We show well-posedness of the Cauchy problem for the delay equation u′(t) = Au(t) + Φut, with initial values in X × Lp(−h, 0; X); X a Banach space, A the generator of a C0-semigroup on X, h ∈ {1,∞}, 1 6 p < ∞. In the first result, Φ is defined by a Stieltjes integral, and p = 1. In the second result, Φ is a continuous linear mapping from W 1 p (−h, 0; X) to the Favard class of A. MSC 2000: 34K06

We prove local well-posedness results for the semi-linear wave equation for data in H , 0 < < n?3 2(n?1) , extending the previously known results for this problem. The improvement comes from an introduction of a two-scale Lebesgue space X r;p k .

Constructions of metrics with special holonomy by methods of exterior differential systems are reviewed and the interpretations of these construction as ‘flows’ on hypersurface geometries are considered. It is shown that these hypersurface ‘flows’ are not generally well-posed for smooth initial data and counterexamples to existence are constructed.

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-intime existence of classical solution to the initial-boundary value problem by the method of energy estimates. keywords: Euler equation; initial-boundary value problem; well-posedness. MSC(2000): 35A05; 35L45.

Considering the Cauchy problem for the modified finite-depthfluid equation ∂tu− Gδ(∂ 2 xu)∓ u 2ux = 0, u(0) = u0, where Gδf = −iF [coth(2πδξ)− 1 2πδξ ]Ff , δ&1, and u is a real-valued function, we show that it is uniformly globally well-posed if u0 ∈ Hs (s ≥ 1/2) with ‖u0‖L2 sufficiently small for all δ&1. Our result is sharp in the sense that the solution map fails to be C in Hs(s < 1/2). More...

We study a class of equilibrium problems which is identified with a complete metric space of functions. For most elements of this space of functions in the sense of Baire category, we establish that the corresponding equilibrium problem possesses a unique solution and is well-posed.

We consider the problem of hedging the loss of a given portfolio of derivatives using a set of more liquid derivative instruments. We illustrate why the typical mathematical formulation for this hedging problem is ill-posed. We propose to determine a hedging portfolio by minimizing a proportional cost subject to an upper bound on the hedge risk; this bound is typically slightly larger than the ...

Reconstruction of a signal from its spectral phase or magnitude is in general an ill-posed problem. Various conditions restricting the class of signals under consideration have been shown to be su cient to regularize the problem so that a unique (or essentially unique) signal corresponds to any given spectral magnitude or spectral phase function. This paper shows that a nite discretetime signal...