Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay

This paper is concerned with the existence and regularity of solutions for a class of neutral partial functional integrodifferential equations with infinite delay in Banach spaces. We use the theory of resolvent operator developed in R. Grimmer (1982) [29] to show the existence of mild solutions. We give sufficient conditions ensuring the existence of strict solutions. The phase space is axiomatically defined. Our results are applied to prove the existence and regularity of solutions to a Lotka–Volterra model with diffusion.     

Existence and regularity of solutions for neutral partial functional integrodifferential equations

In this paper, we study a class of neutral partial functional integrodifferential equations with finite delay by using the theory of resolvent operators. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. As an application, we also consider a diffusive neutral partial functional integrodifferential equation.     

Maximal regularity of second-order equations with delay

We characterize existence and uniqueness of solutions of an inhomogeneous abstract delay equation in Hölder spaces. The method is based on the theory of operator-valued Fourier multipliers.     

Extra regularity for parabolic equations with drift terms

We consider a parabolic equation with a drift term u+bu–u_t=0. Under some natural conditions on the vector valued function b, we prove that solutions possess extra regularity and better qualitative behavior than those provided by standard theory. For example, we show that the fundamental solution has global Gaussian upper bound even for some b with a large singularity in the form of c/|x|. We also show that bounded solutions are Hölder continuous when |b|² is just form bounded and divergence free, a case where not even continuity is expected. A Liouville type theorem is also proven.     

Hypoelliptic regularity in kinetic equations

We establish new regularity estimates, in terms of Sobolev spaces, of the solution f to a kinetic equation. The right-hand side can contain partial derivatives in time, space and velocity, as in classical averaging, and f is assumed to have a certain amount of regularity in velocity. The result is that f is also regular in time and space, and this is related to a commutator identity introduced by Hörmander for hypoelliptic operators. In contrast with averaging, the number of derivatives does not depend on the L^p space considered. Three type of proofs are provided: one relies on the Fourier transform, another one uses Hörmander's commutators, and the last uses a characteristics commutator. Regularity of averages in velocity are deduced. We apply our method to the linear Fokker–Planck operator and recover the known optimal regularity, by direct estimates using Hörmander's commutator.     

Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces

In this work, we study the existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces. We suppose that the undelayed part admits a resolvent operator in the sense given by Grimmer in [R. Grimmer, Resolvent operators for integral equations in a Banach space, Transaction of American Mathematical Society 273 (1982) 333–349]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of the strict solutions.     

Lyapunov regularity of impulsive differential equations

For linear impulsive differential equations, we give a simple criterion for the existence of a nonuniform exponential dichotomy, which includes uniform exponential dichotomies as a very special case. For this we introduce the notion of Lyapunov regularity for a linear impulsive differential equation, in terms of the so-called regularity coefficient. The theory is then used to show that if the Lyapunov exponents are nonzero, then there is a nonuniform exponential behavior, which can be expressed in terms of the Lyapunov exponents of the differential equation and of the regularity coefficient. We also consider the particular case of nonuniform exponential contractions when there are only negative Lyapunov exponents. Having this relation in mind, it is also of interest to provide alternative characterizations of Lyapunov regularity, and particularly to obtain sharp lower and upper bound for the regularity coefficient. In particular, we obtain bounds expressed in terms of the matrices defining the impulsive linear system, and we obtain characterizations in terms of the exponential growth rate of volumes. In addition we establish the persistence of the stability of a linear impulsive differential equation under sufficiently small nonlinear perturbations.     

Local existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces

In this work, we study the existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces. We suppose that the undelayed part admits a resolvent operator in the sense of Grimmer [R. Grimmer, Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society 273 (1982) 333–349]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of strict solutions.     

Ill-posedness for semilinear wave equations with very low regularity

In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where the nonlinear term depends on u and ∂ _t u. We prove a ill-posedness result for the “defocusing” case, and give an alternative proof for the supercritical “focusing” case, which improves the result in Fang and Wang (Chin. Ann. Math. Ser. B 26(3), 361–378, 2005). Supported by NSF of China 10571158.     

Global regularity for d-dimensional micropolar equations with fractional dissipation

This paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.     

Decay and regularity for dispersive equations with constant coefficients

This article discusses some smoothing estimates of the initial value problem for dispersive equations with constant coefficients. In particular, it is shown that a certain condition for the principal part of the symbol (see the assumption (1.3) below, which is equivalent to the one “of principal type” in the paper by Ben-Artzi and Devinatz [2]) is necessary and sufficient for the maximal smoothing in space-time. Dedicated to Professor Norio Shimakura The author was supported in part by Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 13640187).     

Necessary and sufficient conditions for the regularity and stability for some partial functional differential equations with infinite delay

In this work we give necessary and sufficient conditions for the regularity and stability of solutions for some partial functional differential equations with infinite delay. We establish also a new characterization of the infinitesimal generator of the solution semigroup.