메뉴 건너뛰기




Volumn 212, Issue 1, 2014, Pages 93-127

The Hele–Shaw asymptotics for mechanical models of tumor growth

Author keywords

[No Author keywords available]

Indexed keywords


EID: 85027952439     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-013-0704-y     Document Type: Article
Times cited : (205)

References (44)
  • 1
    • 0001395720 scopus 로고
    • Régularité des solutions de l'équation des milieux poreux dans RN.C.R
    • Aronson, D. G., Bénilan, Ph.: Régularité des solutions de l'équation des milieux poreux dans RN. C. R. Acad. Sci. Paris Sér. A-B 288(2), A103-A105 (1979).
    • (1979) Acad. Sci. Paris Sér. A-B , vol.288 , Issue.2 , pp. A103-A105
    • Aronson, D.G.1    Bénilan, P.2
  • 2
    • 0005101459 scopus 로고    scopus 로고
    • Limit behaviour of focusing solutions to nonlinear diffusions
    • Aronson, D. G., Gil, O., Vázquez, J. L.: Limit behaviour of focusing solutions to nonlinear diffusions. Comm. Partial Differ. Equ. 23(1-2), 307-332 (1998).
    • (1998) Comm. Partial Differ. Equ , vol.23 , Issue.1-2 , pp. 307-332
    • Aronson, D.G.1    Gil, O.2    Vázquez, J.L.3
  • 3
    • 44349107599 scopus 로고    scopus 로고
    • On the foundations of cancer modelling: Selected topics, speculations, and perspectives
    • Bellomo, N., Li, N. K., Maini, P. K.: On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 18(4), 593-646 (2008).
    • (2008) Math. Models Methods Appl. Sci , vol.18 , Issue.4 , pp. 593-646
    • Bellomo, N.1    Li, N.K.2    Maini, P.K.3
  • 4
    • 0034254401 scopus 로고    scopus 로고
    • Modelling and mathematical problems related to tumor evolution and its interaction with the immune system
    • Bellomo, N., Preziosi, L.: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Model. 32(3-4), 413-542 (2000).
    • (2000) Math. Comput. Model , vol.32 , Issue.3-4 , pp. 413-542
    • Bellomo, N.1    Preziosi, L.2
  • 6
    • 0000994591 scopus 로고
    • The continuous dependence on ϕ of solutions of ut-Δϕ(u) = 0
    • Bénilan, Ph., Crandall, M. G.: The continuous dependence on ϕ of solutions of ut-Δϕ(u) = 0. Indiana Univ. Math. J. 30(2), 161-177 (1981).
    • (1981) Indiana Univ. Math. J , vol.30 , Issue.2 , pp. 161-177
    • Bénilan, P.1    Crandall, M.G.2
  • 7
    • 0001845039 scopus 로고
    • Solutions of the porous medium equation inRN under optimal conditions on the initial values
    • Bénilan, Ph., Crandall, M. G., Pierre, M.: Solutions of the porous medium equation inRN under optimal conditions on the initial values. Indiana Univ. Math. J. 33(1), 51-87 (1984).
    • (1984) Indiana Univ. Math. J , vol.33 , Issue.1 , pp. 51-87
    • Bénilan, P.1    Crandall, M.G.2    Pierre, M.3
  • 8
    • 0141608309 scopus 로고    scopus 로고
    • Singular limit of perturbed nonlinear semigroups
    • Bénilan, Ph., Igbida, N.: Singular limit of perturbed nonlinear semigroups. Commun. Appl. Nonlinear Anal. 3(4), 23-42 (1996).
    • (1996) Commun. Appl. Nonlinear Anal , vol.3 , Issue.4 , pp. 23-42
    • Bénilan, P.1    Igbida, N.2
  • 9
    • 0141720042 scopus 로고
    • La limite de la solution de ut = Δpum lorsque m →∞
    • Bénilan, Ph., Igbida, N.: La limite de la solution de ut = Δpum lorsque m →∞. C. R. Acad. Sci. Paris Sér. I Math. 321(10), 1323-1328 (1995).
    • (1995) C.R. Acad. Sci. Paris Sér. I Math , vol.321 , Issue.10 , pp. 1323-1328
    • Bénilan, P.1    Igbida, N.2
  • 10
    • 0000469757 scopus 로고
    • Traveling wave solutions to combustion models and their singular limits
    • Berestycki, H., Nicolaenko, B., Scheurer, B.: Traveling wave solutions to combustion models and their singular limits. SIAM J. Math. Anal. 16(6), 1207-1242 (1985).
    • (1985) SIAM J. Math. Anal , vol.16 , Issue.6 , pp. 1207-1242
    • Berestycki, H.1    Nicolaenko, B.2    Scheurer, B.3
  • 11
    • 77949814436 scopus 로고    scopus 로고
    • The Hele-Shaw problem as a "mesa" limit of Stefan problems: Existence, uniqueness, and regularity of the free boundary
    • Blank, I. A., Korten, M. K., Moore, C. N.: The Hele-Shaw problem as a "mesa" limit of Stefan problems: existence, uniqueness, and regularity of the free boundary. Trans. Am. Math. Soc. 361(3), 1241-1268 (2009).
    • (2009) Trans. Am. Math. Soc , vol.361 , Issue.3 , pp. 1241-1268
    • Blank, I.A.1    Korten, M.K.2    Moore, C.N.3
  • 13
    • 0030586186 scopus 로고    scopus 로고
    • Growth of necrotic tumors in the presence and absence of inhibitors
    • Byrne, H. M., Chaplain, M. A.: Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci. 135(15), 187-216 (1996).
    • (1996) Math. Biosci , vol.135 , Issue.15 , pp. 187-216
    • Byrne, H.M.1    Chaplain, M.A.2
  • 14
    • 58349117422 scopus 로고    scopus 로고
    • Individual-based and continuum models of growing cell populations: A comparison
    • Byrne, H. M., Drasdo, D.: Individual-based and continuum models of growing cell populations: a comparison. J. Math. Biol. 58(4-5), 657-687 (2009).
    • (2009) J. Math. Biol , vol.58 , Issue.4-5 , pp. 657-687
    • Byrne, H.M.1    Drasdo, D.2
  • 15
    • 0842281982 scopus 로고    scopus 로고
    • Modelling solid tumour growth using the theory of mixtures
    • Byrne, H. M., Preziosi, L.: Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20(4), 341-366 (2003).
    • (2003) Math. Med. Biol , vol.20 , Issue.4 , pp. 341-366
    • Byrne, H.M.1    Preziosi, L.2
  • 16
    • 0002964937 scopus 로고    scopus 로고
    • Avascular growth, angiogenesis and vascular growth in solid tumours: Themathematical modeling of the stages of tumor development
    • Chaplain, M. A. J.: Avascular growth, angiogenesis and vascular growth in solid tumours: themathematical modeling of the stages of tumor development. Math. Comput. Model. 23(6), 47-87 (1996).
    • (1996) Math. Comput. Model , vol.23 , Issue.6 , pp. 47-87
    • Chaplain, M.A.J.1
  • 18
    • 0005011575 scopus 로고
    • On the weak solution of moving boundary problems
    • Crowley, A. B.: On the weak solution of moving boundary problems. J. Inst. Math. Appl. 24(1), 43-57 (1979).
    • (1979) J. Inst. Math. Appl , vol.24 , Issue.1 , pp. 43-57
    • Crowley, A.B.1
  • 19
    • 33750625405 scopus 로고    scopus 로고
    • Formation of necrotic cores in the growth of tumors: Analytic results
    • Cui, S.: Formation of necrotic cores in the growth of tumors: analytic results. Acta Math. Sci. Ser. B Engl. Ed. 26(4), 781-796 (2006).
    • (2006) Acta Math. Sci. Ser. B Engl. Ed , vol.26 , Issue.4 , pp. 781-796
    • Cui, S.1
  • 20
    • 41849134265 scopus 로고    scopus 로고
    • Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth
    • Cui, S., Escher, J.: Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth. Commun. PartialDiffer. Equ. 33(4-6), 636-655 (2008).
    • (2008) Commun. PartialDiffer. Equ , vol.33 , Issue.4-6 , pp. 636-655
    • Cui, S.1    Escher, J.2
  • 21
    • 84967787229 scopus 로고
    • The ill-posed Hele-Shaw model and the Stefan problem for supercooled water
    • Di Benedetto, E., Friedman, A.: The ill-posed Hele-Shaw model and the Stefan problem for supercooled water. Trans. Am. Math. Soc. 282(1), 183-204 (1984).
    • (1984) Trans. Am. Math. Soc , vol.282 , Issue.1 , pp. 183-204
    • Di Benedetto, E.1    Friedman, A.2
  • 22
    • 79960606367 scopus 로고    scopus 로고
    • Ablative Hele-Shaw model for ICF flows modeling and numerical simulation
    • Egly, H., Després, B., Sentis, R.: Ablative Hele-Shaw model for ICF flows modeling and numerical simulation. Math. Models Methods Appl. Sci. 21(7), 1571-1600 (2011).
    • (2011) Math. Models Methods Appl. Sci , vol.21 , Issue.7 , pp. 1571-1600
    • Egly, H.1    Després, B.2    Sentis, R.3
  • 23
    • 0000016784 scopus 로고
    • The mesa problem: Diffusion patterns for ut = ∇(um∇u) as m →∞
    • Elliot, C. M., Herrero, M. A., King, J. R., Ockendon, J. R.: The mesa problem: diffusion patterns for ut = ∇(um∇u) as m →∞. IMA J. Appl. Math. 37(2), 147-154 (1986).
    • (1986) IMA J. Appl. Math , vol.37 , Issue.2 , pp. 147-154
    • Elliot, C.M.1    Herrero, M.A.2    King, J.R.3    Ockendon, J.R.4
  • 24
    • 84971155357 scopus 로고
    • A variational inequality approach to Hele-Shaw flow with a moving boundary
    • Elliot, C. M., Janovsky, V.: A variational inequality approach to Hele-Shaw flow with a moving boundary. Proc. R. Soc. Edinburgh. Sect. A 88(1-2), 93-107 (1981).
    • (1981) Proc. R. Soc. Edinburgh. Sect. A , vol.88 , Issue.1-2 , pp. 93-107
    • Elliot, C.M.1    Janovsky, V.2
  • 27
    • 70349241484 scopus 로고    scopus 로고
    • Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model
    • Friedman, A., Hu, B.: Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model. Trans. Am. Math. Soc. 360(10), 5291-5342 (2008).
    • (2008) Trans. Am. Math. Soc , vol.360 , Issue.10 , pp. 5291-5342
    • Friedman, A.1    Hu, B.2
  • 28
    • 0010783253 scopus 로고
    • Asymptotic behavior of solutions of ut = Δφm(u) as m → ∞with inconsistent initial values
    • Gauthier-Villars, Paris
    • Friedman, A., Huang, S. Y.: Asymptotic behavior of solutions of ut = Δφm(u) as m → ∞with inconsistent initial values. In: Analyse Mathématique et applications. Gauthier-Villars, Paris, 165-180, 1988.
    • (1988) Analyse Mathématique et applications , pp. 165-180
    • Friedman, A.1    Huang, S.Y.2
  • 29
    • 0035371717 scopus 로고    scopus 로고
    • Convergence of the porous media equation toHele-Shaw
    • Gil, O., Quirós, F.: Convergence of the porous media equation toHele-Shaw. Nonlinear Anal. Ser. A Theory Methods 44(8), 1111-1131 (2001).
    • (2001) Nonlinear Anal. Ser. A Theory Methods , vol.44 , Issue.8 , pp. 1111-1131
    • Gil, O.1    Quirós, F.2
  • 30
    • 0346724405 scopus 로고    scopus 로고
    • Boundary layer formation in the transition from the porous media equation to a Hele-Shaw flow
    • Gil, O., Quirós, F.: Boundary layer formation in the transition from the porous media equation to a Hele-Shaw flow. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(1), 13-36 (2003).
    • (2003) Ann. Inst. H. Poincaré Anal. Non Linéaire , vol.20 , Issue.1 , pp. 13-36
    • Gil, O.1    Quirós, F.2
  • 32
    • 0000128125 scopus 로고
    • Models for the growth of a solid tumor by diffusion
    • Greenspan, H. P. Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. 51(4), 317-340 (1972).
    • (1972) Stud. Appl. Math , vol.51 , Issue.4 , pp. 317-340
    • Greenspan, H.P.1
  • 33
    • 0141720035 scopus 로고    scopus 로고
    • Themesa-limit of the porous medium equation and theHele-Shawproblem
    • Igbida, N.: Themesa-limit of the porous medium equation and theHele-Shawproblem. Differ. Integr. Equ. 15(2), 129-146 (2002).
    • (2002) Differ. Integr. Equ , vol.15 , Issue.2 , pp. 129-146
    • Igbida, N.1
  • 34
    • 0037055768 scopus 로고    scopus 로고
    • Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations
    • Jakobsen, E. R., Karlsen, K. H.: Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations. J. Differ. Equ. 183(2), 497-525 (2002).
    • (2002) J. Differ. Equ , vol.183 , Issue.2 , pp. 497-525
    • Jakobsen, E.R.1    Karlsen, K.H.2
  • 35
    • 0043245213 scopus 로고    scopus 로고
    • Uniqueness and existence results on viscosity solutions of the Hele-Shaw and the Stefan problems
    • Kim, I. C.: Uniqueness and existence results on viscosity solutions of the Hele-Shaw and the Stefan problems. Arch. Rat. Mech. Anal. 168(4), 299-328 (2003).
    • (2003) Arch. Rat. Mech. Anal , vol.168 , Issue.4 , pp. 299-328
    • Kim, I.C.1
  • 36
    • 70349490442 scopus 로고    scopus 로고
    • Homogenization of a Hele-Shaw problem in periodic and random media
    • Kim, I. C., Mellet, A.: Homogenization of a Hele-Shaw problem in periodic and random media. Arch. Ration. Mech. Anal. 194(2), 507-530 (2009).
    • (2009) Arch. Ration. Mech. Anal , vol.194 , Issue.2 , pp. 507-530
    • Kim, I.C.1    Mellet, A.2
  • 38
    • 58349095136 scopus 로고    scopus 로고
    • Multiphase modelling of tumour growth and extracellularmatrix interaction: Mathematical tools and applications
    • Preziosi, L., Tosin, A.: Multiphase modelling of tumour growth and extracellularmatrix interaction: mathematical tools and applications. J. Math. Biol. 58(4-5), 625-656 (2009).
    • (2009) J. Math. Biol , vol.58 , Issue.4-5 , pp. 625-656
    • Preziosi, L.1    Tosin, A.2
  • 40
    • 38249022046 scopus 로고
    • A singular limit problem for the porous medium equation
    • Sacks, P. E.: A singular limit problem for the porous medium equation. J. Math. Anal. Appl. 140(2), 456-466 (1989).
    • (1989) J. Math. Anal. Appl , vol.140 , Issue.2 , pp. 456-466
    • Sacks, P.E.1
  • 41
    • 84951609574 scopus 로고
    • Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations
    • Sánchez-Garduño, F., Maini, P. K.: Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations. J. Math. Biol. 33(2), 163-192 (1994).
    • (1994) J. Math. Biol , vol.33 , Issue.2 , pp. 163-192
    • Sánchez-Garduño, F.1    Maini, P.K.2
  • 44
    • 84894531156 scopus 로고    scopus 로고
    • Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, ISBN: 978-0-19-856903-9
    • Vázquez, J. L.: The porous medium equation. Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. ISBN: 978-0-19-856903-9.
    • (2007) The porous medium equation
    • Vázquez, J.L.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.