In Memory of Daniel Bauman Henry (Dan)

Dan Henry died on May 4, 2002 at the young age of 57. He was an inspiration, mentor and friend to many colleagues throughout the world.

Dan was the true representative of a 'mathematician' with very broad interests and that rare talent to recognize important topics which needed to be pursued in depth. In addition to doing first class research, he was a good listener, a serious critic and shared freely his ideas. Any discussion with him almost always led to additional insights.

Dan was more concerned about understanding problems than furthering his own reputation.

He was not particularly interested in publishing his work and only would do so when he was convinced that a real addition had been made to the subject. The index in MathSciNet shows that he had only 17 published works over the period 1970-2002. However, the range of topics is impressive and the contribution significant. His impact on the development of dynamical systems in infinite dimensions represented by evolutionary equations cannot be over emphasized.

Dan was extremely independent and careful in the direction in which he expended his energy. I was fortunate and, in some sense, lucky to have Dan as a student. Even though he always attended my seminars at Brown University and expressed some interest in the topics and methods, it was not clear that he wanted to work with me. When I took leave from Brown in 1968-69 to visit UCLA and USC for a special year in Applied Mathematics, I asked Dan if he would like to go with me. After careful consideration, he decided to go. In my course on Functional Differential Equations at UCLA, Dan was one of the most active participants and contributed significally to the understanding of the subject. In the spring of 1969, feeling that he had enough material to submit a thesis, I asked him if he were interested in obtaining a Ph.D. this year. He said that he had not thought about it, but returned in two weeks with an excellent hand written original manuscript.

After obtaining his Ph.D. from Brown University in 1969, Dan spent several years at the University of Kentucky and some time as a visitor at Brown University. In 1979, Dan gave a course at Brown which José Carlos de Oliveira of São Paulo attended. José suggested to Chaim Honig , the director of the Instituto de Matemática e Estatística, to invite Dan to visit São Paulo. Chaim agreed and a formal invitation was arranged together with the assistance of Waldyr Oliva, the Dean of USP. It was an ideal environment with many colleagues who appreciated his individuality and encouraged him in every way. In São Paulo, he was able to pursue his research, have students and interact with the faculty.

He gave interesting original courses on many different topics with carefully prepared detailed notes. Some of us were fortunate to have access to parts of his unpublished work to use in our own research. It is exciting to know that his colleagues in São Paulo have prepared these notes as an e-book which will be available to the general public free of charge.

We have lost a good friend and colleague.

It is appropriate to make a few remarks about some of his mathematical contributions.

The early work of Dan was devoted to linear Functional Differential Equations (LFDE). A small solution of such equations is defined to be one which approaches zero faster than any exponential. In 1970, using deep results in complex variables, he showed that there is a constant k , which depends only upon the delay and the dimension of the system, such that any such solution must be identically zero after time = k . This fundamental result was instrumental in the theory of Verduyn-Lunel explaining the role of the eigenvalues and eigenfunctions in the dynamics of LFDE.

In 1971, Dan presented the adjoint theory of boundary value problems for LFDE which unified and clarified earlier results based on a formal adjoint. In (1974) (and more completely in 1987), he characterized the essential spectrum of neutral LFDE in terms of the linear part involving delays only the derivatives. This led to a complete theory of the sensitivity of the spectrum of LFDE to variations in delays and its impact on control in engineering problems.

With Cannon and Kotlow in (1974) (1976), he gave important new results on classical solutions of the two-phase Stefan problem.

In 1981, Dan published the now classical book ' Geometric Theory of Semilinear Parabolic Equations '. This book has served as a basis for this subject since its publication and has been the inspiration for so many new developments in this area as well as other infinite dimensional dynamical systems. This was the culmination of ten years of effort devoted to understanding this subject in the context of a dynamical system. Many topics and ideas in the book have not yet been fully exploited. In 1985, this book was translated into Russian.

With Perez and Wreszenski in (1982) and with Albert and Bona in (1987), he published new results on the stability of solitary waves which include the Korteweg-de Vries and the Benjamin-Ono equations.

In 1985, for parabolic equations in one space dimension, he proved that the stable and unstable manifolds of hyperbolic equilibria always intersect transversally. This showed that one could obtain a Morse-Smale system by only proving hyperbolicity of equilibria. This inspired many generalizations to nonautonomous equations as well as to the characterization of the ordinary differential equations that have this property.

For many years, Dan was interested in the effects that the boundary of the domain of definition for a partial differential equation has on the dynamics. We have a short account of some of this work in a paper in 1986 where he discussed hyperbolicity of equilibria generically with respect to smooth perturbations of the boundary. He had a complete manuscript almost finished and these will be prepared for publication by his friends in São Paulo.

As remarked earlier, Dan had very broad interests. For many years, he was preparing a book on science fiction and told me ' that he would never publish anything that could not be verified scientifically '. The paper on the temperature of an asteroid in 1990 resulted from this philosophy and extended important results of Levinson in the 1930's on integral equations.

The paper with Perisinutto and Lopes in (1993) gave a characterization of the essential spectrum of the semigroup generated by the classical partial differential equations of thermoelasticity.

In 1994, he published his last paper on the basic concepts of exponential dichotomies and hyperbolicity in the dynamics of infinite dimensional systems for which the semigroup is not a homeomorphism. It is pleasing to know that it was published in the first issue of the journal Resenhas of the University of São Paulo.

Jack K. Hale
Regents Professor Emeritus
School of Mathematics
Georgia Institute of Technology
January 2, 2006