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Abstract: In this paper we study the period function of centers of planar polynomial differential systems. With a convenient compactification of the phase portrait, the boundary of the period annulus of the center has two connected components: the center itself and a polycycle. We are interested in the behaviour of the period function near the polycycle. The desingularization of its critical points gives rise to a new polycycle (monodromic as well) with hyperbolic saddles or saddle-nodes at the vertices. In this paper we compute the first terms in the asymptotic development of the time function around any orbitally linearizable saddle that may come from this desingularization process. In addition, we use these developments to study the bifurcation diagram of the period function of the dehomogenized Loud''s centers. More generally, the tools developed here can be used to study the return time function around a monodromic polycycle. This work is a continuation of the results in [P. Mardešić, D. Marín, J. Villadelprat, On the time function of the Dulac map for families of meromorphic vector fields, Nonlinearity 16 (2003) 855–881; P. Mardešić, D. Marín, J. Villadelprat, The period function of reversible quadratic centers, J. Differential Equations 224 (2006) 120–171]. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of positive solutions and sign-changing solutions for the following boundary value problem in the half-space where λ is a positive parameter and the nonlinear term f is superlinear at zero and asymptotically linear at infinity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the vanishing viscosity limit of initial boundary value problems for one-dimensional mixed hyperbolic–parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems: in particular, we assume that an eigenvalue of the inviscid system vanishes uniformly. We prove the stability of boundary layers expansions in small time (i.e before shocks for the inviscid system) as long as the amplitude of the boundary layers remains sufficiently small. In particular, by using Lagrangian coordinates, we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous Dirichlet condition for the velocity at the boundary. [Copyright &y& Elsevier]