Let $\sigma$ denote the Hilbert transform and $\mu\geq0$. We prove that if $u\in C\left( \left[ 0,T\right] ,H^{2}\left( \mathbb{R}\right) \cap L_{2}^{2}\left( \mathbb{R}\right) \right) $ is a solution of \[ \partial_{t}u+2\sigma\partial_{x}^{2}u+u\partial_{x}u=\mu\partial_{x}^{2}u \] such that there are $t_{0} <t_{1}$ $ <t_{2\text{ }}$satisfying $u\left( t_{j}\right) \in H^{4}\left( \mathbb{R}\right) \cap L_{4}^{2}\left( \mathbb{R}\right) $, $j=0,1,2$, then $u\left( t\right) =0$ for all $t\in\left[ 0,T\right] $.

We consider steady flows of slightly compressible viscoelastic fluids for which the extra-stress tensor is given by a differential constitutive equation. We examine the effect, on the flows, of compressibility. In particular, we show the existence of a unique solution to the 3-D steady boundary value problem, in the case of a nonzero Newtonian viscosity (Jeffreys' type fluids).

We are concerned with the problem of determining the sharp regularity of the coefficients with respect to the time variable $t$ in order to have a well posed Cauchy problem in $H^\infty$ or in Gevrey classes for a $p$-evolution operator of Schrödinger type. We use and mix two different scales of regularity of global and local type: the modulus of Hölder continuity and/or the behavior with respect to $|t-t_0|^{-q},\ q\geq 1,$ of the first derivative as $t$ tends to a point $t_0$. Both are ways to weaken the Lipschitz regularity. We give also counterexamples to show that the conditions we find are sharp.

The hydrodynamic-Maxwell equations are studied, as a compressible model of charge transport induced by an electromagnetic field in semiconductors. A local smooth solution theory for the Cauchy problem is established by the author's modification of the classical semigroup resolvent approach of Kato. The author's theory has three noteworthy features: (1) stability under vanishing heat flux, which is not derivable from other theories; (2) accommodation to arbitrarily specified terminal time for the regularized problem; and, (3) constructive in nature, in that it is based upon time semidiscretization, and the solution of these semidiscrete problems determines the localization theory criteria. The regularization is employed to avoid vacuum states, and eliminated for the final results which may contract the admissible time interval. We also provide a symmetrized formulation in matrix form which is useful for applications and simulation. The theory uses the generalized energy estimates of Friedrichs on the ground function space, and leverages them to the smooth space via Kato's commutator estimate.

The exact controllability of two nonlinear population dynamics problems is established. The first problem is age structured, and the control corresponds to a supply of individuals on a small age interval. The second one is age and space structured and the control corresponds to a supply of individuals on a small subdomain $\omega$ of the whole space domain $\Omega$. The method is based on a uniform observability result for the backward nonlocal adjoint system and on Kakutani's fixed-point theorem.

This paper is devoted to recovering a scalar time-dependent function in a source term in an integro-differential parabolic equation with delay. For such a problem existence and uniqueness results as well as continuous dependence upon the data are proved.