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Abstract The present paper is concerned with the asymptotic behaviors of radially symmetric solutions for the multi-dimensional Burgers equation on the exterior domain in R n , n ≥ 3 , where the boundary and far field conditions are prescribed. We show that in some case where the corresponding 1-D Riemann problem for the non-viscous part admits a shock wave, the solution tends toward a linear superposition of stationary and rarefaction waves as time goes to infinity, and also show the decay rate estimates. Furthermore, we improve the results on the asymptotic stability of the stationary waves which are treated in the previous papers [2] , [3]. Finally, for the case of n = 3 , we give the complete classification of the asymptotic behaviors, which includes even a linear superposition of stationary and viscous shock waves. [ABSTRACT FROM AUTHOR]

In this paper, we continue to study a spatially heterogeneous predator–prey system where the interaction is governed by a Holling type II functional response, which has been studied in Du and Shi (2007) [14] . We further study the asymptotic profile of positive solutions and give a complete understanding of coexistence region. Moreover, a good understanding of the number, stability and asymptotic behavior of positive solutions is gained for large m . Finally, we further compare the difference of steady-state solutions between m > 0 and m = 0 . It turns out that the spatial heterogeneity of the environment and the Holling type II functional response play a very important role in this model. [ABSTRACT FROM AUTHOR]

This paper deals with the following linearly coupled nonlinear Kirchhoff-type system: { − ( a 1 + b 1 ∫ R 3 | ∇ u | 2 d x ) Δ u + μ 1 u = f ( u ) + β v in R 3 , − ( a 2 + b 2 ∫ R 3 | ∇ v | 2 d x ) Δ v + μ 2 v = g ( v ) + β u in R 3 , u , v ∈ H 1 ( R 3 ) , where a i > 0 , b i ≥ 0 , μ i > 0 are constants for i = 1 , 2 , β > 0 is a parameter and f , g ∈ C ( R , R ) . Under the general Berestycki–Lions type assumptions on f and g , we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods. We also study the asymptotic behavior of these solutions as β → 0 + . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a free boundary value problem for two-phase liquid–gas model with mass-dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum with a discontinuity. This is an extension of the paper [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid–gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV]. Just as in [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid–gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV], the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions when , which have improved the previous result of Evje and Karlsen, and get the asymptotic behavior result, also we obtain the regularity of the solutions by energy method. [Copyright &y& Elsevier]

Abstract: This paper deals with weak solutions of the one-dimensional viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions, where , and . For these solutions we investigate the convergence to steady states via a Lyapunov functional, which is constructed with a technique developed by Zelenyak. [Copyright &y& Elsevier]

Abstract: In this paper, a multidimensional nonisentropic hydrodynamic model for semiconductors with the nonconstant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. Global existence to the Cauchy problem for the multidimensional nonisentropic hydrodynamic semiconductor model with the small perturbed initial data is established, and the asymptotic behavior of these smooth solutions is investigated, namely, that the solutions converge to the general steady-state solution exponentially fast as is obtained. Moreover, the existence and uniqueness of the stationary solutions are investigated. [Copyright &y& Elsevier]

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution. [Copyright &y& Elsevier]

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given. [Copyright &y& Elsevier]

In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem for p-system with relaxationvt−ux=0,ut+p(v)x=1/ϵ (f(v)−u), (E) with initial data(v, u)(x, 0)=(v 0(x), u 0(x))→(v±, u±), v±>0, as x→± ∞. (I) We are interested to show the solutions of (E), (I) tend also to the equilibrium rarefaction waves and the traveling waves even if the limits (v±, u±) of the initial data at x=±∞ do not satisfy the equilibrium equation; i.e., u±≠f(v±). When the limits of the initial data at infinity satisfy equilibrium states, Liu [9] studied the stability of rarefaction waves and traveling waves for the general 2×2 hyperbolic conservation laws with relaxation. [Copyright &y& Elsevier]