Identifying a space dependent coefficient in a reaction-diffusion equation

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May 2011, 5(2): 285-296. doi: 10.3934/ipi.2011.5.285

Identifying a space dependent coefficient in a reaction-diffusion equation

Elena Beretta ^1, and Cecilia Cavaterra ^2,

1.	Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le Aldo Moro 5, 00185 Roma, Italy
2.	Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy

Received March 2010 Revised September 2010 Published May 2011

Abstract
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We consider a reaction-diffusion equation for the front motion $u$ in which the reaction term is given by $c(x)g(u)$. We formulate a suitable inverse problem for the unknowns $u$ and $c$, where $u$ satisfies homogeneous Neumann boundary conditions and the additional condition is of integral type on the time interval $[0,T]$. Uniqueness of the solution is proved in the case of a linear $g$. Assuming $g$ non linear, we show uniqueness for large $T$.

Keywords: reaction-diffusion equations., Inverse problems.

Mathematics Subject Classification: Primary: 35R30; Secondary: 35K5.

Citation: Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems and Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285

References:

[1]	M. Choulli, An inverse problem for a semilinear parabolic equation, Inverse Problems, 10 (1994), 1123-1132. doi: 10.1088/0266-5611/10/5/009.
[2]	M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition, Inverse Problems, 13 (1997), 19-27. doi: 10.1088/0266-5611/13/1/003.
[3]	M. Choulli and M. Yamamoto, Uniqueness and stability in determining the heat radiative coefficient, the initial temperature and a boundary coefficient in a parabolic equation, Nonlinear Anal., 69 (2008), 3983-3998. doi: 10.1016/j.na.2007.10.031.
[4]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
[5]	V. Isakov, Inverse Parabolic Problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209. doi: 10.1002/cpa.3160440203.
[6]	V. Isakov, "Inverse Problems for Partial Differential Equations," Second Edition, Springer, New York, 2006.
[7]	V. Isakov, Some inverse parabolic problems for the diffusion equation, Inverse Problems, 15 (1999), 3-10. doi: 10.1088/0266-5611/15/1/004.
[8]	V. L. Kamynin, On the unique solvability of an inverse problem for parabolic equations under a final overdetermination conditions, Math. Notes, 73 (2003), 202-211. doi: 10.1023/A:1022107024916.
[9]	V. L. Kamynin, On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination conditions, Math. Notes, 77 (2005), 482-493. doi: 10.1007/s11006-005-0047-6.
[10]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," AMS, Providence, RI, 1968.
[11]	V. Méndez, J. Fort, H. G. Rotstein and S. Fedotov, Speed of reaction-diffusion fronts in spatially heterogeneous media, Phys. Rev. E (3), 68 (2003), 041105. doi: 10.1103/PhysRevE.68.041105.
[12]	C. V. Pao, "Nonlinear Parabolic And Elliptic Equations," Plenum Press, New York, 1992.
[13]	A. I. Prilepko and V. V. Solov'ev, Solvability theorems and the Rothe method in inverse problems for an equation of parabolic type II, Diff. Eq., 23 (1987), 1341-1349.
[14]	A. B. Kostin and A. I. Prilepko, On certain inverse problems for parabolic equations with final and integral observation, Russian Acad. Sci. Sb. Math., 75 (1993), 473-490. doi: 10.1070/SM1993v075n02ABEH003394.
[15]	H. G. Rotstein, A. M. Zhabotinsky and I. R. Epstein, Dynamics of one- and two-dimensional kinds in bistable reaction-diffusion equations with quasidiscrete sources of reaction, Chaos, 11 (2001), 833-842. doi: 10.1063/1.1418459.

show all references

References:

[1]	M. Choulli, An inverse problem for a semilinear parabolic equation, Inverse Problems, 10 (1994), 1123-1132. doi: 10.1088/0266-5611/10/5/009.
[2]	M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition, Inverse Problems, 13 (1997), 19-27. doi: 10.1088/0266-5611/13/1/003.
[3]	M. Choulli and M. Yamamoto, Uniqueness and stability in determining the heat radiative coefficient, the initial temperature and a boundary coefficient in a parabolic equation, Nonlinear Anal., 69 (2008), 3983-3998. doi: 10.1016/j.na.2007.10.031.
[4]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
[5]	V. Isakov, Inverse Parabolic Problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209. doi: 10.1002/cpa.3160440203.
[6]	V. Isakov, "Inverse Problems for Partial Differential Equations," Second Edition, Springer, New York, 2006.
[7]	V. Isakov, Some inverse parabolic problems for the diffusion equation, Inverse Problems, 15 (1999), 3-10. doi: 10.1088/0266-5611/15/1/004.
[8]	V. L. Kamynin, On the unique solvability of an inverse problem for parabolic equations under a final overdetermination conditions, Math. Notes, 73 (2003), 202-211. doi: 10.1023/A:1022107024916.
[9]	V. L. Kamynin, On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination conditions, Math. Notes, 77 (2005), 482-493. doi: 10.1007/s11006-005-0047-6.
[10]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," AMS, Providence, RI, 1968.
[11]	V. Méndez, J. Fort, H. G. Rotstein and S. Fedotov, Speed of reaction-diffusion fronts in spatially heterogeneous media, Phys. Rev. E (3), 68 (2003), 041105. doi: 10.1103/PhysRevE.68.041105.
[12]	C. V. Pao, "Nonlinear Parabolic And Elliptic Equations," Plenum Press, New York, 1992.
[13]	A. I. Prilepko and V. V. Solov'ev, Solvability theorems and the Rothe method in inverse problems for an equation of parabolic type II, Diff. Eq., 23 (1987), 1341-1349.
[14]	A. B. Kostin and A. I. Prilepko, On certain inverse problems for parabolic equations with final and integral observation, Russian Acad. Sci. Sb. Math., 75 (1993), 473-490. doi: 10.1070/SM1993v075n02ABEH003394.
[15]	H. G. Rotstein, A. M. Zhabotinsky and I. R. Epstein, Dynamics of one- and two-dimensional kinds in bistable reaction-diffusion equations with quasidiscrete sources of reaction, Chaos, 11 (2001), 833-842. doi: 10.1063/1.1418459.

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