Remarks on a Backward Parabolic Problem

Abstract

After spending a quarter at the UCLA in the early 1998, I got to know Professor Stanley Osher better. He always seems to have insights and nice ideas to do computations for complicated problems. Thus at those times when I encounter problems for which analysis seem either practically impossible or extremely difficult and, for which some reliable computations may give either a reasonable solution or some hints, I often turn to experts like Stanley to see if they can do anything about them. The present article is of such nature, and I would like to dedicate it to Stanley on the occasion of his 60th birthday. I had some ideas of handling the backward parabolic problem about one year ago, then I learned from P. Lax some earlier works (nearly 50 years ago!) of F. John, see [1]. Though the approach I had is apparently rather different from that of F. John, they seem to have some deep connections. I shall explain some notions (which I found rather amusing) introduced in John's work in the next section. A solution to the problem (*) (or a somewhat more general problem) will be explained in section 3. In the final section, I shall describe a few issues which may be of interest from both theory and computations. I have no intention here to make various statements or estimates more refined. The goal here is to present the problems and certain point views on such problems. I wish to thank P. Lax for bringing John's work to my attention and for several interesting discussions.