Inverse problems for heat equation and space-time fractional diffusion equation with one measurement
Abstract
Given a connected compact Riemannian manifold (M , g) without boundary, dim M ≥ 2, we consider a space-time fractional diffusion equation with an interior source that is supported on an open subset V of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order α ∈ (0 , 1 ], and the space fractional part by (-Δg) β, where β ∈ (0 , 1 ] and Δg is the Laplace-Beltrami operator on the manifold. The case α = β = 1, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on V determines the manifold up to a Riemannian isometry.
- Publication:
-
Journal of Differential Equations
- Pub Date:
- October 2020
- DOI:
- 10.1016/j.jde.2020.05.022
- arXiv:
- arXiv:1903.04348
- Bibcode:
- 2020JDE...269.7498H
- Keywords:
-
- 35R11;
- 35R30;
- Mathematics - Analysis of PDEs;
- 35R11;
- 35R30
- E-Print:
- 34 pages, 1 figure