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The determination of a parabolic equation from initial and final data. (English) Zbl 0644.35093

The author studies an inverse problem relative to the parabolic equation

${u}_{t}-{\Delta }u+a\left(x\right)u=0,\phantom{\rule{1.em}{0ex}}\left(x,t\right)\in {\Omega }×\left(0,T\right),$
$\partial u/\partial n=0,\phantom{\rule{1.em}{0ex}}\left(x,t\right)\in \partial {\Omega }×\left(0,T\right),\phantom{\rule{1.em}{0ex}}u\left(x,0\right)=f\left(x\right)·$

The additional information is $u\left(x,T\right)=g\left(x\right)$. Under certain conditions there is at most one solution pair (a,u) to the problem considered. It is a “folk-theorem” in undetermined coefficient problems that one gets a well-posed problem if the additional data is prescribed in a direction “parallel” to the coefficient but not if it is “perpendicular”. See also S. Handrock-Meyer, Inverse problems for the heat equation (German), to appear in Zeitschrift für Analysis und ihre Anwendungen.

Reviewer: G.Anger

##### MSC:
 35R30 Inverse problems for PDE 35K15 Second order parabolic equations, initial value problems
##### Keywords:
undetermined coefficient; well-posed problem