Dissipative Schrodinger operators are studied in L2(0, ∞) which are extensions of symmetric operators with defect index (2, 2). We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix according to the scheme of Lax and Phillips. With the help of the incoming spectral representation, we construct a functional model of the dissipative operator and construct its characteristic function in terms of solutions of the corresponding differential equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operator.

In previous papers we have studied oscillation properties of Sturm–Liouville problems (−Py′)′ + qy = λry, with λ-dependent boundary conditions, under various ‘definiteness’ conditions. Here we present a new, unified, approach which also covers cases previously untreated, e.g. of semidefinite weight, and also the fully indefinite problem.

In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem

where the data f and u0 belong to L1(Ω × (0, T)) and L1 (Ω), and where the function a:(0, T) × Ω × ℝN → ℝN is monotone (but not necessarily strictly monotone) and defines a bounded coercive continuous operator from the space into its dual space. The renormalised solution is an element of C0 ([ 0, T] L1 (Ω)) such that its truncates TK(u) belong to with

this solution satisfies the equation formally obtained by using in the equation the test function S(u)φ, where φ belongs to and where S belongs to C∞(ℝ) with

A fundamental prerequisite for the numerical computation of optimal controls is to show that sequences of suboptimal (that is, close-to-optimal) controls converge. We show this in a version that applies to hyperbolic and parabolic distributed parameter systems, the latter including the Navier–Stokes equations. The optimal problems include control and state constraints; in the parabolic case, the constraints may be pointwise on the solution and the gradient.

In this paper, the existence and uniqueness of the global smooth solution are proved for an evolutionary Ginzburg–Landau model for superconductivity under the Coulomb and Lorentz gauge.

In this paper, we prove the global existence and uniqueness of solutions to the Cauchy problem of a hyperbolic system, which probably contains so-called δ-waves.

We study the bifurcation of limit cycles in general quadratic perturbations of the particular quadratic system which represents one of two codimension-five components in the intersection of two strata in the centre manifold, and Q4. The study of limit cycles for this degenerate case requires us to investigate not the first but the second variation M2 of the displacement function. We prove that up to three limit cycles can emerge from the period annulus surrounding the centre. This implies that the cyclicity of period annuli of nearby systems in and Q4 is at most three as well. Our approach relies upon the possibility of deriving appropriate Picard–Fuchs equations satisfied by the four independent integrals included in M2.

We investigate critical points of the free energy Eε(u) of the Cahn–Hilliard model over the unit square under the constraint of a mean value ü. We show that for any fixed value ü in the so-called spinodal region and to any mode of an infinite class, there are critical points of Eε(u) having the characteristic symmetries of that mode provided ε > 0 is small enough. As ε tends to zero, these critical points have singular limits forming characteristic patterns for each mode. Furthermore, any singular limit is a stable critical point of E0(u)). Our method consists of a global bifurcation analysis of critical points of the energy Eε(u) where the bifurcation parameter is the mean value ü.

In this paper, we study the Orr–Sommerfeld problem on a finite interval. It is shown that the eigenfunctions and associated functions form a Bari basis in a suitable Hilbert space if the unperturbed velocity profile u is sufficiently smooth. To this end, the Orr–Sommerfeld problem is considered as a bounded perturbation of a certain self-adjoint spectral problem.

Let λε be a Dirichlet eigenvalue of the ‘periodically, rapidly oscillating’ elliptic operator –∇·(a(x/ε)∇) and let ∇ be a corresponding (simple) eigenvalue of the homogenised operator –∇·(A∇). We characterise the possible limit points of the ratio (λε–λ)/ε as ε→0. Our characterisation is quite explicit when the underlying domain is a (planar) convex, classical polygon with sides of rational or infinite slopes. In particular, in this case it implies that there is often a continuum of such limit points.

The set of points (α, β) ∈ℝ2 for which the problem − δu = αu+ − βu− in Ω, u = 0 on ∂Ω(u+ = max {u, o} and u− = min {−u, 0}) has nontrivial solutions is important for the study of certain nonlinear problems. It is shown that for ‘most’ bounded domains Ω in ℝn, such a set is locally the union of a finite number of curves.

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the C0-norm of the initial data is sufficiently small, while the C1-norm of the initial data can be large. The analysis is based on a priori estimates, which are obtained by a generalised Lax transformation.