The space of continuous states of perturbative interacting quantum field theories in globally hyperbolic curved spacetimes is determined. Following Brunetti and Fredenhagen, we first define an abstract algebra of observables which contains the Wick-polynomials of the free field as well as their time-ordered products, and hence, by the well-known rules of perturbative quantum field theory, also the observables (up to finite order) of interest for the interacting quantum field theory. We then determine the space of continuous states on this algebra. Our result is that this space consists precisely of those states whose truncated n-point functions of the free field are smooth for all n = 2, and whose two-point function has the singularity structure of a Hadamard fundamental form. A crucial role in our analysis is played by the positivity property of states. On the technical side, our proof involves functional analytic methods, in particular the methods of microlocal analysis.

This paper deals with a coupled system of reaction-diffusion equations modeling the competition of two species with cross diffusion. We discuss the existence of positive steady-state solutions in relation to the intrinsic birth rates a , a of sufficiently large. ᮊ

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i (u i ) may have the property D i (0) = 0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a twocomponent competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.

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