernels as time tends to infinity. The proof is based on the detailed pointwise estimates of solutions in the Fourier space.

We are concerned with the initial value problem for a damped wave equation with a nonlinear convection term which is derived from a semilinear hyperbolic system with relaxation. We show the global existence and asymptotic decay of solutions in W-1,W-p (1 <= p <= infinity) under smallness condition on the initial data. Moreover, we show that the solution approaches in W-1,W-p (1 <= p <= infinity) the nonlinear diffusion wave expressed in terms of the self-similar solution of the Burgers equation as time tends to infinity. Our results are based on the detailed pointwise estimates for the fundamental solutions to the linearlized equation.

We discuss the global solvability and asymptotic behavior of solutions to the Cauchy problem for some nonlinear hyperbolic-elliptic system with a fourth-order elliptic part. This system is a modified version of the simplest radiating gas model and verifies a decay property of regularity-loss type. Such a dissipative structure also appears in the dissipative Timoshenko system studied by Rivera and Racke. This dissipative property is very weak in high frequency region and causes the difficulty in deriving the desired a priori estimates for global solutions to the nonlinear problem. In fact, it turns out that the usual energy method does not work well. We overcome this difficulty by employing a time-weighted energy method which is combined with the optimal decay for lower order derivatives of solutions, and we establish a global existence and asymptotic decay result. Furthermore, we show that the solution has an asymptotic self-similar profile described by the Burgers equation as time tends to infinity.

Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space R-+(n)(n >= 2) under outflow boundary condition is investigated. It is shown that the planar stationary solution is stable with respect to small perturbations in H-s (R-+(n)) with s >= [n/2] + 1 and the perturbations decay in L-infinity norm as t --> infinity, provided that the magnitude of the stationary solution is sufficiently small. The stability result is proved by the energy method. In the proof an energy functional based on the total energy of the system plays an important role.

This paper investigates the solvability of initial boundary value problem for a quasilinear hyperbolic-parabolic system which consists of a transport equation and strongly parabolic system. The characteristics of the transport equation are assumed to be outward on the boundary of the domain. The unique local (in time) existence of solutions is shown in the class of continuous functions with values in H-s, where s is an integer satisfying s >= [n/2] + 1.

In this paper, we introduce an entropy condition for hyperbolic systems of balance laws. Under this condition, we use the Chapman-Enskog expansion to derive the corresponding viscous conservation laws. Further structural conditions are discussed in order to develop (local and global) existence theories for the balance laws and viscous conservation laws.

We introduce a new L-p energy method for multi-dimensional viscous conservation laws. Our energy method is useful enough to derive the optimal decay estimates of solutions in the W-1,W-p space for the Cauchy problem. It is also applicable to the problem for the stability of planar waves in the whole space or in the half space, and gives the optimal convergence rate toward the planar waves as time goes to infinity. This energy method makes use of several special interpolation inequalities.

We are concerned with the asymptotic behavior of a solution to the initial value problem for a system of hyperbolic conservation laws coupled with elliptic equations. This kind of problem was first considered in our previous paper. In the present paper, we generalize the previous results to a broad class of hyperbolic-elliptic coupled systems.
Assuming the existence of the entropy function and the stability condition, we prove the global existence and the asymptotic decay of the solution for small initial data in a suitable Sobolev space. Then, it is shown that the solution is well approximated, for large time, by a solution to the corresponding hyperbolic-parabolic coupled system. The first result is proved by deriving a priori estimates through the standard energy method. The spectral analysis with the aid of the a priori estimate gives the second result.

We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier-Stokes equation in a half space. The main concern is to analyze the phenomena that happens when the fluid blows out through the boundary. Thus, it is natural to consider the problem in the Eulerian coordinate. We have obtained the two results for this problem. The first result is concerning the existence of the stationary solution. We present the necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if an initial perturbation is small in the suitable Sobolev space. The second result is proved by using an L-2-energy method with the aid of the Poincare type inequality.

We investigate the asymptotic stability of a stationary solution to an initial boundary value problem for a 2-dimensional viscous conservation law in half plane. Precisely, we show that under suitable boundary and spatial asymptotic conditions, a solution converges to the corresponding stationary solution as time tends to infinity. The proof is based on an a priori estimate in the H-2-Sobolev space, which is obtained by a standard energy method. In this computation, we utilize the Poincare type inequality. In addition, we obtain a convergence rate under the assumption that the initial data converges to a spatial asymptotic state algebraically fast. This result is obtained by a weighted energy estimate.

The present paper is concerned with stationary solutions for discrete velocity models of the Boltzmann equation with reflective boundary condition in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of stationary solutions satisfying the reflective boundary condition as well as the spatially asymptotic condition given by a Maxwellian state. First, the sufficient condition is obtained for the linearized system. Then, this result is applied to prove the existence theorem for the nonlinear equation through the contraction mapping principle. Also, it is shown that the stationary solution approaches the asymptotic Maxwellian state exponentially as the spatial variable tends to infinity. Moreover, we show the time asymptotic stability of the stationary solutions. In the proof, we employ the standard energy method to obtain a priori estimates for nonstationary solutions. The exponential convergence at the spatial asymptotic state of the stationary solutions gives essential information to handle some error terms. Then we discuss some concrete models of the Boltzmann type as an application of our general theory.

We-study the existence and the uniqueness of stationary-solutions for discrete velocity models of the Boltzmann equation in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of solutions-connecting the given boundary data and the Maxwellian state at a spatially asymptotic point. First, a sufficient condition is obtained for the Linearized system. Then this result as well as the contraction mapping principle is applied to prove the existence theorem for the nonlinear equation.
Also, we show that the stationary wave approaches the Maxwellian state exponentially at a spatially asymptotic point. We also discuss some concrete models of Boltzmann type as an application of our general theory. Here, it turns out that our sufficient condition is general enough to cover many concrete models.

This paper deals with the global existence and the time asymptotic state of solutions to the initial value problems for the system derived from approximating a one-dimensional model of a radiating gas. When the spatial derivative of the initial data is larger than a certain negative critical value, a unique solution exists globally in time. But if it is smaller than another negative critical value, the spatial derivative of the corresponding solution blows up in a finite time. Thus it is natural to think about weak solutions in a suitable sense. As a prototype of weak solutions, we consider the Cauchy problem with the Riemann initial data of which the left state is larger than the right state. This condition ensures the existence of the corresponding traveling wave, connecting the left state and the right state asymptotically. This Riemann problem admits a global weak solution, provided that the magnitude of the initial discontinuity is smaller than 1/2. Although the solution has a discontinuity, we have the uniqueness of a solution in weak sense by imposing the entropy condition. Furthermore, the magnitude of the discontinuity contained in the solution decays to zero with an exponential rate as the time t goes to infinity. Also, the solution approaches the corresponding traveling wave with the rate t(-1/4) uniformly. The first result is obtained by the maximal principles. To show the second result, we have used an energy method with some estimates, which are also obtained through maximal principles.

This paper is concerned with the existence and the asymptotic stability of traveling waves for a model system derived from approximating the one-dimensional system of the radiating gas. We show the existence of smooth or discontinuous traveling waves and also prove the uniqueness of these traveling waves under the entropy condition, in the class of piecewise smooth functions with the first kind discontinuities. Furthermore, we show that the C-3-smooth traveling waves are asymptotically stable and that the rate of convergence toward these waves is t(?1/4),which looks optimal. The proof of stability is given by applying the standard energy method to the integrated equation of the original one.

A reaction-diffusion system which is related to a simple irreversible chemical reaction between two chemical substances is considered. When a non-negative global solution for the system converges uniformly to zero with polynomial rate as time goes to infinity, large-time approximation of the solution is studied. It is shown that the difference of the solution and its spatial average tends to zero with exponential rate via a global solution for the corresponding system of ordinary differential equations.

In this paper, we shall discuss the stability of smooth monotone travelling wave solutions for viscoelastic materials with memory. It is known that a smooth monotone travelling wave solution exists for (1.1) if the end states are close and satisfy the Rankine-Hugoniot condition. For such a travelling wave, we shall show that if the initial data are close to a travelling wave solution, the solutions to (1.1) will approach the travelling wave solution in sup norm as the rime goes to infinity, For the constitutive relations, we shall discuss two cases: convex and nonconvex. (C) 1996 Academic Press, Inc.

In this paper we discuss the necessary and sufficient conditions for the existence of smooth monotone shock profiles for viscoelastic materials with memory. We also discuss the uniqueness. We consider both convex and nonconvex constitutive relations. In the case of nonconvex constitutive relations, we include a degenerate case where the speed of the shock profile is equal to the speed of the equilibrium characteristics at one of the end states. This was not discussed in previous literature.

Large time behavior of the solution to some simple reaction-diffusion system is studied. It is proved that the solution behaves like the solution to the corresponding system of ordinary differential equations as time goes to infinity. The proof is based on an energy method combined with the L(p)-L(q) estimate for the associated semigroup.

The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain of n-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions as t tending to infinity is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.

The initial-boundary value problems and the corresponding stationary problems of the discrete Boltzmann equation are studied. It is shown that stationary solutions exist for any boundary data. These stationary solutions are unique in a neighborhood of a given constant Maxwellian. Furthermore, it is proved that if both initial and boundary data are close to a given constant Maxwellian, then unique solutions to the initial-boundary value problems exist globally in time and converge to the corresponding unique stationary solutions exponentially as time goes to infinity. The stability condition plays an essential role in proving the uniqueness and the time-asymptotic stability results. © 1991 JJIAM Publishing Committee.

The initial value problem for the nonlinear Boltzmann equation is studied. For the existence of global solutions near a Maxwellian, it is important to obtain a desired decay estimate for the linearized equation. In previous works, such a decay estimate was obtained by a method based on the spectral theory for the linearized Boltzmann operator. The aim of this paper is to show the same decay estimate by a new method. Our method is the so-called energy method and makes use of a Ljapunov function for the ordinary differential equation obtained by taking the Fourier transform. Our Ljapunov function is constructed explicitly by using some property of the equations for thirteen moments. © 1990 JJAM Publishing Committee.

This chapter discusses the initial value problem for the Boltzmann equation. The problem concerning the existence of global solutions of the problem in a neighborhood of a Maxwellian is studied. The key of the problem is to get a suitable decay estimate for the linearized Boltzmann equation around the Maxwellian. © 1987, Publishing Committee of Lecture Notes in Numerical and Applied Analysis

This chapter reviews a recent work on the classical solutions of the compressible Euler equation and discusses the existence of global classical solutions. © 1987 Publishing Committee of Lecture Notes in Numerical and Applied Analysis.

The linearized equations of the electrically conducting compressible viscous fluids are studied. It is shown that the decay estimate (1+t)-3/4 in L2(R3) holds for solutions of the above equations, provided that the initial data are in L2(R3)∩L1(R3). Since the systems of equations are not rotationally invariant, the perturbation theory for one parameter family of matrices is not useful enough to derive the above result. Therefore, by exploiting an energy method, we show that the decay estimate holds for the solutions of a general class of equations of symmetric hyperbolic-parabolic type, which contains the linearized equations in both electro-magneto-fluid dynamics and magnetohydrodynamics. © 1984 JJAM Publishing Committee.

The equations of an electrically conducting compressible fluid in electro-magneto-fluid dynamics are studied. It is proved that in a certain case of two-dimensional flow, the equations of the fluid become a symmetric hyperbolic-parabolic system in both of the viscous and non-viscous cases. Therefore, the initial value problem is well posed in the Sobolev spaces at least for short time interval. Furthermore, in the viscous case, the solution exists globally in time and tends to the constant state as time goes to infinity, provided the initial data are closed to the constant state. The proof is based on a technical energy method, which makes use of a quadratic function associated with the total energy of the fluid. © 1984 JJAM Publishing Committee.