Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems
Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems
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DOI：
10.1016/j.jmaa.2011.04.058
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发表时间：
2011-10
影响因子：
1.3
通讯作者：
Kenichi Sakamoto;Masahiro Yamamoto
中科院分区：
文献类型：
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作者：
Kenichi Sakamoto;Masahiro Yamamoto
考虑分数阶扩散波方程的初值/边值问题：α u（x，t）= Lu（x，t），其中0< α <$2，其中L是具有t-独立光滑系数的对称一致椭圆算子.首先我们建立了弱解的唯一存在性和当时间t趋于∞时的渐近性态，并利用特征函数展开式进行了证明。其次，对α∈（0，1），利用特征函数展开式，证明了（i）倒向问题的时间稳定性，（ii）初值确定的唯一性，（iii）当t→∞时，解的衰减率唯一性，（iv）在（0，T）上一点的观测确定源的t相关因子的源反问题的稳定性。
We consider initial value/boundary value problems for fractional diffusion-wave equation:∂ t α u (x, t)= L u (x, t), where 0< α⩽ 2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to∞ and the proofs are based on the eigenfunction expansions. Second for α∈(0, 1), we apply the eigenfunction expansions and prove (i) stability in the backward problem in time,(ii) the uniqueness in determining an initial value and (iii) the uniqueness of solution by the decay rate as t→∞,(iv) stability in an inverse source problem of determining t-dependent factor in the source by observation at one point over (0, T).