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Abstract: In this paper, we study a free boundary problem for compressible spherically symmetric Navier–Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in -norm and weighted -norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics. [Copyright &y& Elsevier]

Abstract: We let Ω be a smooth bounded domain of and a sequence of functions such that in . We consider a sequence of functions such that in Ω for all . We address in this paper the question of the asymptotic behavior of the ''s when . The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the ''s are radially symmetrical, we push further the analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. [Copyright &y& Elsevier]

In this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations xn+1=A+∑i=1kaixn−pi/∑j=1mbjyn−qj,yn+1=B+∑i=1kciyn−pi/∑j=1mdjxn−qj, yn+1=B+∑i=1kciyn−pi/∑j=1mdjxn−qj, where k,m∈{1,2,…}, A,B,ai,ci,bj,dj, i∈{1,…,k}, j∈{1,…,m}, are positive constants, pi, qj, i∈{1,…,k}, j∈{1,…,m}, are positive integers such that p1, q1 and the initial values xi,yi, i∈{−π,−π+1,…,0}, π=max{pk,qm} are positive numbers. [Copyright &y& Elsevier]