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In this paper, we shall investigate the large-time behavior of the solution to an outflow problem of the one-dimensional bipolar quantum Navier-Stokes-Poisson system in the half space. Under some suitable assumptions on the boundary data and the space-asymptotic states, we successfully construct a nonlinear wave which is the superposition of the stationary solution and the 2-rarefaction wave. Then, by means of the $ L^2 $-energy method, we prove that this nonlinear wave is asymptotically stable provided that the initial perturbation and the strength of the stationary solution are small enough, while the strength of the 2-rarefaction wave can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects $ -\frac {\mu }{(1+t)^{\lambda }}n_iu_i(i=1, 2) $ − μ (1 + t) λ n i u i (i = 1 , 2) for $ -1 − 1 < λ < 1 , μ > 0. Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of solutions to an outflow problem for the one-dimensional bipolar quantum Navier-Stokes-Poisson equations in the half space. First, by means of the manifold theory and the center manifold theorem, we show the existence and spatial decay rate of the stationary solution provided the boundary strength is small enough. Next, based on elaborate energy estimates, we prove that the stationary solution is asymptotically stable in the case that the boundary strength and the initial perturbation around the stationary solution are sufficiently small. [ABSTRACT FROM AUTHOR]

We shall investigate the asymptotic stability, toward a nonlinear wave, of the solution to an outflow problem for the one-dimensional compressible Navier-Stokes-Poisson equations. First, we construct this nonlinear wave which, under suitable assumptions, is the superposition of a stationary solution and a rarefaction wave. Then it is shown that the nonlinear wave is asymptotically stable in the case that the initial data are a suitably small perturbation of the nonlinear wave. The main ingredient of the proof is the -energy method that takes into account both the effect of the self-consistent electrostatic potential and the spatial decay of the stationary part of the nonlinear wave. [ABSTRACT FROM AUTHOR]

Abstract: In this article we study, by the vanishing viscosity method, the sensitivity analysis of an optimal control problem for 1-D scalar conservation laws in the presence of shocks. It is reduced to investigate the vanishing viscosity limit for the nonlinear conservation law, the corresponding linearized equation and its adjoint equation, respectively. We employ the method of matched asymptotic expansions to construct approximate solutions to those equations. It is then proved that the approximate solutions, respectively, satisfy those viscous equations in the asymptotic sense, and converge to the solutions of the corresponding inviscid problems with certain convergent rates. A new equation for the variation of shock positions is derived. It is also discussed how to identify descent directions to find the minimizer of the viscous optimal control problem in the quasi-shock case. [Copyright &y& Elsevier]

Taking into account the occurrence of a zero of the surface diffusion current and the requirement of the Ehrlich-Schwoebel effect, Siegert et al. formulated a model of Langevin type that describes the growth of pyramid-like structures on a surface under conditions of molecular beam epitaxy and that the slope of these pyramids is selected by the crystalline symmetries of the growing film. In this article, the existence and uniqueness of weak solution to an initial boundary value problem for this model is proved, in the case that the noise is neglected. The regularity of the weak solution to models, with/without slope selection, is also investigated. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We prove the global-in-time existence of spherically symmetric solutions to an initial-boundary value problem for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, non-uniformly parabolic equation of second order. This problem models the evolutional behavior of materials in which martensitic phase transitions, driven by configurational forces, take place. Moreover, it can be considered to be a regularization of the corresponding sharp interface model. By assuming that the solutions are spherically symmetric, we reduce the original multi-dimensional problem to the one in one space dimension, then prove the existence of spherically symmetric solutions. Our proof is valid due to the essential feature that the resulting problem is one-dimensional. [ABSTRACT FROM AUTHOR]

Abstract: We study an initial boundary value problem of a model describing the evolution in time of diffusive phase interfaces in solid materials, in which martensitic phase transformations driven by configurational forces take place. The model was proposed earlier by the authors and consists of the partial differential equations of linear elasticity coupled to a nonlinear, degenerate parabolic equation of second order for an order parameter. In a previous paper global existence of weak solutions in one space dimension was proved under Dirichlet boundary conditions for the order parameter. Here we show that global solutions also exist for Neumann boundary conditions. Again, the method of proof is only valid in one space dimension. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Existence of weak solutions is proved for a phase field model describing an interface in an elastically deformable solid, which moves by diffusion of atoms along the interface. The volume of the different regions separated by the interface is conserved, since no exchange of atoms across the interface occurs. The diffusion is driven only by reduction of the bulk free energy. The evolution of the order parameter in this model is governed by a degenerate parabolic fourth-order equation. If a regularizing parameter in this equation tends to zero, then solutions tend to solutions of a sharp interface model for interface diffusion. The existence proof is valid only for a 1½-dimensional situation. [ABSTRACT FROM PUBLISHER]

Examines the multiplicity of solutions to a nonlinear boundary value problem of ordinary differential equation. Evolutionary system in shape memory alloys; Number of solutions; Standard theory of dynamical system.

In this paper, we investigate the L[sup 2] decay rates for the solution to a class system of non-Newtonian fluid in R[sup n](n>=2). The decay rates are optimal in the sense that they coincide with the decay rates of the solution to the heat system. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]