The Cauchy problem for parabolic equations with quadratic nonlin-earity is studied. We investigate the existence of global-in-time solutions and their large-time behavior assuming some scaling property of the equation as well as of the norm of the Banach space in which the solutions are constructed. Some applications of these abstract results are indicated.

Abstract. We use a new variational method—based on the theory of anti-selfdual Lagrangians developed in [2] and [3]—to establish the existence of solutions of convex Hamiltonian systems that connect two given Lagrangian submanifolds in R . We also consider the case where the Hamiltonian is only semi-convex. A variational principle is also used to establish existence for the corresponding Cauchy...

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We prove Hölder-continuous dependence results for the difference between solutions of certain ill-posed and approximate well-posed problems in both Hilbert and Banach spaces. We use operator-theoretic methods, including C-semigroups, to treat the abstract Cauchy problem du dt = Au, u(0) = χ, 0 ≤ t < T, where the operator −A is the infinitesimal generator of a holomorphic semigroup.

Abstract The Cauchy problem for the three-dimensional Euler–Boltzmann equations of a polytropic, ideal and isentropic fluid in radiation hydrodynamics is considered. The formation of singularities in smooth solutions is studied. It is proved that some C1 solutions, regardless of the size of the initial disturbance, will develop singularities in a finite time provided that the initial disturbanc...

We study computability of the abstract linear Cauchy problem du(t)/dt = Au(t), u(0) = x ∈ X, (1) where A is a linear operator, possibly unbounded, on a Banach space X. We give necessary and sufficient conditions for A such that the solution operator K : x → u of the problem (1) is computable. For studying computability we use the representation approach to Computable Analysis developed by Weihr...

We consider a fractional order integro-differential equation with a weakly singular convolution kernel. The equation with homogeneuos Dirichlet boundary conditions is reformulated as an abstract Cauchy problem, and well-posedness is verified in the context of linear semigroup theory. Then we formulate a continuous Galerkin method for the problem, and we prove stability estimates. These are then...

Abstract: A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is prese...

The details are presented of a new evolution algorithm for the characteristic initial-boundary value problem based upon an affine parameter rather than the areal radial coordinate used in the BondiSachs formulation. The advantages over the Bondi-Sachs version are discussed, with particular emphasis on the application to the characteristic extraction of the gravitational waveform from Cauchy sim...

The posterior moments of parameters specifying distributions are minimum mean square Bayesian estimators for the corresponding moments of those parameters, and as such are ubiquitous in the Bayesian approach to statistical inference of distributions. The Cauchy distribution is most notable for its wide tails, decided absence of high-order moments, and non-existence of less-than-data dimension s...