We investigate an ill-posed Cauchy problem for a linear functional-differential equation of retarded type on the negative half-line. Using a step-by-step procedure this problem is replaced by an inverse problem for an operator equation of the first kind in a functional space. The Tikhonov's method is then used for finding solutions. We also construct special boundary value problems for the functional-differential equations. Solutions of these boundary value problems determine regularized solutions of the ill-posed Cauchy problem by the step-by-step procedure.

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In 1990, Gérard-Tahara [4] introduced the Briot-Bouquet type partial differential equation t∂_tu = F(t,x,u,∂_xu), and they determined the structure of holomorphic and singular solutions provided that the characteristic exponent ρ(x) satisfies ρ(0) ∉ {1,2,...}. In this paper the author shows existences of holomorphic and singular solutions of the following type of difference-differential equations tD_qu = F(t,x,u,∂_xu).

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We consider the nonstationary Navier-Stokes system in a smooth bounded domain Ω ⊂ R³ with initial value u₀∈ L_σ²(Ω). It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin's condition. In this paper, we introduce a weighted Serrin condition that yields a necessary and sufficient initial value condition to guarantee the existence of local strong solutions u(·) contained in the weighted Serrin class ∫₀^T(τ^α||u(τ)||_q)^sdτ < ∞ with 2/s + 3/q = 1 − 2α, 0 < α < 1/2. Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class.

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The probability content of a convex polyhedron with a multivariate normal distribution can be regarded as a real analytic function. We give a system of linear partial differential equations with polynomial coefficients for the function and show that the system induces a holonomic module. The rank of the holonomic module is equal to the number of nonempty faces of the convex polyhedron, and we provide an explicit Pfaffian equation (an integrable connection) that is associated with the holonomic module. These are generalizations of results for the Schläfli function that were given by Aomoto.

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In this paper, we prove the maximal L_p-L_q regularity of the compressible and incompressible two phase flow with phase transition in the model problem case with the help of $\mathcal R$-bounded solution operators corresponding to generalized resolvent problem. The problem arises from the mathematical study of the motion of two-phase flows having gaseous phase and liquid phase separated by a sharp interface with phase transition.

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Consider any essentially self-adjoint Schrödinger operator S₀ = −Δ + q₁(x) + q₂(x), Dom(S₀) = C₀^∞(R^N) where q₁(x) ∈ L_loc² satisfies q₁(x) ≥ 0 and q₂(x) ∈ L_loc² is a (−Δ)-bounded real-valued multiplication operator with bound less than 1. Let now q₃(x) be a rapidly oscillating potential, e.g., q₃(x) = |x|³ sin |x|⁵ or (1 + |x|²)⁻¹e^|x| cos(e^|x|) with a singularity near |x| = ∞. In this paper, it is guaranteed that the perturbation T₀ = S₀ + q₃(x) is also essentially self-adjoint. Moreover, their Friedrichs extensions T and S have the same essential spectrum, i.e., σ_ess(T) = σ_ess(S). In fact, we study these problems more generally, i.e., for complex-valued potentials.

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