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February  2021, 14(2): 695-721. doi: 10.3934/dcdss.2020362

Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation

Patrick Martinez and  Judith Vancostenoble ,

Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier Toulouse Ⅲ, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France

* Corresponding author: Judith Vancostenoble

Received  November 2019 Revised  February 2020 Published  February 2021 Early access  May 2020

We consider a reaction-diffusion model of biological invasion in which the evolution of the population is governed by several parameters among them the intrinsic growth rate $ \mu(x) $. The knowledge of this growth rate is essential to predict the evolution of the population, but it is a priori unknown for exotic invasive species. We prove uniqueness and unconditional Lipschitz stability for the corresponding inverse problem, taking advantage of the positivity of the solution inside the spatial domain and studying its behaviour near the boundary with maximum principles. Our results complement previous works by Cristofol and Roques [11,13].

Keywords: Parabolic partial differential equation, nonlinear equation, reaction-diffusion model, biological invasion model, inverse problem, Carleman estimates.
Mathematics Subject Classification: Primary: 92, 35; Secondary: 92D25, 92D40, 35F20, 35K55, 35Q92, 35R30.
Citation: Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362
References:
[1]

L. Baudouin and J. P. Puel, An inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554.  doi: 10.1088/0266-5611/18/6/307.

[2]

A. Benabdallah, P. Gaitan and J. Le Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim., 46 (2007), 1849-1881.  doi: 10.1137/050640047.

[3]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl., 336 (2007), 865-887.  doi: 10.1016/j.jmaa.2007.03.024.

[4]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I–Species persistence, J. Math. Biol., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential equations, Springer, New York, 2011.

[6]

A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. 

[7]

P. Cannarsa, G. Floridia, F. Gölgeleyen and M. Yamamoto, (Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 105013, 22 pp. doi: 10.1088/1361-6420/ab1c69.

[8]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc., 239 (2016), no. 1133. doi: 10.1090/memo/1133.

[9]

P. Cannarsa, J. Tort, and M. Yamamoto, Determination of a source term in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003, 20 pp. doi: 10.1088/0266-5611/26/10/105003.

[10]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[11]

M. Cristofol and L. Roques, Biological invasions: Deriving the regions at risk from partial measurements, Math. Biosci., 215 (2008), 158-166.  doi: 10.1016/j.mbs.2008.07.004.

[12]

M. Cristofol, J. Garnier, F. Hamel and L. Roques, Uniqueness from pointwise observations in a multi-parameter inverse problem, Commun. Pure Appl. Anal., 11 (2011), 173-188.  doi: 10.3934/cpaa.2012.11.173.

[13]

M. Cristofol and L. Roques, Stable estimation of two coefficients in a nonlinear Fisher-KPP equation, Inverse Problems, 29 (2013), 095007, 18 pp. doi: 10.1088/0266-5611/29/9/095007.

[14]

M. Cristofol, P. Gaitan and H. Ramoul, Inverse problem for a two by two reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573.  doi: 10.1088/0266-5611/22/5/003.

[15]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.

[16]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Seoul, 1996.

[17]

F. Hamel, J. Fayard and L. Roques, Spreading speeds in slowly oscillating environments., Bull. Math. Biol., 72 (2010), 1166-1191.  doi: 10.1007/s11538-009-9486-7.

[18]

F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media, Indiana Univ. Math. J., 60 (2011), 1229-1247.  doi: 10.1512/iumj.2011.60.4370.

[19]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.

[20]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, Vol. 127, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[21]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[22]

M. V. Klibanov and A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[23]

J. Li, M. Yamamoto and J. Zou, Conditional stability and numerical reconstruction of initial temperature, Commun. Pure Appl. Anal., 8 (2009), 361-382.  doi: 10.3934/cpaa.2009.8.361.

[24]

P. Martinez, J. Tort and J. Vancostenoble, Lipschitz stability for an inverse problem for the 2D-Sellers model on a manifold, Riv. Mat. Univ. Parma, (N.S.), 7 (2016), 351-389. 

[25] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[26]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1967.

[27]

J. P. Puel and M. Yamamoto, Applications of Exact Controllability to Some Inverse Problems for the Wave Equation. Control of partial differential equations and applications, Lecture Notes in Pure and Appl. Math., Vol. 174, Dekker, New York, 1996,241–249.

[28] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. 
[29]

L. Roques, Species persistence decreases with habitat fragmentation: An analysis in periodic stochastic environments, J. Math. Biol., 55 (2007), 189-205.  doi: 10.1007/s00285-007-0076-8.

[30]

L. Roques and M. Cristofol, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity, 23 (2010), 675-686.  doi: 10.1088/0951-7715/23/3/014.

[31]

L. Roques and M. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math., 68 (2007), 133-153.  doi: 10.1137/060676994.

[32]

S. Saitoh and M. Yamamoto, Stability of Lipschitz type in determination of initial heat distribution, J. Inequal. Appl., 1 (1997), 73-83.  doi: 10.1155/S1025583497000052.

[33]

J. Tort and J. Vancostenoble, Determination of the insolation function in the nonlinear Sellers climate model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 683-713.  doi: 10.1016/j.anihpc.2012.03.003.

[34]

J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761.

[35]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1287-1317.  doi: 10.1080/03605302.2011.587491.

show all references

References:
[1]

L. Baudouin and J. P. Puel, An inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554.  doi: 10.1088/0266-5611/18/6/307.

[2]

A. Benabdallah, P. Gaitan and J. Le Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim., 46 (2007), 1849-1881.  doi: 10.1137/050640047.

[3]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl., 336 (2007), 865-887.  doi: 10.1016/j.jmaa.2007.03.024.

[4]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I–Species persistence, J. Math. Biol., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential equations, Springer, New York, 2011.

[6]

A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. 

[7]

P. Cannarsa, G. Floridia, F. Gölgeleyen and M. Yamamoto, (Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 105013, 22 pp. doi: 10.1088/1361-6420/ab1c69.

[8]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc., 239 (2016), no. 1133. doi: 10.1090/memo/1133.

[9]

P. Cannarsa, J. Tort, and M. Yamamoto, Determination of a source term in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003, 20 pp. doi: 10.1088/0266-5611/26/10/105003.

[10]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[11]

M. Cristofol and L. Roques, Biological invasions: Deriving the regions at risk from partial measurements, Math. Biosci., 215 (2008), 158-166.  doi: 10.1016/j.mbs.2008.07.004.

[12]

M. Cristofol, J. Garnier, F. Hamel and L. Roques, Uniqueness from pointwise observations in a multi-parameter inverse problem, Commun. Pure Appl. Anal., 11 (2011), 173-188.  doi: 10.3934/cpaa.2012.11.173.

[13]

M. Cristofol and L. Roques, Stable estimation of two coefficients in a nonlinear Fisher-KPP equation, Inverse Problems, 29 (2013), 095007, 18 pp. doi: 10.1088/0266-5611/29/9/095007.

[14]

M. Cristofol, P. Gaitan and H. Ramoul, Inverse problem for a two by two reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573.  doi: 10.1088/0266-5611/22/5/003.

[15]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.

[16]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Seoul, 1996.

[17]

F. Hamel, J. Fayard and L. Roques, Spreading speeds in slowly oscillating environments., Bull. Math. Biol., 72 (2010), 1166-1191.  doi: 10.1007/s11538-009-9486-7.

[18]

F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media, Indiana Univ. Math. J., 60 (2011), 1229-1247.  doi: 10.1512/iumj.2011.60.4370.

[19]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.

[20]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, Vol. 127, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[21]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[22]

M. V. Klibanov and A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[23]

J. Li, M. Yamamoto and J. Zou, Conditional stability and numerical reconstruction of initial temperature, Commun. Pure Appl. Anal., 8 (2009), 361-382.  doi: 10.3934/cpaa.2009.8.361.

[24]

P. Martinez, J. Tort and J. Vancostenoble, Lipschitz stability for an inverse problem for the 2D-Sellers model on a manifold, Riv. Mat. Univ. Parma, (N.S.), 7 (2016), 351-389. 

[25] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[26]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1967.

[27]

J. P. Puel and M. Yamamoto, Applications of Exact Controllability to Some Inverse Problems for the Wave Equation. Control of partial differential equations and applications, Lecture Notes in Pure and Appl. Math., Vol. 174, Dekker, New York, 1996,241–249.

[28] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. 
[29]

L. Roques, Species persistence decreases with habitat fragmentation: An analysis in periodic stochastic environments, J. Math. Biol., 55 (2007), 189-205.  doi: 10.1007/s00285-007-0076-8.

[30]

L. Roques and M. Cristofol, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity, 23 (2010), 675-686.  doi: 10.1088/0951-7715/23/3/014.

[31]

L. Roques and M. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math., 68 (2007), 133-153.  doi: 10.1137/060676994.

[32]

S. Saitoh and M. Yamamoto, Stability of Lipschitz type in determination of initial heat distribution, J. Inequal. Appl., 1 (1997), 73-83.  doi: 10.1155/S1025583497000052.

[33]

J. Tort and J. Vancostenoble, Determination of the insolation function in the nonlinear Sellers climate model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 683-713.  doi: 10.1016/j.anihpc.2012.03.003.

[34]

J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761.

[35]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1287-1317.  doi: 10.1080/03605302.2011.587491.

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