This paper offers the maximal estimates of the solutions of some initial value problems on modular spaces. Our results include the estimates for the solutions of Schrödinger equation.

This paper is concerned with the blowup phenomena for the initial value problem of the wave equation with a critical space-dependent damping term V₀|x|^-1 and a p-th order power nonlinearity on the Euclidean space R^N, where N ≥ 3 and V₀ ∈ [0,(N-1)²/(N+1)). The main result of the present paper is to prove existence of a unique local solution of the problem and to provide a sharp estimate for lifespan for such a solution for small data with a compact support when N/(N-1) < p ≤ p_S(N + V₀), where p_S(N) is the Strauss exponent for the semilinear wave equation without damping. The main idea of the proof is due to the technique of test functions for the classical wave equation originated by Zhou-Han (2014). Consequently, the result poses whether the value V₀ = (N-1)²/(N + 1) is a threshold for diffusive structure of the singular damping |x|^-1 or not.

Time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system is studied. Concerning the linearized problem, the decay estimate with diffusion wave property for an initial data is derived. As an application, the time decay estimate of solutions to the nonlinear problem is given. In contrast to the compressible Navier-Stokes system, for linear system regularities of the initial data are lower and independent of the order of derivative of solutions owing to smoothing effect from the Korteweg tensor. Furthermore, for the nonlinear system diffusion wave property is obtained with an initial data having lower regularity than that of study of the compressible Navier-Stokes system.

The essential support of the symbol of a semiclassical pseudodifferential operator is characterized by semiclassical wavefront sets of distributions. The proof employs a coherent state whose center in phase space is dependent on Planck's constant.

For spherically symmetric repulsive Hamiltonians we prove the limiting absorption principle bound, the radiation condition bounds and the limiting absorption principle. The Sommerfeld uniqueness result also follows as a corollary of these. In particular, the Hamiltonians considered in this paper cover the case of inverted harmonic oscillator. In the proofs of our theorems, we mainly use a commutator argument invented recently by Ito and Skibsted. This argument is simple and elementary, and dose not employ energy cut-offs or the microlocal analysis.

The multiple elliptic gamma function includes q-analogues of the multiple sine function, introduced by Kurokawa. Thus it is natural to consider whether properties of the multiple sine function can be extended to that of the multiple elliptic gamma function. Kurokawa and Wakayama showed that, when the period is "rational", the multiple sine function satisfies an algebraic differential equation. In this paper, we investigate the differential algebraicity of the multiple elliptic gamma function. We show that the multiple elliptic gamma function G_r(z, τ) satisfies an algebraic differential equation when the period τ is "rational".

In this paper, we consider the initial value problem in some Orlicz spaces for the heat equation with a nonlinearity having an exponential growth at infinity and vanishing at zero. Under some smallness conditions on the initial data and appropriate behavior near the origin for the nonlinearity, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time. We show in particular that the decay depends on the behavior of the nonlinearity near the origin.