An inequality whose origins date to the work of G. H. Hardy is presented. This Hardy-type inequality applies to derivatives of arbitrary order of functions whose domain is a subset of ℝn. The Friedrichs inequality is a corollary. The result is then used to establish lower bounds on the essential spectra of even-order elliptic partial differential operators on unbounded domains.

We study the phenomenon of the infinite-time stabilization of classical global solutions of nonlinear reaction–diffusion equations to an unbounded (singular) stationary state and we present a new case of such an asymptotic singularity pattern formation. We concentrate on the most famous parabolic model, namely the semi-linear Frank-Kamenetskii equation from combustion theory,where B is a ball in RN. Our goal is to show that a new asymptotic problem arises precisely in dimension 10, not being available in other dimensions (which were studied earlier). For N = 10, we fix the ball B = {|x| < 4} and take bounded initial data u0 below the singular stationary solution Us(x) = ln(16/|x|2), which is unbounded at the origin x = 0.

We establish a sharp estimate on the rate of convergence u(x, t) → Us(x) as t → ∞ on compact subsets bounded away from x = 0 and also at the singularity point. We show that u(0, t) = α0t + O(ln t) → ∞, where the positive constant α0 is given by the first eigenvalue of the associated linear differential operator. We present formal asymptotic results showing that a detailed asymptotic analysis depends on a quite involved balance between various linear and nonlinear terms. Moreover, similar critical asymptotic behaviour is shown to exist in various related nonlinear second- and higher-order parabolic equation.

We consider the possible disentanglements of holomorphic map germs f: (ℂn, 0) → (ℂN, 0), 0 < n < N, with nonisolated locus of instability Inst (f). The aim is to achieve lower bounds for their (homological) connectivity in terms of dim Inst (f). Our methods apply in the case of corank 1.

We study the possibility of perturbing a matrix A by a diagonal matrix so that an eigenvalue problem with leading matrix A has specifiedeigenvalues when A is replaced by A+D. The particular cases presented are the one-parameter generalized eigenvalue problem (A× = λB μ×, a two-parameter eigenvalue problem (A + λB + μC)× = 0, a linked system ofsuch two-parameter problems and a quadratic eigenvalue problem (A + λB + λ2C)× = 0. The work extends results of Hadeler for the classical problem A× = λ×.

We compare solutions of a class of degenerate parabolic equations on a Riemannian manifold $M$ with solutions of the equation on a model manifold. The class of equations under consideration contains both the parabolic $p$-Laplace equation and the porous medium equation. We prove that, under curvature conditions, solutions on model manifolds induce sub- or supersolutions on $M$. Using this result, we obtain curvature-dependent estimates for the speed of propagation of solutions.

An elementary matrix has ones down the main diagonal and at most one element off the diagonal that differs from zero. We study the subgroups of SL2ℤ generated by sets of elementary matrices. Specifically, we give a stringent condition that the entries of a matrix belonging to such a group must satisfy.

We give a simple description of the wave operators appearing in the Lax-Phillips scattering theory. This is used to derive a relation between the scattering matrix and a kind of time delay operator and to characterize all scattering systems having the same scattering operator.

In this paper, the formally J-symmetric Sturm-Liouville operator with complex-valued coefficients is considered and a generalisation of the Weyl limit-point, limit-circle dichotomy is sought by means of m (λ )-functions. These functions are then used to give an explicit description of all the associated J-selfadjoint operators with separated boundary conditions in the limit-circle case. A formulation of the eigenvalues of these operators, and a characterisation of which extensions are non-well-posed, are also found. Finally, the limit-point case is studied, mainly by means of an example.

We discuss travelling wavefronts of a degenerate and singular parabolic equation in non-divergent form with changing sign sources. Necessary and sufficient conditions will be given for the existence of smooth or non-smooth and non-decreasing or non-increasing solutions. We also study the regularity of such solutions.

We study the Lane–Emden equation in strips.

Second-order expressions –d2/dx2+q whose squares are of limit-3 type are constructed. The construction is simpler than others in the literature. The function q can be chosen to be infinitely differentiable and decreasing and to satisfy q(x)≧ –x2(log x)2+ε (x→∞) for a given ε>0.

We consider the free linear Schrödinger equation on a torus 𝕋d, perturbed by a Hamiltonian nonlinearity, driven by a random force and subject to a linear damping:

Here u = u(t, x), x ∈ 𝕋d, 0 < ν ≪ 1, q∗ ∈ ℕ, f is a positive continuous function, ρ is a positive parameter and are standard independent complex Wiener processes. We are interested in limiting, as ν → 0, behaviour of distributions of solutions for this equation and of its stationary measure. Writing the equation in the slow time τ = νt, we prove that the limiting behaviour of them both is described by the effective equation

where the nonlinearity F(u) is made out of the resonant terms of the monomial |u|2q∗u.

For a class of Fuchsian groups, which includes integral automorphs of quadratic forms and unit groups of indefinite quaternion algebras, it is shown that the geometry of a suitably chosen fundamental region leads to explicit bounds for a complete set of generators.

In this paper we consider the initial-boundary value problem for the semihnear diffusion equation ul=uxx+f(u) on the half-line x>0, when for 0<a<1 f(0)=f(a)=f(1)=0 and f(u)<0 on (0, a), f(u)>0 on (a, 1). For a wide class of initial and boundary values a uniformly valid asymptotic expression is given to which the solution converges exponentially. This expression is composed of a travelling wave and a solution of the stationary problem.

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.

We continue with the work of earlier papers concerning the use of partial dilferential equations to prove the uniform convergence of the eigenfunction expansion associated witha left definite two-parameter system of ordinary differential equations of the second order.

We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability.

We prove that several definitions of integral multilinear mappings given in the literature are equivalent. We show that what we call S-factorizable multilinear mappings are integral, but that the converse is not true (contrary to earlier claims). Using S-factorizable polynomials we give characterizations of L∞-spaces, Asplund spaces, spaces not containing ℓ1 and spaces with the compact range property. Some of these characterizations seem to be new even for linear operators.

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampère type equation and we obtain a new existence result of solutions to the Dual Orlicz-Minkowski problem for smooth measures, especially for even smooth measures.

We consider the maximal operator defined on the real line by which is related to the Cesàro convergence of the singular integrals. We characterize the weights w for which Mα is of weak type, strong type and restricted weak type (p, p) with respect to the measure w(x) dx.