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Inverse parabolic problems with the final overdetermination. (English) Zbl 0729.35146

Let ${\Omega }$ be a bounded (smooth) domain in ${ℝ}^{n}$ and let I be either interval (-$\infty ,T\right)$ or (0,T) $\left(T>0\right)$. The author considers the following identification problems ${P}_{\sigma }$ $\left(\sigma =0,1\right):$ determine a pair of functions u: ${\Omega }×I\to ℝ$ and f: $\overline{{\Omega }}\to ℝ$ such that

$\left(1\right)\phantom{\rule{1.em}{0ex}}{D}_{t}u\left(x,t\right)+Au\left(x,t\right)=\alpha \left(x,t\right)f\left(x\right)+F\left(x,t\right),\phantom{\rule{1.em}{0ex}}\left(x,t\right)\in Q:={\Omega }×I,$
$\left(2\right)\phantom{\rule{1.em}{0ex}}{B}_{\sigma }u\left(x,t\right)=g\left(x,t\right),\phantom{\rule{1.em}{0ex}}\left(x,t\right)\in {\Gamma }:=\partial {\Omega }×I,\phantom{\rule{1.em}{0ex}}\left(3\right)\phantom{\rule{1.em}{0ex}}u\left(x,0\right)={h}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{1.em}{0ex}}\left(4\right)\phantom{\rule{1.em}{0ex}}u\left(x,T\right)={h}_{T}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in {\Omega }·$

Here $A={a}^{jk}\left(x,t\right){D}_{j}{D}_{k}+{a}^{j}\left(x,t\right){D}_{j}+a\left(x,t\right)$ $\left({D}_{j}=\partial /\partial {x}_{j}$, $j=1,···,n\right)$ is a uniformly elliptic operator with negative definite principal part, while ${B}_{0}u=u$ and ${B}_{1}u=\partial u/\partial \ell +bu$ with $\ell \left(x,t\right)·N\left(x\right)>ϵ$ for some positive constant $ϵ$, N(x) denoting the outward normal at a point (x,t)$\in {\Gamma }$. Finally, $\alpha ,F,g,{h}_{0},{h}_{T}$ are prescribed functions in suitable (anisotropic) Hölder spaces.

Under suitable hypotheses on A,B and $\alpha$, the author shows that problem (1)-(4) is Fredholm with zero index. Then, by restriction of the classes of admissible unknowns f and weights $\alpha$, the author proves the uniqueness (and, consequently, the existence and the stability) of the solution to problem (1)-(4). Two basic assumptions for this are the positivity and the strict monotonicity of $\alpha$ with respect to time $\left(\alpha \left(x,t\right)>0$, ${D}_{t}\alpha \left(x,t\right)>0\forall \left(x,t\right)\in Q\right)$. In connection with such hypotheses, the author shows that, when ${D}_{t}\alpha$ fails to be strictly positive in $\overline{Q},$ there exist positive ${C}^{\infty }$-weight functions $\alpha$ for which problem (1)-(4) admits nontrivial solutions.

A further identification problem is studied in the case where $\alpha =F=0$ and the coefficient a in A is supposed to be unknown and to depend only on x. Also for such a problem existence and uniqueness results are proved.

The paper concludes with the proof of a stability result for the solution to problem (1)-(4), when Q is a non-cylindrical domain.

##### MSC:
 35R30 Inverse problems for PDE 35K20 Second order parabolic equations, initial boundary value problems