We present the sufficient conditions for the L²-well-posedness of the initial value problem for third and fourth order dispersive equations in one space dimension. In third order case and particular fourth order case, these conditions are also necessary. Tarama's pseudodifferential operators play crucial role in our proof.

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We show the global existence theorem for the two-dimensional Boussinesq equations in the entire plane with non-decaying initial data. To this end, we give an estimate of the vorticity in the uniformly local Lebesgue space.

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In discussing time slicing approximations of Feynman's path integrals, one can use stationary phase method for some oscillatory integrals over a space of large dimension. Previously estimates of remainder term of such stationary phase method were given in [2] and [6]. In the present note an improved remainder estimate is given for such stationary phase method. The estimate is sharp enough to give us the second term of semiclassical asymptotic expansion if it is applied to Feynman path integrals of some kind discussed in [6] and [4]. See our forthcoming paper [3].

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In this paper, we consider periodic solutions of the functional-differential equation x″ + x(t−kx) = 0. The structure of the set A_k (k ∈ (0, ∞)) of all its nontrivial periodic solutions x satisfying x′ < 1/k on R is described. It is proved that for each k ∈ (0, ∞) and T_* ∈ (2π, ∞), there exists x ∈ A_k having the period T_* and for each k ∈ (0, ∞) and a ∈ (0,1/k), there exists a unique x ∈ A_k such that x(0) = 0 and x′(0) = a.

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We consider a transmission problem for a model describing the evolution of sound in a compressible fluid. Assuming that dissipative mechanisms of memory type are effective in part of the boundary, the coefficients are piecewise constants and suitable geometric conditions on the domain and the interfaces, we prove that the total energy associated with the model decays exponentially fast as t → + ∞.

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We find and study coupled Painlevé V systems in dimension 4, which are different from the systems of type A₅⁽¹⁾. We also give the augmented phase spaces for these systems.

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