The hypothesis α ∈ Lp (−r, 0) made in the paper by G. Di Blasio and A. Lorenzi  on identification problems for integro-differential delay equations in Banach spaces is relaxed to a weaker one α ∈ L1 (−r, 0). The method makes analogous relaxation possible for equations in Hilbert spaces investigated by G. Di Blasio, K. Kunisch and E. Sinestrari .
We consider an infinite system of quasilinear first-order partial differential equations, generalized to contain spacial integration, which describes an incompressible fluid mixture of infinite components in a line segment whose motion is driven by unbounded and space-time dependent evaporation rates. We prove unique existence of the solution to the initial-boundary value problem, with conservation-of-fluid condition at the boundary. The proof uses a map on the space of collection of characteristics, and a representation based on a non-Markovian point process with last-arrival-time dependent intensity.
We consider evolution differential equations in Fréchet spaces that possess unconditional Schauder basis and construct a version of the majorant functions method to obtain existence theorems for Cauchy problems. Applications to PDE and ODE have been considered.
We consider the mixed problem for weakly damped modified Boussinesq-Beam equations on the one dimensional half line (0, + ∞). We shall derive fast decay results of the total energy and L2-norm of solutions based on the idea due to , which is an essential modification of that developed by Morawetz . In order to apply that idea due to  to the one dimensional exterior mixed problem, one also constructs an important Hardy-Sobolev type inequality, which holds only in the 1-D half line case.
In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation. In particular, we prove norm inflation based at every initial condition in negative Sobolev spaces below or at the scaling critical regularity.