American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An effective design method to produce stationary chemical reaction-diffusion patterns
  • CPAA Home
  • This Issue
  • Next Article
    Approximation of nonlinear parabolic equations using a family of conformal and non-conformal schemes
January  2012, 11(1): 173-188. doi: 10.3934/cpaa.2012.11.173

Uniqueness from pointwise observations in a multi-parameter inverse problem

Michel Cristofol 1, , Jimmy Garnier 2, , François Hamel 3, and  Lionel Roques 4,

1. 

Aix-Marseille Université, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France
2. 

UR 546 Biostatistique et Processus Spatiaux, INRA, F-84000 Avignon, France, and Aix-Marseille Université, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France
3. 

Aix-Marseille Université & Institut Universitaire de France, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20
4. 

UR 546 Biostatistique et Processus Spatiaux, INRA, F-84000 Avignon, France

Received  March 2010 Revised  November 2010 Published  September 2011

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\varepsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases $N=2$ and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.
Keywords: multi-parameter, heterogeneous media, inverse problem, uniqueness., Reaction-diffusion.
Mathematics Subject Classification: 65M32, 35K20, 35K55, 35K5.
Citation: Michel Cristofol, Jimmy Garnier, François Hamel, Lionel Roques. Uniqueness from pointwise observations in a multi-parameter inverse problem. Communications on Pure & Applied Analysis, 2012, 11 (1) : 173-188. doi: 10.3934/cpaa.2012.11.173
References:
[1]

W. C. Allee, "The Social Life of Animals,", Norton, (1938).   Google Scholar

[2]

M. Bellassoued and M. Yamamoto, Inverse source problem for a transmission problem for a parabolic equation,, J. Inverse Ill-Posed Probl., 14 (2006), 47.  doi: 10.1163/156939406776237456.  Google Scholar

[3]

A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component,, Appl. Anal., 88 (2009), 683.  doi: 10.1080/00036810802555490.  Google Scholar

[4]

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

[5]

A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems,, Soviet Math. Doklady, 24 (1981), 244.   Google Scholar

[6]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", John Wiley & Sons Ltd, (2003).  doi: 10.1002/0470871296.  Google Scholar

[7]

M. Choulli, E. M. Ouhabaz and M. Yamamoto, Stable determination of a semilinear term in a parabolic equation,, Commun. Pure Appl. Anal., 5 (2006), 447.  doi: 10.3934/cpaa.2006.5.447.  Google Scholar

[8]

M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2\times 2$ reaction-diffusion system using a carleman estimate with one observation,, Inverse Problems, 22 (2006), 1561.  doi: 10.1088/0266-5611/22/5/003.  Google Scholar

[9]

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

[10]

B. Dennis, Allee effects: population growth, critical density, and the chance of extinction,, Natur. Resource Modeling, 3 (1989), 481.   Google Scholar

[11]

P. DuChateau and W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation,, J. Differential Equations, 59 (1985), 155.   Google Scholar

[12]

H. Egger, H. W. Engl and M. V. Klibanov, Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation,, Inverse Problems, 21 (2005), 271.  doi: 10.1088/0266-5611/21/1/017.  Google Scholar

[13]

M. El Smaily, F. Hamel and L. Roques, Homogenization and influence of fragmentation in a biological invasion model,, Discrete Contin. Dyn. Syst. - A, 25 (2009), 321.  doi: 10.3934/dcds.2009.25.321.  Google Scholar

[14]

R. A. Fisher, The wave of advance of advantageous genes,, Annals of Eugenics, 7 (1937), 335.   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

T. H. Keitt, M. A. Lewis and R. D. Holt, Allee effects, invasion pinning, and species' borders,, American Naturalist, 157 (2001), 203.  doi: 10.1086/318633.  Google Scholar

[19]

M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,", Inverse And Ill-Posed Series, (2004).   Google Scholar

[20]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bull. Univ. \'Etat Moscou, 1 (1937), 1.   Google Scholar

[21]

M. A. Lewis and P. Kareiva, Allee dynamics and the speed of invading organisms,, Theor. Population Biol., 43 (1993), 141.  doi: 10.1006/tpbi.1993.1007.  Google Scholar

[22]

A. Lorenzi, An inverse problem for a semilinear parabolic equation,, Ann. Mat. Pura Appl., 131 (1982), 145.  doi: 10.1007/BF01765150.  Google Scholar

[23]

H. Matano, K.-I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit,, Netw. Heterog. Media, 1 (2006), 537.   Google Scholar

[24]

J. D. Murray, "Mathematical Biology,", 3$^{rd}$ edition, (2002).  doi: 10.1007/b98868.  Google Scholar

[25]

S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation,, Nihonkai Math. J., 12 (2001), 71.   Google Scholar

[26]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,", Plenum Press, (1992).   Google Scholar

[27]

M. S. Pilant and W. Rundell,, An inverse problem for a nonlinear parabolic equation,, Comm. Partial Differential Equations, 11 (1986), 445.  doi: 10.1080/03605308608820430.  Google Scholar

[28]

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

[29]

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

[30]

L. Roques and F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence,, Math. Biosci., 210 (2007), 34.  doi: 10.1016/j.mbs.2007.05.007.  Google Scholar

[31]

L. Roques, A. Roques, H. Berestycki and A. Kretzschmar, A population facing climate change: joint influences of Allee effects and environmental boundary geometry,, Population Ecology, 50 (2008), 215.  doi: 10.1007/s10144-007-0073-1.  Google Scholar

[32]

L. Roques and R. S. Stoica, Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments,, J. Math. Biol., 55 (2007), 189.  doi: 10.1007/s00285-007-0076-8.  Google Scholar

[33]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford Series in Ecology and Evolution, (1997).   Google Scholar

[34]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic-waves in heterogeneous environments,, Theoret. Population Biol., 30 (1986), 143.  doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[35]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.   Google Scholar

[36]

S. Soubeyrand, L. Held, M. Hohle and I. Sache, Modelling the spread in space and time of an airborne plant disease,, J. Roy. Statist. Soc. Ser. C, 57 (2008), 253.  doi: 10.1111/j.1467-9876.2007.00612.x.  Google Scholar

[37]

S. Soubeyrand, S. Neuvonen and A. Penttinen, Mechanical-statistical modeling in ecology: from outbreak detections to pest dynamics,, Bull. Math. Biol., 71 (2009), 318.  doi: 10.1007/s11538-008-9363-9.  Google Scholar

[38]

P. Turchin, "Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants,", Sinauer Associates, (1998).   Google Scholar

[39]

A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Royal Soc. London - B, 237 (1952), 37.   Google Scholar

[40]

R. R. Veit and M. A. Lewis, Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America,, American Naturalist, 148 (1996), 255.  doi: 10.1086/285924.  Google Scholar

[41]

C. K. Wikle, Hierarchical models in environmental science,, Intern. Stat. Rev., 71 (2003), 181.  doi: 10.1111/j.1751-5823.2003.tb00192.x.  Google Scholar

[42]

M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient,, Inverse Problems, 17 (2001), 1181.  doi: 10.1088/0266-5611/17/4/340.  Google Scholar

show all references

References:
[1]

W. C. Allee, "The Social Life of Animals,", Norton, (1938).   Google Scholar

[2]

M. Bellassoued and M. Yamamoto, Inverse source problem for a transmission problem for a parabolic equation,, J. Inverse Ill-Posed Probl., 14 (2006), 47.  doi: 10.1163/156939406776237456.  Google Scholar

[3]

A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component,, Appl. Anal., 88 (2009), 683.  doi: 10.1080/00036810802555490.  Google Scholar

[4]

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

[5]

A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems,, Soviet Math. Doklady, 24 (1981), 244.   Google Scholar

[6]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", John Wiley & Sons Ltd, (2003).  doi: 10.1002/0470871296.  Google Scholar

[7]

M. Choulli, E. M. Ouhabaz and M. Yamamoto, Stable determination of a semilinear term in a parabolic equation,, Commun. Pure Appl. Anal., 5 (2006), 447.  doi: 10.3934/cpaa.2006.5.447.  Google Scholar

[8]

M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2\times 2$ reaction-diffusion system using a carleman estimate with one observation,, Inverse Problems, 22 (2006), 1561.  doi: 10.1088/0266-5611/22/5/003.  Google Scholar

[9]

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

[10]

B. Dennis, Allee effects: population growth, critical density, and the chance of extinction,, Natur. Resource Modeling, 3 (1989), 481.   Google Scholar

[11]

P. DuChateau and W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation,, J. Differential Equations, 59 (1985), 155.   Google Scholar

[12]

H. Egger, H. W. Engl and M. V. Klibanov, Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation,, Inverse Problems, 21 (2005), 271.  doi: 10.1088/0266-5611/21/1/017.  Google Scholar

[13]

M. El Smaily, F. Hamel and L. Roques, Homogenization and influence of fragmentation in a biological invasion model,, Discrete Contin. Dyn. Syst. - A, 25 (2009), 321.  doi: 10.3934/dcds.2009.25.321.  Google Scholar

[14]

R. A. Fisher, The wave of advance of advantageous genes,, Annals of Eugenics, 7 (1937), 335.   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

T. H. Keitt, M. A. Lewis and R. D. Holt, Allee effects, invasion pinning, and species' borders,, American Naturalist, 157 (2001), 203.  doi: 10.1086/318633.  Google Scholar

[19]

M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,", Inverse And Ill-Posed Series, (2004).   Google Scholar

[20]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bull. Univ. \'Etat Moscou, 1 (1937), 1.   Google Scholar

[21]

M. A. Lewis and P. Kareiva, Allee dynamics and the speed of invading organisms,, Theor. Population Biol., 43 (1993), 141.  doi: 10.1006/tpbi.1993.1007.  Google Scholar

[22]

A. Lorenzi, An inverse problem for a semilinear parabolic equation,, Ann. Mat. Pura Appl., 131 (1982), 145.  doi: 10.1007/BF01765150.  Google Scholar

[23]

H. Matano, K.-I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit,, Netw. Heterog. Media, 1 (2006), 537.   Google Scholar

[24]

J. D. Murray, "Mathematical Biology,", 3$^{rd}$ edition, (2002).  doi: 10.1007/b98868.  Google Scholar

[25]

S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation,, Nihonkai Math. J., 12 (2001), 71.   Google Scholar

[26]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,", Plenum Press, (1992).   Google Scholar

[27]

M. S. Pilant and W. Rundell,, An inverse problem for a nonlinear parabolic equation,, Comm. Partial Differential Equations, 11 (1986), 445.  doi: 10.1080/03605308608820430.  Google Scholar

[28]

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

[29]

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

[30]

L. Roques and F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence,, Math. Biosci., 210 (2007), 34.  doi: 10.1016/j.mbs.2007.05.007.  Google Scholar

[31]

L. Roques, A. Roques, H. Berestycki and A. Kretzschmar, A population facing climate change: joint influences of Allee effects and environmental boundary geometry,, Population Ecology, 50 (2008), 215.  doi: 10.1007/s10144-007-0073-1.  Google Scholar

[32]

L. Roques and R. S. Stoica, Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments,, J. Math. Biol., 55 (2007), 189.  doi: 10.1007/s00285-007-0076-8.  Google Scholar

[33]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford Series in Ecology and Evolution, (1997).   Google Scholar

[34]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic-waves in heterogeneous environments,, Theoret. Population Biol., 30 (1986), 143.  doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[35]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.   Google Scholar

[36]

S. Soubeyrand, L. Held, M. Hohle and I. Sache, Modelling the spread in space and time of an airborne plant disease,, J. Roy. Statist. Soc. Ser. C, 57 (2008), 253.  doi: 10.1111/j.1467-9876.2007.00612.x.  Google Scholar

[37]

S. Soubeyrand, S. Neuvonen and A. Penttinen, Mechanical-statistical modeling in ecology: from outbreak detections to pest dynamics,, Bull. Math. Biol., 71 (2009), 318.  doi: 10.1007/s11538-008-9363-9.  Google Scholar

[38]

P. Turchin, "Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants,", Sinauer Associates, (1998).   Google Scholar

[39]

A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Royal Soc. London - B, 237 (1952), 37.   Google Scholar

[40]

R. R. Veit and M. A. Lewis, Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America,, American Naturalist, 148 (1996), 255.  doi: 10.1086/285924.  Google Scholar

[41]

C. K. Wikle, Hierarchical models in environmental science,, Intern. Stat. Rev., 71 (2003), 181.  doi: 10.1111/j.1751-5823.2003.tb00192.x.  Google Scholar

[42]

M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient,, Inverse Problems, 17 (2001), 1181.  doi: 10.1088/0266-5611/17/4/340.  Google Scholar

[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]

Markus Gahn. Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6511-6531. doi: 10.3934/dcdsb.2019151
[3]

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
[4]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
[5]

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
[6]

Hiroshi Matsuzawa. On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments. Conference Publications, 2009, 2009 (Special) : 516-525. doi: 10.3934/proc.2009.2009.516
[7]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173
[8]

Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007
[9]

Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems & Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048
[10]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85
[11]

Georg Hetzer. Global existence for a functional reaction-diffusion problem from climate modeling. Conference Publications, 2011, 2011 (Special) : 660-671. doi: 10.3934/proc.2011.2011.660
[12]

Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128
[13]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020033
[14]

Aníbal Rodríguez-Bernal, Alejandro Vidal–López. Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 537-567. doi: 10.3934/dcds.2007.18.537
[15]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55
[16]

Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multi-spot patterns in shadow systems of reaction-diffusion equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 717-736. doi: 10.3934/cpaa.2015.14.717
[17]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343
[18]

Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010
[19]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[20]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences