In this paper, we perform an asymptotic study on slightly viscous flows between two immiscible incompressible fluids. The motion is governed by linearized Navier-Stokes equations together with interfacial conditions. A second-order asymptotic expansion with respect to viscosity is obtained by using the method of multiple scales. In particular, viscous decay rate for the interfacial wave amplitude and viscous correction for the phase speed are explicitly identified.

We consider a class of non-local boundary value problems of the type used to model a variety of physical and biological processes, from Ohmic heating to population dynamics. Of particular relevance therefore is the existence of positive solutions. We are interested in the existence of such solutions that arise as a direct consequence of the non-local interactions in the problem. Conditions are therefore imposed that preclude the existence of a positive solution for the related local problem. Under these conditions, we prove that there exists a unique positive solution to the boundary value problem for all sufficiently strong non-local interactions and no positive solutions exists otherwise.

Polynomials satisfying a certain twin asymptotic periodic recurrence relation are considered. It is assumed that the coefficients of the recurrence formula are unbounded but vary regularly and have different behaviour for even and odd indices. The asymptotic behaviour of the ratio of contiguous polynomials is analyzed.

We study the large-time behavior of the solution to an initial boundary value problem on the half line for scalar conservation law, where the data on the boundary and also at the far field are prescribed. In the case where the flux is convex and the corresponding Riemann problem for the hyperbolic part admits the transonic rarefaction wave (which means its characteristic speed changes the sign), it is known by the work of Liu-Matsumura-Nishihara (’98) that the solution tends toward a linear superposition of the stationary solution and the rarefaction wave of the hyperbolic part. In this paper, it is proved that even for a quite wide class of flux functions which are not necessarily convex, such the superposition of the stationary solution and the rarefaction wave is asymptotically stable, provided the rarefaction wave is weak. The proof is given by a technical $L^2$-weighted energy method.

This paper study the stability of moving invariant manifolds of nonlinear impulsive integro-differential equations. The obtain results are based on the method of piecewise continuous Lyapunov’s functions and the comparison principle.