# American Institute of Mathematical Sciences

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

## Identifying a space dependent coefficient in a reaction-diffusion equation

 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

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 $T$.
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.

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##### 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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