Jerome A. Goldstein

THE HEAT EQUATION WITH A SINGULAR POTENTIAL

Of concern is the singular problem du/dt = au + (c/|x|2)iz + /((, x), tz(x,0) = u0(x), and its generalizations. Here c > 0, x G RN, t > 0, and/and t/0 are nonnegative and not both identically zero. There is a dimension dependent constant C (N) such that the problem has no solution for c> Ct(N). For c =S Ct(A/) necessary and sufficient conditions are found for / and u0 so that a nonnegative solution exists.

Fractional telegraph equations

We investigate several aspects of the fractional telegraph equations, in an effort to better understand the anomalous diffusion processes observed in blood flow experiments. In the earlier work Eckstein et al. [Electron. J. Differential Equations Conf. 03 (1999) 39–50], the telegraph equation D 2 u + 2aDu + Au = 0 was used, where D = d/dt ,a nd it was shown that, as t tends to infinity, u is approximated by v ,w here 2aDv + Av = 0; here A =− d 2 /dx 2 on L 2 (R) ,o rA can be a more general nonnegative selfadjoint operator. In this paper the concern is with the fractional telegraph equation E 2 u + 2aEu + Au = 0, where E = D γ and 0

Linear parabolic equations with strong singular potentials

Using an extension of a recent method of Cabre and Martel (1999), we extend the blow-up and existence result in the paper of Baras and Goldstein (1984) to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials. This problem has been left open in Baras and Goldstein. These potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail. Even in the special case when the leading operator is the Laplacian, we extend a recent result in Cabre and Martel from bounded smooth domains to unbounded nonsmooth domains.

Singular nonlinear parabolic boundary value problems in one space dimension

Global existence and uniqueness are established for the mixed initial-boundary problem for the nonlinear parabolic equation ∂u∂t = φ(x, ∂u∂x) ∂2u∂x2 (0 ⩽ x ⩽ 1, t ⩽ 0), where φ(x, ξ) ⩾φ0(x) > 0 for 0 < x < 1 and ∝01 φ0(x)−1 dx < ∞. The boundary conditions can be either linear (e.g., Dirichlet, Neumann, or periodic) or nonlinear, in which case they take the form (− 1)j u(j, t) ϵ βj(ux(J, t)) for j = 0, 1, where βj is a maximal monotone graph in R × R containing the origin.

Remarks on the Inverse Square Potential in Quantum Mechanics

Publisher Summary This chapter discusses the inverse square potential in quantum mechanics. The chapter presents the Schrodinger operator (L 2 (ℝ N )), because it has the curious property of being well behaved or poorly behaved depending on the value of the constant c . By maximum principle and comparison arguments, the chapter presents good existence results for c .

Chaotic solution for the Black-Scholes equation

The Black-Scholes semigroup is studied on spaces of continuous functions on (0,∞) which may grow at both 0 and at ∞, which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces Y s,τ := {u ∈ C((0,∞)) : lim x→∞ u(x) 1 + xs = 0, lim x→0 u(x) 1 + x−τ = 0} with norm ‖u‖Y s,τ = sup x>0 ∣ ∣ ∣ u(x) (1+xs)(1+x−τ ) ∣ ∣ ∣ 1, τ ≥ 0 with sν > 1, where √ 2ν is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion.

Semigroup-theoretic proofs of the central limit theorem and other theorems of analysis

The theory of one parameter semigroups of bounded linear operators on Banach spaces has deep and far reaching applications to partial differential equations and Markov processes. Here we present some known elementary applications of operator semigroups to approximation theory, a new proof of the central limit theorem, and we give E. Nelsons rigorous interpretation of Feynman integrals. Our main tools are (i) a special case of the Trotter-Neveu-Kato approximation theorem, of which we give a new elementary proof, and (ii) P. Chernoffs product formula.

On the convergence and approximation of cosine functions

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Some remarks on infinitesimal generators of analytic semigroups

It is well known that many infinitesimal generators of strongly continuous one parameter semigroups of bounded linear operators on a Banach space actually generate semigroups which are analytic in a sector (0) = {z: I arg zj <0} where 0<0?<r/2 (see [4, p. 260]). However, the infinitesimal generator of an analvtic group is necessarily a bounded operator. In fact, the following stronger result is valid.

Degenerate Second Order Differential Operators Generating Analytic Semigroups inLp andW1,p

We deal with the problem of analyticity for the semigroup generated by the second order differential operator Au ≔ αu″ + βu′ (or by some restrictions of it) in the spaces Lp(0, 1), with or without weight, and in W1,p(0, 1), 1 0 in (0, 1), and the domain of A is determined by the generalized Neumann boundary conditions and by Wentzell boundary conditions.