It is proved that mild solutions of Cauchy problems associated to weakly continuous semigroups $P(t)$ of infinitesimal generator $\mathcal{A}$ are the limit, uniformly on compact sets of $[0,T]\times H$, of classical solutions $u_{n}$ of approximating Cauchy problems associated to an operator $\mathcal{A}_{0}$ which is the restriction of $\mathcal{A}$ to a suitable subspace $D(\mathcal{A}_{0})$ (easier to describe in the applications). An application is given to transitions semigroups associated to Kolmogorov equations.

We prove that the solution of the Vlasov-Fokker-Planck equation converges to the unique stationary solution with same mass as time tends to infinity. The same result holds in the repulsive coulombic case for the Vlasov-Poisson-Fokker-Planck system; the newtonian attractive case is also studied. We establish positive and negative answers to the question of existence of a stationary solution for the last problem by examining the Poisson-Boltzmann equation.

In this work we study the spectrum of some linear weakly coupled noncooperative elliptic systems with two species subject to homogeneous Dirichlet boundary conditions. When the domain supporting the species is a ball or an annulus we prove that zero is never an eigenvalue. As a consequence of this result we also show that the classical Lotka-Volterra predator-prey model with diffusion and radial function coefficients has a unique component-wise radial coexistence state with no zero eigenvalue of the linearization on the space $W^{2,p}(D)$, $p > {N\over 2}$. Therefore, a radially symmetric coexistence state only may lose stability by a Hopf bifurcation when we vary parameters.

This paper deals with classical solutions of quasilinear equations of the form $$ [\rho(\mathbf{x},t,u,\nabla u)u,_i],_i = u,_t $$ in a semi--infinite cylinder in $\Bbb R^3$ with homogeneous initial data and with homogeneous Dirichlet data prescribed on the lateral surface for all time. Under appropriate assumptions on the form of $\rho$, Phragmen--Lindelöf type growth--decay estimates are derived.

We consider the existence of generalized solutions of a nonlinear functional integro-differential equation of type $$ x'(t)+A(t, x_t)x(t)\ni G(t,x_t,\int^t_0k(t,s,x_s)ds),\quad t\in [0,T),\quad x_0=\phi $$ in general Banach spaces. Our study is performed by means of the concept of the Method of Lines and well known Banach fixed point theorems. We extend the results of Kartsatos and Parrott, Tanaka to an abstract nonlinear integro-differential equation.

We consider an infinite dimensional linear controlled Ito system with almost periodic coefficients and two--sided average cost. Under stabilizability and detectability assumptions we prove that the unique bounded solution of the associated Riccati equation is almost periodic.

We consider damped second order in time systems such as those arising in structures with piezoceramic actuators and sensors. These systems are naturally formulated as abstract second order systems with unbounded nonhomogeneous term. Existence, uniqueness and continuous dependence of solutions in a weak or variational setting are given. A semigroup formulation is presented and conditions under which the variational solutions and semigroup solutions are the same are discussed.

By constructing so called stable set and unstable set, the global existence and blowing-up property of solutions are discussed to some nonlinear wave equations with dissipative terms.

This paper considers a reaction-diffusion system that models the situation in which two predators feed on one prey species. A sufficient condition for the existence of a positive steady-state solution and a positive global attractor is given in terms of the natural growth rates and the relative reaction rates of the three species. We prove that when the natural growth rates are in a certain unbounded parameter set $\Lambda$, the corresponding elliptic system has a pair of positive upper-lower solutions which ensures the existence of a positive solution. Two converging monotone sequences are generated from the upper and lower solutions by an iteration scheme and their limits enclose a positive attractor for the reaction-diffusion system. The present paper also gives some explicit information on the limiting behavior of the time-dependent solution in relation to the semitrivial and trivial steady states. Finally, numerical evidence of the long-term coexistence is demonstrated by solving the corresponding finite-difference system.

In this paper we consider the problem of constructing a well-posed state space model for a class of singular integro-differential equations of neutral type. The work is motivated by the need to develop a framework for the analysis of numerical methods for designing control laws for aeroelastic systems. Semigroup theory is used to establish existence and well-posedness results for initial data in weighted $L^{2}-$spaces. It is shown that these spaces lead naturally to the dissipativeness of the basic dynamic operator. The dissipativeness of the solution generator combined with the Hilbert space structure of these weighted spaces make this choice of a state space more suitable for use in the design of computational methods for control than previously used product spaces.

We are concerned with classical solutions to the initial value problem for $\square u=|u|^p$ or $\square u=|u_t|^p$ in $\Bbb R^n\times[0,\infty)$ with \lq\lq small" data. If the data have compact support, it is partially known that there is a critical number $p_0(n)$ such that most solutions blow-up in finite time for $1<p\le p_0(n)$ and a global solution exists for $p>p_0(n)$. In this paper, we shall show for all $n\ge2$ that, if the support of data is noncompact, there are blowing-up solutions even for $p>p_0(n)$ because of the \lq\lq bad" spatial decay of the initial data. Moreover, critical decays for each equation are conjectured. The proof lies in the pointwise estimates of the fundamental solution of $\square$.

We derive a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with a nonlinear dissipation $ \rho (u_{t}) $, where $ \rho (v) $ is a function like $ v/\sqrt{ 1+v^{2}}$. Since our dissipation is weak as $\vert u_{t}\vert $ tends to $ \infty $ we treat strong solutions rather than usual energy finite solutions.

We establish sharp $L^{\infty}$ bounds for functions defined on arbitrary open sets in $\Bbb R^2$ and $\Bbb R^3$, which vanish on the boundary and have $L^2$ Laplacians. All functions corresponding to the best possible constants are explicitly given. The proof is based on integral representations using the Green's function for the Helmholtz equation in arbitrary domains.

The generalized eigenfunctions for a class of highly singular Schrödinger operators that includes the important delta-shell potential of nuclear physics is shown to have a meromorphic continuation into the upper half-plane.