In this paper we analyze the monotonicity of the time to peak response $T_{\rm max}$ with respect to the drug dose $D$ for the four different turnover models I - IV, as introduced by Dayneka et al. [2]. We do this for the situation when the drug is supplied through an initial bolus, and eliminated according to a single exponential function and stimulation or inhibition takes place through a Hill function. We show that in Models I and III, in which the drug impacts the production term, the function $T_{\rm max}(D)$ is increasing for all values of the system- and the drug parameters. For Model II (inhibition of the loss term) the situation is more delicate. Here we prove monotonicity of $T_{\rm max}(D)$ for a substantial range of values of the rate- and drug constants, but leave the question of monotonicity open for some values. Finally, in Model IV (stimulation of the loss term) the function $T_{\rm max}(D)$ is known not to be monotone for some values of the rate constants and $I_{\rm max}$ [12].

We prove some uniqueness results for the Cauchy problem for the 3-D time-dependent Ginzburg-Landau (TDGL) model for superconductivity with the choice of the Lorentz gauge in the multiplier spaces (Morrey spaces) and in the inhomogeneous Besov spaces, respectively.

We study large-time behavior of solutions to the initial-value problem for a parabolic system of chemotaxis in one space dimension. Chemotaxis is the directed movement of amoebae in response to chemical gradients. The aim of this paper is to obtain the first and second asymptotic profiles of solutions to the parabolic system. Then we are able to find out the optimal asymptotic rate of the first asymptotic profiles.

We consider a coupled system of Kuramoto-Sivashinsky equations depending on a suitable parameter $\nu > 0$ and study its asymptotic behavior for $t$ large, as $\nu\rightarrow 0$. Introducing appropriate boundary conditions we show that the energy of the solutions decays exponentially uniformly with respect to the parameter $\nu$. In the limit, as $\nu\rightarrow 0$, we obtain a coupled system of Korteweg-de Vries equations known to describe strong interactions of two long internal gravity waves in a stratified fluid for which the energy tends to zero exponentially as well. The decay fails when the length of the space interval $L$ lies in a set of critical lengths.

We consider the initial-boundary-value problem for the Ostrovskiy type equation on a half-line. We study traditionally important problems of a theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary-value problem and the asymptotic behavior of solutions for large time in the critical case, when the time decay of the nonlinearity has the same rate as that of the linear part of the equation.

We prove global well posedness and scattering for the nonlinear Schröodinger equation with power-type nonlinearity $$ \begin{cases} i u_t +\Delta u = |u|^p u, \quad \frac{4}{n} < p < \frac{4}{n-2},\\ u(0,x) = u_0(x)\in H^s({\mathbb R}^n), \quad n\geq 3, \end{cases} $$ below the energy space, i.e., for $s<1$. In [15], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao established polynomial growth of the $H^s_x$-norm of the solution, and hence global well posedness for initial data in $H^s_x$, provided $s$ is sufficiently close to $1$. However, their bounds are insufficient to yield scattering. In this paper, we use the a priori interaction Morawetz inequality to show that scattering holds in $H^s({\mathbb R}^n)$ whenever $s$ is larger than some value $0<s_0(n,p)<1$.

The paper deals with the well posedness of linear hyperbolic second-order systems with homogeneous boundary conditions, in the half-space $\Omega = {\bf R}^{d-1} \times (0,\infty)$. At first we consider operators with constant coefficients: by performing a Fourier-Laplace transform and by studying the space of the solutions, we prove a sufficient condition for the well posedness of the boundary-value problem, by means of the Hille-Yosida theorem. Subsequently, we study the case of evolution boundary-value problems for second-order systems with coefficients that depend on the space variable. Proving that the linear second-order operator associated with the system, under suitable assumptions, turns out to be maximal and monotone, we establish the well posedness of the problem by applying the Hille-Yosida theorem.

We prove that if the space $Y$ of Henstock-Lebesgue integrable functions from a compact real interval to a Banach space $E$ is normed by the Alexiewicz norm and ordered by a regular cone $E_+$, then the $E_+$-valued functions of $Y$ form a regular order cone of $Y$. This property is shown to hold also when $Y$ is the space of Henstock-Kurzweil integrable functions if $E$ is weakly sequentially complete and $E_+$ is normal. As an application of the obtained results we prove existence and comparison results for least and greatest solutions of nonlocal implicit initial-value problems of discontinuous functional differential equations containing non-absolutely integrable functions.

In this paper, we study existence and exponential stability problems of mild solutions associated with stochastic delay evolution equations in Hilbert spaces. These problems are resolved using the properties of a stochastic convolution integral and a comparison principle by exploiting Holder-type conditions on the drift and diffusion terms. The results obtained here generalize the corresponding main results from [1, 3, 4, 9, 13, 16].

In this article we study the asymptotic behavior of the solution to the compressible Reynolds equation which describes the flow of fluid (air) on the hard disk as the distance between the read-write head and disk tends to zero. We apply these results to show the existence of equilibruim solutions for this system.