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The existence and smoothness of centre manifolds and a reduction principle are proven for impulsive delay differential equations. Several intermediate results of theoretical interest are developed, including a variation of constants formula for linear equations in the phase space of right-continuous regulated functions, linear variational equation and smoothness of the nonautonomous process, and a Floquet theorem for periodic systems. Three examples are provided to illustrate the results.

I present new techniques and results concerning the stability of travelling waves to semilinear parabolic equations. I am concerned with those problems in which the essential spectrum of the linearization about the wave contains λ = 0. In these circumstances, the stability results of D. Henry ( in "Lecture Notes in Mathematics," Vol. 840, Springer-Verlag, New York, 1981) and P. Bates and C. Jones ( Dynamics Rep. 2 (1989), 1-38) are not applicable, as the spectral information does not yield an exponential time decay estimate on the semigroup. I show that under certain (relatively) easily computable conditions the wave is stable in polynomially weighted L ∞ spaces. Furthermore, the rate at which the perturbation decays to the wave as t → ∞ depends on the growth rate of the polynomial near infinity. The method of proof uses the semigroup of the linearized equation, with solutions of the full problem expressed by the variation of constants formula. Estimates for the semigroup are derived through a new technique for estimating the resolvent.

The study of (1.2) is rather classic and a great number of research papers and monographs exist (see [5, 9, 10, 15, 161 and their references). In these works, various types of existence, uniqueness, differentiability and continuous dependence theorems are established on the basis of the construction and the representation of the semi-group T(t) relating to (1.2). Our purpose here is to give two representations of the fundamental solution of (1.1) in terms of Z(t) and A, and establish a variation of constants formula for (1.1). Such expressions are useful to obtain the fundamental theorems for (1.1) and some system theoretical results [ 11, 121. We briefly explain the content of this paper. In Section 2, the notations and notions used throughout the paper and the system description are given. A definition of the mild solution is also given in Section 2 and the treatment is based on the perturbation theory (i.e., Eq. (1.1) is considered as a pertur349 0022-039618 l/090349-20$02.00/O

Two timing, an ad hoc method for studying periodic evolution equations, can be given a rigorous justification when the problem is in standard form, u = ϵf ( t , u ). First solve dw dσ = ϵ(I − M) f(σ, w) for w(σ, v) , where M is the mean value operator and v is any initial value. Then w ( σ , v ) is periodic in σ but does not satisfy the original equation. Now, force a solution u ( t ), using nonlinear variation of constants, in the form w ( σ , v ( τ )), where σ = t is the fast time and τ = ϵt is the slow time. With the resulting differential equation for v , one reads off from its nonconstant solutions the approximate transient behavior of u ( t ) for times of order ϵ −1 . On the other hand, the equilibrium points (constant solutions) v 0 correspond to steady state (periodic solutions) of the original system. Interesting applications, such as to one-dimensional wave equations with cubic damping, can be given.

The classical Duhamel principle, established nearly 200 years ago by Jean-Marie-Constant Duhamel, reduces the Cauchy problem for an inhomogeneous partial differential equation to the Cauchy problem for the corresponding homogeneous equation. In this paper we generalize this famous principle to a wide class of fractional order differential-operator equations.

By means of a nonlinear variation of constants formula it is shown that, under suitable assumptions, there exists a global invariant manifold for semilinear hyperbolic evolution equations with a retarded perturbation, provided that the time-delay is small or the magnitude and the Lipschitz constant of the perturbation are small. By construction, this invariant manifold is weaved by trajectories of an associated nonretarded evolution equation; it is also locally exponentially attracting. As applications of the theory, we treat retarded perturbutions of the Sine-Gordon equation and the dissipative Klein-Gordon equation.

We treat a class of equations given by e 2 u ″ ( x ) = u ( x ) ( q ( x , e ) − u ( x ) ) , u ( − 1 ) = α − , u ( 1 ) = α + , and obtain rigorous uniform asymptotic expansions of the solutions as e → 0 . A key tool is a new formula of variation of constants that works for such quadratic equations. Included are solutions with one or more spikes. One example of this class of problems is a famous problem studied by Carrier and discussed formally by Bender and Orszag in the book “Advanced Mathematical Methods for Scientists and Engineers”. In this paper we give the first rigorous derivation for the well-known asymptotics for that problem. Another more applied example is also covered by our theory.

We will solve the following boundary value problem in linear viscoelasticity: given the value of the stress on (a part of) the boundary of the domain find the stress in the whole body at all positive times. Using a state space setting we show that the stress field inside the body can be obtained from the boundary stress by a variation-of-parameters formula involving an analytic semigroup. The relation between the regularities of the boundary stress and the stress inside the body is therefore characterized by the well-known and rich regularity theory for analytic semigroups.