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This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain Ω : = R × T 2 with T 2 = (R / Z) 2. The stability problem of viscous shock under small 1-dimensional perturbations was solved in Duan-Liu-Zhang [7]. In this paper, we prove the viscous shock is still stable under small 3-d perturbations. Firstly, we decompose the perturbation into the zero mode and non-zero mode. Then we can show that both the perturbation and zero-mode time-asymptotically tend to zero by the anti-derivative technique and crucial estimates on the zero-mode. Moreover, we can further prove that the non-zero mode tends to zero with exponential decay rate. The key point is to estimate the non-zero mode of nonlinear terms involving electronic potential, see Lemma 6.1 below. [ABSTRACT FROM AUTHOR]

In this paper, we prove some qualitative properties of pushed fronts for the periodic reaction-diffusion-equation with general monostable nonlinearities. Especially, we prove the exponential behavior of pushed fronts when they are approaching their unstable state. The proof also allows us to get the exponential behavior of pulsating fronts with speed c larger than the minimal speed. Through the exponential behavior, we finally prove the stability of pushed fronts. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the large time behavior of the three-dimensional full compressible Navier-Stokes-Korteweg equations, which is used to model compressible viscous and heat-conductive fluids with internal capillarity, i.e., the liquid-vapor phase mixtures endowed with a variable internal capillarity. First, we construct the planar rarefaction wave to the three-dimensional full compressible Navier-Stokes-Korteweg equations, which can be derived by the fact that the rarefaction wave is nonlinearly stable to the one-dimensional full compressible Navier-Stokes-Korteweg equations. Then it is shown that the planar rarefaction wave is asymptotically stable provided that the initial data are a suitably small perturbation of the planar rarefaction wave and the strength of the rarefaction wave is small. The proof is based on the delicate energy method. [ABSTRACT FROM AUTHOR]

We consider the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions on an interval. Our previous papers [2,3] obtained the global bifurcation diagram of stationary solutions, which includes the secondary bifurcation from the odd symmetric solution due to the symmetric breaking effect. This paper derives the stability/instability of all symmetric solutions and instability of a part of asymmetric solutions. To do so, we use the exact representation of symmetric solutions and show the distribution of eigenvalues of the linearized eigenvalue problem around these solutions. And we show the instability of asymmetric solutions by the SLEP method. Finally, our results with respect to stability are supported by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper we prove the well-posedness to the wave equation of Kirchhoff type. Under a portion of the boundary, we consider the acoustic boundary conditions. We also prove the exponential stability of the energy associated to the problem. Our result generalize the previous literature where only the Carrier model was considered (when the nonlinearity involves the L 2 (Ω) norm) or the Kirchhoff model with porous acoustic boundary conditions. The main tool is the Faedo-Galerkin method. Due to the presence of acoustic boundary conditions, we can not use special basis and new estimates are necessary. To prove the stability we use integral estimates. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in investigating stability results for generalized ordinary differential equations (generalized ODEs in short), and their applications to measure differential equations and dynamic equations on time scales. First, we establish stability, asymptotic and exponential stability for the trivial solution of generalized ODEs. Secondly, we use the well known correspondence between solutions of generalized ODEs and solutions of measure differential equations, obtaining analogues results for the last equations. Finally, we apply some of these results for dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when there is only one viscous damping acting either on the beam equation or one of the wave equations. For a few special cases, optimal orders are confirmed. We also study the synchronization of the model with viscous damping on the transversal displacement. Our results reveal that the order of the polynomial decay rate is sensitive to various boundary conditions and to the damping locations. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with well-posedness of an anisotropic parabolic equation with the convection term. When some diffusion coefficients are degenerate on the boundary ∂Ω and the others are positive on Ω ‾ , we propose a novel partial boundary value condition to study the stability of the solutions for the anisotropic parabolic equation. A new concept, the general characteristic function of the domain Ω, is introduced and applied. The existence and stability of the solutions is established under the given partial boundary value conditions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the stability results for measure differential equations, considering more general conditions under the Lyapunov functionals and concerning the functions f and g. Moreover, we prove these stability results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction–diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption. [Copyright &y& Elsevier]

Abstract: This paper is concerned with existence and stability of traveling curved fronts for the Allen–Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front. [Copyright &y& Elsevier]