INVERSE PROBLEM ON DETERMINING MANY UNKNOWNS FROM SCHWARTZ-TYPE DISTRIBUTIONS

Abstract

We find the sufficient conditions for the unique (local in time) solvability of an inverse problem of finding m unknown functions $R_l(x)$, $l\in \{1,\dots,m\}$ from the Schwartz-type distributions $S'_{\gamma,(a)}(\Bbb R^n)$ in a source term of a diffusion equation \[D^{\beta}_t u-A(x,D)u=\sum\limits_{l=1}^mR_l(x)g_l(t)+F(x,t), \;\; (x,t) \in Q=\Bbb R^n\times (0,T] \] with the Djrbasian-Nersesian-Caputo time-fractional derivative of the order $\beta\in (0,1)$ where $A(x,D)$ is an elliptic differential operator of the second order, \[S_{\gamma,(a)}(\Bbb R^n)=\{v\in C^\infty(\Bbb R^n): ||v||_{k,(a)}=\sup\limits_{|\alpha|\le k,x\in \Bbb R^n}e^{a(1-\frac{1}{k}) |x|^{\frac{1}{\gamma}}}|D^{\alpha}v(x)|<+\infty\;\;\forall k\in \Bbb N, k\ge 2\}.\] We use time-integral over-determination conditions \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_l(t)dt=\Phi_l(x), \;\;x\in \Bbb R^n, \;\;l\in \{1,\dots,m\}\] with the given $\eta_l\in C^1[0,T]$ and Schwartz-type distributions $\Phi_l(x)$, $l\in \{1,\dots,m\}$. Note that time-integral over-determination conditions were used in the study of various inverse problems in various functional spaces. By properties of the Green vector-function the problem boils down to solving linear operator equation of the second kind with respect to the unknown solution $u$ of the Cauchy problem, continuous with values in Schwartz-type distributions, and a linear inhomogeneous algebraic system of equations for finding expressions of unknown functions $R_l(x)$, $l\in \{1,\dots,m\}$ through it. We generalize the results of [11] on the classical solvability of a problem with two unknown functions from Schwartz-type spaces of rapidly decreasing functions at infinity on the right-hand side of such an equation.