We consider a time-independent variable coefficients fractional porous medium equation and formulate an associated inverse problem. We determine both the conductivity and the absorption coefficient from exterior partial measurements of the Dirichlet-to-Neumann map. Our approach relies on a time-integral transform technique as well as the unique continuation property of the fractional operator.
V.Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Science & Business Media, 2010.
2.
S.Bhattacharyya, T.Ghosh and G.Uhlmann, Inverse problems for the fractional-Laplacian with lower order non-local perturbations, Transactions of the American Mathematical Society374(5) (2021), 3053–3075. doi:10.1090/tran/8151.
3.
M.Bonforte, Y.Sire and J.L.Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete and Continuous Dynamical Systems35(12) (2015), 5725–5767. doi:10.3934/dcds.2015.35.5725.
4.
L.Caffarelli and L.Silvestre, An extension problem related to the fractional Laplacian, Communications in partial differential equations32(8) (2007), 1245–1260. doi:10.1080/03605300600987306.
5.
C.Cârstea, A.Feizmohammadi, Y.Kian, K.Krupchyk and G.Uhlmann, The Calderón inverse problem for isotropic quasilinear conductivities, Advances in Mathematics391 (2021), 107956. doi:10.1016/j.aim.2021.107956.
6.
C.Cârstea, T.Ghosh and G.Nakamura, An inverse boundary value problem for the inhomogeneous porous medium equation, 2021, arXiv preprint arXiv:2105.01368.
7.
C.Cârstea, T.Ghosh and G.Uhlmann, An inverse problem for the porous medium equation with a possibly singular absorption term, 2021, arXiv preprint arXiv:2108.12970.
8.
G.Covi, An inverse problem for the fractional Schrödinger equation in a magnetic field, Inverse Problems36(4) (2020), 045004. doi:10.1088/1361-6420/ab661a.
9.
G.Covi, Uniqueness for the fractional Calderón problem with quasilocal perturbations, 2021, arXiv preprint arXiv:2110.11063.
10.
G.Covi, M.Á.García-Ferrero and A.Rüland, On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols, 2021, arXiv preprint arXiv:2109.14976.
11.
G.Covi, K.Mönkkönen, J.Railo and G.Uhlmann, The higher order fractional Calderón problem for linear local operators: Uniqueness, 2020, arXiv preprint arXiv:2008.10227.
12.
T.Ghosh, Y.-H.Lin and J.Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communications in Partial Differential Equations42(12) (2017), 1923–1961. doi:10.1080/03605302.2017.1390681.
13.
T.Ghosh, A.Rüland, M.Salo and G.Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, Journal of Functional Analysis (2020), 108505. doi:10.1016/j.jfa.2020.108505.
14.
T.Ghosh, M.Salo and G.Uhlmann, The Calderón problem for the fractional Schrödinger equation, Analysis & PDE13(2) (2020), 455–475. doi:10.2140/apde.2020.13.455.
15.
T.Ghosh and G.Uhlmann, The Calderón problem for nonlocal operators, 2021, arXiv preprint arXiv:2110.09265.
16.
Y.Kian and G.Uhlmann, Recovery of nonlinear terms for reaction diffusion equations from boundary measurements, 2020, arXiv preprint arXiv:2011.06039.
17.
K.Krupchyk and G.Uhlmann, Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities, 2019, arXiv preprint arXiv:1909.08122.
18.
R.-Y.Lai, Y.-H.Lin and A.Rüland, The Calderón problem for a space–time fractional parabolic equation, SIAM Journal on Mathematical Analysis52(3) (2020), 2655–2688. doi:10.1137/19M1270288.
19.
R.-Y.Lai and T.Zhou, An inverse problem for non-linear fractional magnetic Schrodinger equation, 2021, arXiv preprint arXiv:2103.08180.
20.
L.Li, Determining the magnetic potential in the fractional magnetic Calderón problem, Communications in Partial Differential Equations46(6) (2021), 1017–1026. doi:10.1080/03605302.2020.1857406.
21.
L.Li, A fractional parabolic inverse problem involving a time-dependent magnetic potential, SIAM Journal on Mathematical Analysis53(1) (2021), 435–452. doi:10.1137/20M1359638.
22.
L.Li, An inverse problem for a fractional diffusion equation with fractional power type nonlinearities, Inverse Problems and Imaging, 16(3) (2022), 613–624. doi:10.3934/ipi.2021064.
23.
L.Li, On inverse problems arising in fractional elasticity, 2021, arXiv preprint arXiv:2109.03387.
24.
A.Rüland and M.Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Analysis193 (2020), 111529. doi:10.1016/j.na.2019.05.010.
25.
R.Shankar, Recovering a quasilinear conductivity from boundary measurements, Inverse problems37(1) (2020), 015014. doi:10.1088/1361-6420/abced7.
26.
J.L.Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press on Demand, 2007.
27.
J.L.Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete and Continuous Dynamical Systems. Series S7(4) (2014), 857–885. doi:10.3934/dcdss.2014.7.857.
