We study the initial-boundary value problem for the coupled Klein-Gordon-Schrödinger equations in a domain in R^N with N ≤ 4. Under natural assumptions on the initial data, we prove the existence and uniqueness of global solutions in H² ⊕ H² ⊕ H¹. The method of the construction of global strong solutions depends on the proof that solutions of regularized systems by the Yosida approximation form a bounded sequence in H² ⊕ H² ⊕ H¹ and a convergent sequence in H¹ ⊕ H¹ ⊕ L². The method of proof is independent of the Brezis-Gallouet technique and a compactness argument.

In this paper, we consider the higher order parabolic Schrödinger type operator with nonnegative potentials. We assume that the nonnegative potential belongs to both the reverse Hölder class and the Gaussian class. Using the pointwise estimates of Hardy-Littlewood maximal operator, we establish L^p boundedness of Riesz transforms associated with the higher order parabolic Schrödinger type operator.

We study the unitarity of monodromies of rank two Fuchsian systems of SL type with (n + 1) regular singularities on the Riemann sphere, namely, we give a sufficient and necessary condition for the monodromy group to be conjugate to a subgroup of a special unitary group SU(p,q). When n ≥ 3, the moduli space of irreducible monodromies can be realized as an affine algebraic set in C^m for some m ∈ N. In this paper, we give a characterization and construction of unitary monodromies in terms of this affine algebraic set. The signatures of unitary monodromies are also classified.

We establish necessary and sufficient conditions for the existence of asymptotic periodic solutions of delay evolution equations on half line. These conditions are determined by the spectrum of forcing term and spectrum of evolution semigroups associated with the equations. Our goal is to extend on Massera's theorem by recalling a new concept of solutions, namely, asymptotic solutions for evolution equations in Banach spaces, which have potential applications to partial differential equations as well as abstract functional differential equations.

It is well known that a time delay can cause the instability of the equilibrium. In the case of pathogen dynamics models, the absorption effect (when pathogens invade target cells, they are absorbed into cells and disappear from blood) also can cause the instability. Besides the two effects above, introducing an immune response variable into a pathogen dynamics model can cause the instability. This paper shows, for the models lacking one or more of the three effects, the positive equilibrium is always stable. Moreover, it shows that, for the model incorporating all the three effects, the positive equilibrium can be unstable.