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.