Simultaneous determination of initial value and source term for time-fractional wave-diffusion equations

OpenAlex · İngilizce

We consider initial boundary value problems for time fractional diffusion-wave equations: $ d_t^{\alpha} u(x,t) = -Au(x,t) + \mu(t)f(x) $ in a bounded domain $ \Omega \subset \mathbb{R}^d $, where $ \alpha \in (0,1) \cup (1,2) $, $ \mu(t)f(x) $ describes a source, and $ -A $ is a symmetric elliptic operator:$ -Av(x) = \sum\limits^d_{i,j = 1} \partial_i(a_{ij}(x) \partial_jv(x)) + c(x)v(x) $for $ x \in \Omega $. We assume that there exists $ T>0 $ such that $ \mu(t) = 0 $ for $ t > T $. For $ T_2>T_1>T $, we prove the uniqueness in simultaneously determining $ f $ in $ \Omega $, $ \mu $ in $ (0,T) $, and initial values of $ u $ by data $ u\vert_{\omega\times (T_1,T_2)} $, provided that the order $ \alpha $ does not belong to an at most countably infinite set in $ (0,1) \cup (1,2) $ which is characterized by $ \mu $. The proof is based on the asymptotic expansion of the solution $ u $ by means of the Mittag-Leffler functions.