|The interplay of rough intuition and mathematical precision - my 1972 parabolic backwards nonuniqueness results
Mathematical discovery often involves the interplay - back and
forth - between rough intuition and more precise mathematics.
Unfortunately, only the smoothed-out (and perhaps over-generalized)
mathematical version gets published, and the original rough intuitive
model which led to the discovery goes forever unexplained and
unpublished. In 1972, while on sabbatical in Firenze, I constructed
an example of backwards nonuniqueness for a uniformly parabolic
equation in self-adjoint divergence form with space and time varying
conductivity coefficients. The solution u(x,y,t) begins proportional
to cos(x)exp(-1t) and then, after an intricate 7-phase dance of
varying the conductivity matrix, becomes later proportional to
cos(2x)exp(-4t). In later steps one makes the solution oscillate
faster and faster in space and thus decay faster and faster in time
until it vanishes in finite time and continues as zero thereafter.
But the crucial discovery, that one can vary the conductivity so as
to make a solution go from a cos(x) form to a cos(2y) form, did not
first occur to me in its published mathematically precise 7-phase form.
Instead it occurred as a rough physically motivated construction,
never published, which I would like to now explain.