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 & 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. 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. 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. doi: 10.1016/j.na.2007.10.031.

[4]

A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall, (1964).

[5]

V. Isakov, Inverse Parabolic Problems with the final overdetermination,, Comm. Pure Appl. Math., 44 (1991), 185. doi: 10.1002/cpa.3160440203.

[6]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second Edition, (2006).

[7]

V. Isakov, Some inverse parabolic problems for the diffusion equation,, Inverse Problems, 15 (1999), 3. 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. 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. 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, (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). doi: 10.1103/PhysRevE.68.041105.

[12]

C. V. Pao, "Nonlinear Parabolic And Elliptic Equations,", Plenum Press, (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.

[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. 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. 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. 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. 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. doi: 10.1016/j.na.2007.10.031.

[4]

A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall, (1964).

[5]

V. Isakov, Inverse Parabolic Problems with the final overdetermination,, Comm. Pure Appl. Math., 44 (1991), 185. doi: 10.1002/cpa.3160440203.

[6]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second Edition, (2006).

[7]

V. Isakov, Some inverse parabolic problems for the diffusion equation,, Inverse Problems, 15 (1999), 3. 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. 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. 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, (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). doi: 10.1103/PhysRevE.68.041105.

[12]

C. V. Pao, "Nonlinear Parabolic And Elliptic Equations,", Plenum Press, (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.

[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. 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. doi: 10.1063/1.1418459.

[1]

Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369

[2]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43

[3]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[4]

Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907

[5]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495

[6]

José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299

[7]

Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193

[8]

Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526

[9]

Antoine Mellet, Jean-Michel Roquejoffre, Yannick Sire. Generalized fronts for one-dimensional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 303-312. doi: 10.3934/dcds.2010.26.303

[10]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001

[11]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253

[12]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[13]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427

[14]

Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011

[15]

Filipa Caetano, Martin J. Gander, Laurence Halpern, Jérémie Szeftel. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations. Networks & Heterogeneous Media, 2010, 5 (3) : 487-505. doi: 10.3934/nhm.2010.5.487

[16]

Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784

[17]

Masaharu Taniguchi. Traveling fronts in perturbed multistable reaction-diffusion equations. Conference Publications, 2011, 2011 (Special) : 1368-1377. doi: 10.3934/proc.2011.2011.1368

[18]

Michele V. Bartuccelli, K. B. Blyuss, Y. N. Kyrychko. Length scales and positivity of solutions of a class of reaction-diffusion equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 25-40. doi: 10.3934/cpaa.2004.3.25

[19]

Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks & Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79

[20]

Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]