论文标题
卡尔曼估计的逆抛物线问题与数据初始或最终观察时刻获取
Inverse parabolic problems by Carleman estimates with data taken initial or final time moment of observation
论文作者
论文摘要
我们考虑在一个时间间隔$（0，t）$的情况下，在有界域$ \ ooo $的抛物线方程中使用均匀的neumann边界条件。我们任意选择一个子结合$γ\ subset \ ppp \ oooo $。然后，我们讨论一个逆问题，即通过解决方案的额外数据$ u $：$ \ u \ vert_ {γ\ times（0，t）} $和$ u（\ cdot，t_0）$在$ \ ooo $ in $ t_0 = 0 $ t_ = 0 $或$ t = t $。首先，我们建立了有条件的Lipschitz稳定性估算以及情况的唯一性$ t_0 =t。$ second，在$γ$的额外条件下，我们证明了case $ t_0 = 0 $的唯一性。第二个结果调整了M.V.的唯一性。 Klibanov（逆问题{\ bf 8}（1992）575-596）在有限域$ \ oooo $中的反问题。我们修改了他的方法，该方法将逆抛物线问题降低到反相反的问题，因此，即使对于逆抛物线问题，我们也必须假设适用于相应的逆双曲线问题的唯一性。此外，我们证明了$ t_0 = 0 $的抛物线方程的某些反源问题的唯一性，整个$ \ ppp \ oooo $ n of Boundary条件没有边界条件。
We consider a parabolic equation in a bounded domain $\OOO$ over a time interval $(0,T)$ with the homogeneous Neumann boundary condition. We arbitrarily choose a subboundary $Γ\subset \ppp\OOO$. Then, we discuss an inverse problem of determining a zeroth-order spatially varying coefficient by extra data of solution $u$: $u\vert_{Γ\times (0,T)}$ and $u(\cdot,t_0)$ in $\OOO$ with $t_0=0$ or $t=T$. First we establish a conditional Lipschitz stability estimate as well as the uniqueness for the case $t_0=T.$ Second, under additional condition for $Γ$, we prove the uniqueness for the case $t_0=0$. The second result adjusts the uniqueness by M.V. Klibanov (Inverse Problems {\bf 8} (1992) 575-596) to the inverse problem in a bounded domain $\OOO$. We modify his method which reduces the inverse parabolic problem to an inverse hyperbolic problem, and so even for the inverse parabolic problem, we have to assume conditions for the uniqueness for the corresponding inverse hyperbolic problem. Moreover we prove the uniqueness for some inverse source problem for a parabolic equation for $t_0=0$ without boundary condition on the whole $\ppp\OOO$.
