We consider a nonlinear diffusion equation describing the planar spreading of a viscous fluid injected between an elastic sheet and an underlying rigid plane. The dynamics depends sensitively on the physical conditions at the contact line where the sheet is lifted off the plane by the fluid. We explore two possibilities for these conditions (or “regularisations”): a pre-wetted film and a constant-pressure fluid lag (a gas-filled gap between the fluid edge and the contact line). For both flat and inclined planes, we compare numerical and asymptotic solutions, identifying the distinct stages of evolution and the corresponding characteristic rates of spreading.

In this paper, Chebyshev series and rigorous numerics are combined to compute solutions of the Euler-Lagrange equations for the one-dimensional Ginzburg-Landau model of superconductivity. The idea is to recast solutions as fixed points of a Newton-like operator defined on a Banach space of rapidly decaying Chebyshev coefficients. Analytic estimates, the radii polynomials and the contraction mapping theorem are combined to show existence of solutions near numerical approximations. Coexistence of as many as seven nontrivial solutions is proved.

This paper is concerned with the spatial dynamics of a monostable delayed age-structured population model in a 2D lattice strip. When there exists no positive equilibrium, we prove the global attractivity of the zero equilibrium. Otherwise, we give some sufficient conditions to guarantee the global attractivity of the unique positive equilibrium by establishing a series of comparison arguments. Furthermore, when those conditions do not hold, we show that the system is uniformly persistent. Finally, the spreading speed, including the upward convergence, is established for the model without the monotonicity of the growth function. The linear determinacy of the spreading speed and its coincidence with the minimal wave speed are also proved.

Recently Biscari, DiCarlo and Turzi [Anisotropic wave propagation in nematic liquid crystals, Soft Matter 10, 8296–8307.] proposed a theory for nematoacustics which models nematic liquid crystals as nematic elastomers with molecular relaxation. Here, we extend the analysis of this theory to account for the director motion possibly induced by the propagation of a sound wave. We find that the director vibration is related to the - usually small - anisotropy of the molecular distribution, thus providing a justification to the relative high ultrasonic intensities required to observe non-negligible acousto-optic responses.

We consider and analyse sampling theories in the reproducing kernel Hilbert space (RKHS) in this paper. The reconstruction of a function in an RKHS from a given set of sampling points and the reproducing kernel of the RKHS is discussed. Firstly, we analyse and give the optimal approximation of any function belonging to the RKHS in detail. Then, a necessary and sufficient condition to perfectly reconstruct the function in the corresponding RKHS of complex-valued functions is investigated. Based on the derived results, another proof of the sampling theorem in the linear canonical transform (LCT) domain is given. Finally, the optimal approximation of any band-limited function in the LCT domain from infinite sampling points is also analysed and discussed.