Inverse source problems for positive operators. I. Hypoelliptic diffusion and subdiffusion equations

Abstract

A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod.

arXiv comment26 pages, 7 figures. arXiv admin note: text overlap with arXiv:1812.01336

References in corpus (6)

• A Tutorial on Inverse Problems for Anomalous Diffusion Processes
• The polyanalytic Ginibre ensembles
• Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field
• Wave equation for operators with discrete spectrum and irregular propagation speed
• Discrete coherent states for higher Landau levels
• Heat content and horizontal mean curvature on the Heisenberg group

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• On the existence of eigenvalues of a boundary value problem with transmitting condition of the integral form for parabolic-hyperbolic equation
• Inverse $t$-source problem and a strict positivity property for coupled subdiffusion systems