Inverse problem with two unknown time-dependent functions for $2b$-order differential equation with fractional derivative

Abstract

We study the inverse problem for a differential equation of order $2b$ with a Riemann-Liouville fractional derivative over time and given Schwartz-type distributions in the right-hand sides of the equation and the initial condition. The generalized (time-continuous in a certain sense) solution $u$ of the Cauchy problem for such an equation, the time-dependent continuous young coefficient and a part of a source in the equation are unknown.
 In addition, we give the time-continuous values $\Phi_j(t)$ of desired generalized solution $u$ of the problem on a fixed test functions $\varphi_j(x)$, $x\in \mathbb R^n$, namely $(u(\cdot,t),\varphi_j(\cdot))=\Phi_j(t)$, $t\in [0,T]$, $j=1,2$.
 We find sufficient conditions for the uniqueness of the generalized solution of the inverse problem throughout the layer $Q:=\mathbb R^n\times [0,T]$ and the existence of a solution in some layer $\mathbb R^n\times [0,T_0]$, $T_0\in (0,T]$.

Coefficient inverse problem for the strongly degenerate parabolic equation

“…where t ∈ [0, T] and the constants C 13 , C 14 depend on the problem data. Let us consider the equations ( 25), (26). Using ( 30), ( 31), we derive…”

“…These problems are widely studied within the past decades (see [2,10,21]) due to its applications in medicine, geophysics, tomography, acoustics, ecology, financial mathematics, electrodynamics, etc. Various statements of coefficient inverse problems for parabolic equation are investigated in [1,4,7,9,[11][12][13]18,[24][25][26][27]31]. In these papers, the authors studied both the inverse problems of recovering of the time-dependent major coefficients in the parabolic equations without degenerations and the minor coefficients or source terms in them.…”