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Inverse parabolic problems with the final overdetermination. (English) Zbl 0729.35146
Let $\Omega$ be a bounded (smooth) domain in ${\bbfR}\sp n$ and let I be either interval (-$\infty,T)$ or (0,T) $(T>0)$. The author considers the following identification problems $P\sb{\sigma}$ $(\sigma =0,1):$ determine a pair of functions u: $\Omega\times I\to {\bbfR}$ and f: ${\bar \Omega}\to {\bbfR}$ such that $$ (1)\quad D\sb tu(x,t)+Au(x,t)=\alpha (x,t)f(x)+F(x,t),\quad (x,t)\in Q:=\Omega \times I, $$ $$ (2)\quad B\sb{\sigma}u(x,t)=g(x,t),\quad (x,t)\in \Gamma:=\partial \Omega \times I,\quad (3)\quad u(x,0)=h\sb 0(x),\quad x\in \Omega,\quad (4)\quad u(x,T)=h\sb T(x),\quad x\in \Omega. $$ Here $A=a\sp{jk}(x,t)D\sb jD\sb k+a\sp j(x,t)D\sb j+a(x,t)$ $(D\sb j=\partial /\partial x\sb j$, $j=1,...,n)$ is a uniformly elliptic operator with negative definite principal part, while $B\sb 0u=u$ and $B\sb 1u=\partial u/\partial \ell +bu$ with $\ell (x,t)\cdot N(x)>\epsilon$ for some positive constant $\epsilon$, N(x) denoting the outward normal at a point (x,t)$\in \Gamma$. Finally, $\alpha,F,g,h\sb 0,h\sb T$ 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 $(\alpha (x,t)>0$, $D\sb t\alpha (x,t)>0\forall (x,t)\in Q)$. In connection with such hypotheses, the author shows that, when $D\sb t\alpha$ fails to be strictly positive in $\bar Q,$ there exist positive $C\sp{\infty}$-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.

35R30Inverse problems for PDE
35K20Second order parabolic equations, initial boundary value problems
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