Paul Sacks

Continuity of solutions of a singular parabolic equation

which is of parabolic divergence form since

Reconstruction techniques for classical inverse Sturm-Liouville problems

’ 3 0 a.e. However it is the solution u of (0.1) which will be continuous, while the solution u of (0.2) need not be. In certain physical applications u corresponds to temperature while u is the enthalpy. The difficulty in the analysis of this equation, written in the form (0.2), stems from the fact that c#+ need not be bounded or bounded away from zero. Thus the equation may be of degenerate or singular parabolic type. As particular cases of the equations covered by our results we mention the three model problems

Some existence and nonexistence theorems for solutions of degenerate parabolic equations

This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.

Phaseless inverse scattering and the phase problem in optics

Abstract The initial and boundary value problem for the degenerate parabolic equation v t = Δ ( ϑ ( v )) + F ( v ) in the cylinder Ω × ¦0, ∞), Ω ⊂ R n bounded, for a certain class of point functions ϑ satisfying ϑ ′( v ) ⩾ 0 (e.g., ϑ(v) = ¦v¦ m sign v ) is considered. In the case that F ( v ) sign v ⩽ C(1 + ¦ϑ(v)¦ α ), α , the equation has a global time solution. The same is true for α = 1 provided the measure of Ω is sufficiently small. In the case that F(v) ϑ(v) is nondecreasing a condition is given on the initial state v ( x , 0) which implies that the solution must blow up in finite time. The existence of such initial states is discussed.

SomeL 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions

Two related problems are considered: (i) the inverse scattering problem for a potential V(x) supported on the half‐line {x≥0}, when the given data is ‖R−(k)‖, the amplitude of the reflection coefficient and (ii) determination of a function g(t) supported on the half‐line {t≥0} when the given data is ‖g(k)‖, the amplitude of the Fourier transform of g. Under certain conditions on V or g, uniqueness theorems are proved and computational methods are developed. A numerical example of recovery of V(x) from ‖R−(k)‖ is given.

Reconstruction of steplike potentials

In this paper we study questions of existence, uniqueness, and continuous dependence for semilinear elliptic equations with nonlinear boundary conditions. In particular, we obtain results concerning the continuous dependence of the solutions on the nonlinearities in the problem, which in turn implies analogous results for a related parabolic problem. Such questions arise naturally in the study of potential theory, flow through porous media, and obstacle problems.

Inverse problem on the line without phase information

In this article we study some numerical methods for the determination of a potential V(x) in the one-dimensional Schrodinger equation. We assume that V(x) = 0 for x <0, and tends to a nonnegative constant as x tends to positive infinity. We suppose also that there are no bound states. The approach pursued here is a based on a transformation to an equivalent ‘time domain’ problem, namely the determination of an unknown coefficient in a wave equation. We also discuss some advantages of replacing the unknown potential by an equivalent unknown impedance.

Analysis of a convective reaction-diffusion equation II

The one-dimensional Schrodinger equation is considered for real potentials that are integrable, have finite first moment, and contain no bound states. The recovery of a potential with support in a right half-line is studied in terms of the scattering data consisting of the magnitude of the reflection coefficient, a known potential placed to the left of the unknown potential, and the magnitude of the reflection coefficient of the combined potential. Several kinds of methods are described for retrieval of the reflection coefficient corresponding to the unknown potential. Some illustrative examples are provided.

Global Behavior for a Class of Nonlinear Evolution Equations

We study the large time behavior of positive solutions of the semilinear parabolic equation

The Number of Peaks of Positive Solutions of Semilinear Parabolic Equations

u_t = u_{xx} + \varepsilon (g(u))_x + f(u)