In this note we concern with the wave maps from the Lorentzian manifold with the periodic in time metric into the Riemannian manifold, which belongs to the one-parameter family of Riemannian manifolds. That family contains as a special case the Poincaré upper half-plane model. Our interest to such maps is motivated with some particular type of the Robertson-Walker spacetime arising in the cosmology. We show that small periodic in time perturbation of the Minkowski metric generates parametric resonance phenomenon. We prove that, the global in time solvability in the neighborhood of constant solutions is not a stable property of the wave maps.

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In this paper we deal with the Cauchy problem for second-order hyperbolic operators with the coefficients of their principal parts depending only on the time variable. We shall give outer estimates of the wave front sets of solutions to the Cauchy problem when the Cauchy problem is C^∞ well-posed. More precisely, we shall show that the wave front sets of solutions are included in the union of broken null bicharacteristics emanating from the singularities of data in non-trivial cases.

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We discuss a free boundary problem for a diffusion equation in a one-dimensional interval which models the spreading of invasive or new species. Moreover, the free boundary represents a spreading front of the species and its dynamical behavior is determined by a Stefan-like condition. This problem has been proposed by Du and Lin (2010) and, recently, Kaneko and Yamada have studied a free boundary problem for a general reaction-diffusion equation under Dirichlet boundary conditions. The main purpose of this paper is to define "spreading" and "vanishing" of species for a free boundary problem with general nonlinearity and study the underlying principle to determine the spreading or vanishing behavior as time tends to infinity. It will be proved that vanishing occurs if and only if the free boundary stays in a bounded interval, and that, when vanishing occurs, the population decreases exponentially to zero in large time.

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It is shown that cyclic differential systems of the forms (A) x′i = −pi(t)xαii+1, i = 1, ..., n (xn + 1 = x1), and (B) x′i = −pi(t)x−αii+1, i = 1, ..., n (xn+1 = x1), where αi > 0, i = 1, ..., n, are constants and pi(t) > 0, i = 1, ..., n, are continuous functions on [0,∞) may possess singular solutions of extinct type, that is, those positive solutions (x1(t), ..., xn(t)) of (A) (resp. (B)) which are defined on some finite interval [t0,T), 0 ≤ t0 < T < ∞, and satisfy xi(t) > 0, t ∈ [t0, T), and limt→T−0 xi(t) = 0, i = 1, ..., n.

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We prove that a formal Laurent series satisfying a rational linear q-difference equation of first order does not satisfy any nontrivial algebraic differential equation over the rational function field unless it represents a rational function. This also provides an algebraic proof of Ishizaki's theorem on differential transcendency for a meromorphic function satisfying a linear q-difference equation.

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We study a quasilinear elliptic problem of Kirchhoff-type, where the differential operator is defined by partial derivatives with different powers, so the functional framework involves anisotropic Sobolev spaces. By using notion of monotone operators and variational methods, we prove some existence results.

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