An inverse problem for a fractional diffusion equation with fractional power type nonlinearities

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June 2022, 16(3): 613-624. doi: 10.3934/ipi.2021064

An inverse problem for a fractional diffusion equation with fractional power type nonlinearities

Institute for Pure and Applied Mathematics, University of California, Los Angeles, CA 90095, USA

^*Corresponding author: Li Li

Received April 2021 Revised August 2021 Published June 2022 Early access October 2021

We study the well-posedness of a semi-linear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.

Keywords: Inverse problem, nonlinear fractional parabolic operator, Runge approximation property.

Mathematics Subject Classification: Primary: 35R11, 35R30.

Citation: Li Li. An inverse problem for a fractional diffusion equation with fractional power type nonlinearities. Inverse Problems and Imaging, 2022, 16 (3) : 613-624. doi: 10.3934/ipi.2021064

References:

[1]	B. Barrios, I. Peral, F. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Arch. Ration. Mech. Anal., 213 (2014), 629-650. doi: 10.1007/s00205-014-0733-1.
[2]	S. Bhattacharyya, T. Ghosh and G. Uhlmann, Inverse problems for the fractional-Laplacian with lower order non-local perturbations, Trans. Amer. Math. Soc., 374 (2021), 3053-3075. doi: 10.1090/tran/8151.
[3]	M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations, 59 (2020), 46pp. doi: 10.1007/s00526-020-01740-6.
[4]	G. Covi, An inverse problem for the fractional Schrödinger equation in a magnetic field, Inverse Problems, 36 (2020), 24pp. doi: 10.1088/1361-6420/ab661a.
[5]	G. Covi, K. Mönkkönen and J. Railo, Unique continuation property and Poincaré inequality for higher order fractional laplacians with applications in inverse problems, Inverse Probl. Imaging, 15 (2021), 641-681. doi: 10.3934/ipi.2021009.
[6]	G. Covi, K. Mönkkönen, J. Railo and G. Uhlmann, The higher order fractional Calderón problem for linear local operators: Uniqueness, preprint, arXiv2008.10227.
[7]	S. Dipierro, O. Savin and E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, J. Geom. Anal., 29 (2019), 1428-1455. doi: 10.1007/s12220-018-0045-z.
[8]	M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211.
[9]	X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas ${{F}^{\cdot \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}}}s$. Nat. Ser. A Mat. RACSAM, 110 (2016), 49-64. doi: 10.1007/s13398-015-0218-6.
[10]	T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Comm. Partial Differential Equations, 42 (2017), 1923-1961. doi: 10.1080/03605302.2017.1390681.
[11]	T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 42pp. doi: 10.1016/j.jfa.2020.108505.
[12]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475. doi: 10.2140/apde.2020.13.455.
[13]	A. Greco and A. Iannizzotto, Existence and convexity of solutions of the fractional heat equation, Commun. Pure Appl. Anal., 16 (2017), 2201-2226. doi: 10.3934/cpaa.2017109.
[14]	Y. Kian and G. Uhlmann, Recovery of nonlinear terms for reaction diffusion equations from boundary measurements, preprint, arXiv2011.06039.
[15]	K. Krupchyk and G. Uhlmann, A remark on partial data inverse problems for semilinear elliptic equations, Proc. Amer. Math. Soc., 148 (2020), 681-685. doi: 10.1090/proc/14844.
[16]	Y. Kurylev, M. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math., 212 (2018), 781-857. doi: 10.1007/s00222-017-0780-y.
[17]	R.-Y. Lai and Y.-H. Lin, Inverse problems for fractional semilinear elliptic equations, preprint, arXiv{2004.00549}.
[18]	R.-Y. Lai, Y.-H. Lin and A. Rüland, The Calderón problem for a space-time fractional parabolic equation, SIAM J. Math. Anal., 52 (2020), 2655-2688. doi: 10.1137/19M1270288.
[19]	R.-Y. Lai and T. Zhou, An inverse problem for non-linear fractional magnetic schrodinger equation, preprint, arXiv2103.08180.
[20]	M. Lassas, T. Liimatainen, Y.-H. Lin and M. Salo, Inverse problems for elliptic equations with power type nonlinearities, J. Math. Pures Appl., 145 (2021), 44-82. doi: 10.1016/j.matpur.2020.11.006.
[21]	L. Li, Determining the magnetic potential in the fractional magnetic Calderón problem, Comm. Partial Differential Equations, 46 (2021), 1017-1026. doi: 10.1080/03605302.2020.1857406.
[22]	L. Li, A fractional parabolic inverse problem involving a time-dependent magnetic potential, SIAM J. Math. Anal., 53 (2021), 435-452. doi: 10.1137/20M1359638.
[23]	L. Li, On an inverse problem for a fractional semilinear elliptic equation involving a magnetic potential, J. Differential Equations, 296 (2021), 170-185. doi: 10.1016/j.jde.2021.06.003.
[24]	T. Liimatainen, Y.-H. Lin, M. Salo and T. Tyni, Inverse problems for elliptic equations with fractional power type nonlinearities, preprint, arXiv2012.04944.
[25]	C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484. doi: 10.1016/j.na.2006.11.011.
[26]	A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Anal., 193 (2020), 56pp. doi: 10.1016/j.na.2019.05.010.
[27]	A. Rüland and M. Salo, Quantitative approximation properties for the fractional heat equation, Math. Control Relat. Fields, 10 (2020), 1-26. doi: 10.3934/mcrf.2019027.
[28]	L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 843-855.
[29]	Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797. doi: 10.1353/ajm.1997.0027.
[30]	M. Taylor, Partial Differential Equations III: Nonlinear Equations, 2$^{nd}$ edition, Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.
[31]	J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.

show all references

References:

[1]	B. Barrios, I. Peral, F. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Arch. Ration. Mech. Anal., 213 (2014), 629-650. doi: 10.1007/s00205-014-0733-1.
[2]	S. Bhattacharyya, T. Ghosh and G. Uhlmann, Inverse problems for the fractional-Laplacian with lower order non-local perturbations, Trans. Amer. Math. Soc., 374 (2021), 3053-3075. doi: 10.1090/tran/8151.
[3]	M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations, 59 (2020), 46pp. doi: 10.1007/s00526-020-01740-6.
[4]	G. Covi, An inverse problem for the fractional Schrödinger equation in a magnetic field, Inverse Problems, 36 (2020), 24pp. doi: 10.1088/1361-6420/ab661a.
[5]	G. Covi, K. Mönkkönen and J. Railo, Unique continuation property and Poincaré inequality for higher order fractional laplacians with applications in inverse problems, Inverse Probl. Imaging, 15 (2021), 641-681. doi: 10.3934/ipi.2021009.
[6]	G. Covi, K. Mönkkönen, J. Railo and G. Uhlmann, The higher order fractional Calderón problem for linear local operators: Uniqueness, preprint, arXiv2008.10227.
[7]	S. Dipierro, O. Savin and E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, J. Geom. Anal., 29 (2019), 1428-1455. doi: 10.1007/s12220-018-0045-z.
[8]	M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211.
[9]	X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas ${{F}^{\cdot \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}}}s$. Nat. Ser. A Mat. RACSAM, 110 (2016), 49-64. doi: 10.1007/s13398-015-0218-6.
[10]	T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Comm. Partial Differential Equations, 42 (2017), 1923-1961. doi: 10.1080/03605302.2017.1390681.
[11]	T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 42pp. doi: 10.1016/j.jfa.2020.108505.
[12]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475. doi: 10.2140/apde.2020.13.455.
[13]	A. Greco and A. Iannizzotto, Existence and convexity of solutions of the fractional heat equation, Commun. Pure Appl. Anal., 16 (2017), 2201-2226. doi: 10.3934/cpaa.2017109.
[14]	Y. Kian and G. Uhlmann, Recovery of nonlinear terms for reaction diffusion equations from boundary measurements, preprint, arXiv2011.06039.
[15]	K. Krupchyk and G. Uhlmann, A remark on partial data inverse problems for semilinear elliptic equations, Proc. Amer. Math. Soc., 148 (2020), 681-685. doi: 10.1090/proc/14844.
[16]	Y. Kurylev, M. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math., 212 (2018), 781-857. doi: 10.1007/s00222-017-0780-y.
[17]	R.-Y. Lai and Y.-H. Lin, Inverse problems for fractional semilinear elliptic equations, preprint, arXiv{2004.00549}.
[18]	R.-Y. Lai, Y.-H. Lin and A. Rüland, The Calderón problem for a space-time fractional parabolic equation, SIAM J. Math. Anal., 52 (2020), 2655-2688. doi: 10.1137/19M1270288.
[19]	R.-Y. Lai and T. Zhou, An inverse problem for non-linear fractional magnetic schrodinger equation, preprint, arXiv2103.08180.
[20]	M. Lassas, T. Liimatainen, Y.-H. Lin and M. Salo, Inverse problems for elliptic equations with power type nonlinearities, J. Math. Pures Appl., 145 (2021), 44-82. doi: 10.1016/j.matpur.2020.11.006.
[21]	L. Li, Determining the magnetic potential in the fractional magnetic Calderón problem, Comm. Partial Differential Equations, 46 (2021), 1017-1026. doi: 10.1080/03605302.2020.1857406.
[22]	L. Li, A fractional parabolic inverse problem involving a time-dependent magnetic potential, SIAM J. Math. Anal., 53 (2021), 435-452. doi: 10.1137/20M1359638.
[23]	L. Li, On an inverse problem for a fractional semilinear elliptic equation involving a magnetic potential, J. Differential Equations, 296 (2021), 170-185. doi: 10.1016/j.jde.2021.06.003.
[24]	T. Liimatainen, Y.-H. Lin, M. Salo and T. Tyni, Inverse problems for elliptic equations with fractional power type nonlinearities, preprint, arXiv2012.04944.
[25]	C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484. doi: 10.1016/j.na.2006.11.011.
[26]	A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Anal., 193 (2020), 56pp. doi: 10.1016/j.na.2019.05.010.
[27]	A. Rüland and M. Salo, Quantitative approximation properties for the fractional heat equation, Math. Control Relat. Fields, 10 (2020), 1-26. doi: 10.3934/mcrf.2019027.
[28]	L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 843-855.
[29]	Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797. doi: 10.1353/ajm.1997.0027.
[30]	M. Taylor, Partial Differential Equations III: Nonlinear Equations, 2$^{nd}$ edition, Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.
[31]	J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.

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