Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluidlike tissue under the action of inhibitors
Authors:
Junde Wu and Fujun Zhou
Journal:
Trans. Amer. Math. Soc. 365 (2013), 41814207
MSC (2010):
Primary 35B40, 35R35; Secondary 35Q92, 92C37
Published electronically:
January 28, 2013
MathSciNet review:
3055693
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Abstract: In this paper we study a free boundary problem modeling the growth of tumors with fluidlike tissue under the action of inhibitors. The model includes two elliptic equations describing the concentration of nutrients and inhibitors, respectively, and a Stokes equation for the fluid velocity and internal pressure. By employing the functional approach, analytic semigroup theory and Cui's local phase theorem for parabolic differential equations with invariance, we prove that if a radial stationary solution is asymptotically stable under radial perturbations, then there exists a nonnegative threshold value such that if , then it keeps asymptotically stable under nonradial perturbations. While if , then the radial stationary solution is unstable and, in particular, there exists a centerstable manifold such that if the transient solution exists globally and is contained in a sufficiently small neighborhood of the radial stationary solution, then it converges exponentially to this radial stationary solution (modulo translations) and its translation lies on the centerstable manifold. The result indicates an interesting phenomenon that an increasing inhibitor uptake has a positive effect on the tumor's treatment and can promote the tumor's stability.
 1.
Herbert
Amann, Linear and quasilinear parabolic problems. Vol. I,
Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc.,
Boston, MA, 1995. Abstract linear theory. MR 1345385
(96g:34088)
 2.
Borys
Bazaliy and Avner
Friedman, Global existence and asymptotic stability for an
ellipticparabolic free boundary problem: an application to a model of
tumor growth, Indiana Univ. Math. J. 52 (2003),
no. 5, 1265–1304. MR 2010327
(2004j:35302), 10.1512/iumj.2003.52.2317
 3.
Helen
M. Byrne, A weakly nonlinear analysis of a model of avascular solid
tumour growth, J. Math. Biol. 39 (1999), no. 1,
59–89. MR
1705626 (2000i:92011), 10.1007/s002850050163
 4.
H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151181.
 5.
H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187216.
 6.
Xinfu
Chen, Shangbin
Cui, and Avner
Friedman, A hyperbolic free boundary problem
modeling tumor growth: asymptotic behavior, Trans. Amer. Math. Soc. 357 (2005), no. 12, 4771–4804 (electronic). MR 2165387
(2006d:34144), 10.1090/S0002994705037840
 7.
Shangbin
Cui, Analysis of a mathematical model for the growth of tumors
under the action of external inhibitors, J. Math. Biol.
44 (2002), no. 5, 395–426. MR 1908130
(2003f:92019), 10.1007/s002850100130
 8.
Shangbin
Cui, Wellposedness of a multidimensional free boundary problem
modelling the growth of nonnecrotic tumors, J. Funct. Anal.
245 (2007), no. 1, 1–18. MR 2310801
(2009d:35349), 10.1016/j.jfa.2006.12.020
 9.
Shangbin
Cui, Lie group action and stability analysis of stationary
solutions for a free boundary problem modelling tumor growth, J.
Differential Equations 246 (2009), no. 5,
1845–1882. MR 2494690
(2010e:35305), 10.1016/j.jde.2008.10.014
 10.
Shangbin
Cui and Joachim
Escher, Asymptotic behaviour of solutions of a multidimensional
moving boundary problem modeling tumor growth, Comm. Partial
Differential Equations 33 (2008), no. 46,
636–655. MR 2424371
(2009k:35331), 10.1080/03605300701743848
 11.
Joachim
Escher, Classical solutions to a moving boundary problem for an
ellipticparabolic system, Interfaces Free Bound. 6
(2004), no. 2, 175–193. MR 2079602
(2006a:35316), 10.4171/IFB/96
 12.
S.
J. Franks, H.
M. Byrne, J.
R. King, J.
C. E. Underwood, and C.
E. Lewis, Modelling the early growth of ductal carcinoma in situ of
the breast, J. Math. Biol. 47 (2003), no. 5,
424–452. MR 2029006
(2004j:92030), 10.1007/s002850030214x
 13.
S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the growth of comedo ductal carcinoma in situ, Math. Med. Biol., 20 (2003), 277308.
 14.
S.
J. Franks, H.
M. Byrne, J.
C. E. Underwood, and C.
E. Lewis, Biological inferences from a mathematical model of comedo
ductal carcinoma in situ of the breast, J. Theoret. Biol.
232 (2005), no. 4, 523–543. MR
2125830, 10.1016/j.jtbi.2004.08.032
 15.
S. J. H. Franks and J. P. King, Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties, Math. Med. Biol., 20 (2003), 4789.
 16.
Avner
Friedman, A free boundary problem for a coupled system of elliptic,
hyperbolic, and Stokes equations modeling tumor growth, Interfaces
Free Bound. 8 (2006), no. 2, 247–261. MR 2256843
(2007e:35299), 10.4171/IFB/142
 17.
Avner
Friedman, Mathematical analysis and challenges arising from models
of tumor growth, Math. Models Methods Appl. Sci. 17
(2007), no. suppl., 1751–1772. MR 2362763
(2009d:92028), 10.1142/S0218202507002467
 18.
Avner
Friedman and Bei
Hu, Bifurcation for a free boundary problem modeling tumor growth
by Stokes equation, SIAM J. Math. Anal. 39 (2007),
no. 1, 174–194. MR 2318381
(2008f:35407), 10.1137/060656292
 19.
Avner
Friedman and Bei
Hu, Bifurcation from stability to instability for a free boundary
problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl.
327 (2007), no. 1, 643–664. MR 2277439
(2008e:35213), 10.1016/j.jmaa.2006.04.034
 20.
Avner
Friedman and Fernando
Reitich, Analysis of a mathematical model for the growth of
tumors, J. Math. Biol. 38 (1999), no. 3,
262–284. MR 1684873
(2001f:92011), 10.1007/s002850050149
 21.
Avner
Friedman and Fernando
Reitich, Symmetrybreaking bifurcation of
analytic solutions to free boundary problems: an application to a model of
tumor growth, Trans. Amer. Math. Soc.
353 (2001), no. 4,
1587–1634 (electronic). MR 1806728
(2002a:35208), 10.1090/S000299470002715X
 22.
H.
P. Greenspan, On the growth and stability of cell cultures and
solid tumors, J. Theoret. Biol. 56 (1976),
no. 1, 229–242. MR 0429164
(55 #2183)
 23.
E.
L. Hill, The theory of vector spherical harmonics, Amer. J.
Phys. 22 (1954), 211–214. MR 0061226
(15,792h)
 24.
J.
S. Lowengrub, H.
B. Frieboes, F.
Jin, Y.L.
Chuang, X
Li, P.
Macklin, S.
M. Wise, and V.
Cristini, Nonlinear modelling of cancer: bridging the gap between
cells and tumours, Nonlinearity 23 (2010),
no. 1, R1–R91. MR
2576370, 10.1088/09517715/23/1/001
 25.
Alessandra
Lunardi, Analytic semigroups and optimal regularity in parabolic
problems, Progress in Nonlinear Differential Equations and their
Applications, 16, Birkhäuser Verlag, Basel, 1995. MR 1329547
(96e:47039)
 26.
G.
N. Watson, A Treatise on the Theory of Bessel Functions,
Cambridge University Press, Cambridge, England; The Macmillan Company, New
York, 1944. MR
0010746 (6,64a)
 27.
Junde
Wu and Shangbin
Cui, Asymptotic behaviour of solutions of a free boundary problem
modelling the growth of tumours in the presence of inhibitors,
Nonlinearity 20 (2007), no. 10, 2389–2408. MR 2356116
(2009b:35446), 10.1088/09517715/20/10/007
 28.
Junde
Wu and Shangbin
Cui, Asymptotic behavior of solutions of a free boundary problem
modelling the growth of tumors with Stokes equations, Discrete Contin.
Dyn. Syst. 24 (2009), no. 2, 625–651. MR 2486594
(2010g:35349), 10.3934/dcds.2009.24.625
 29.
Junde
Wu and Shangbin
Cui, Asymptotic stability of stationary solutions of a free
boundary problem modeling the growth of tumors with fluid tissues,
SIAM J. Math. Anal. 41 (2009), no. 1, 391–414.
MR
2505864 (2010m:35578), 10.1137/080726550
 30.
Junde
Wu and Shangbin
Cui, Asymptotic behavior of solutions for parabolic differential
equations with invariance and applications to a free boundary problem
modeling tumor growth, Discrete Contin. Dyn. Syst. 26
(2010), no. 2, 737–765. MR 2556506
(2011d:35528), 10.3934/dcds.2010.26.737
 31.
Fujun
Zhou and Shangbin
Cui, Wellposedness and stability of a multidimensional moving
boundary problem modeling the growth of tumor cord, Discrete Contin.
Dyn. Syst. 21 (2008), no. 3, 929–943. MR 2399443
(2009c:35487), 10.3934/dcds.2008.21.929
 32.
Fujun
Zhou and Junde
Wu, Regularity of solutions to a free boundary problem modeling
tumor growth by Stokes equation, J. Math. Anal. Appl.
377 (2011), no. 2, 540–556. MR 2769156
(2012b:35378), 10.1016/j.jmaa.2010.11.028
 1.
 H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I, Birkhäuser, Basel, 1995. MR 1345385 (96g:34088)
 2.
 B. Bazaliy and A. Friedman, Global existence and asymptotic stability for an ellipticparabolic free boundary problem: An application to a model of tumor growth, Indiana Univ. Math. J., 52 (2003), 12651304. MR 2010327 (2004j:35302)
 3.
 H. M. Byrne, A weakly nonlinear analysis of a model of avascular solid tumor growth, J. Math. Biol., 39 (1999), 5989. MR 1705626 (2000i:92011)
 4.
 H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151181.
 5.
 H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187216.
 6.
 X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Trans. Amer. Math. Soc., 357 (2005), 47714804. MR 2165387 (2006d:34144)
 7.
 S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395426. MR 1908130 (2003f:92019)
 8.
 S. Cui, Wellposedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors, J. Funct. Anal., 245 (2007), 118. MR 2310801 (2009d:35349)
 9.
 S. Cui, Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth, J. Differential Equations, 246 (2009), 18451882. MR 2494690 (2010e:35305)
 10.
 S. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636655. MR 2424371 (2009k:35331)
 11.
 J. Escher, Classical solutions to a moving boundary problem for an ellipticparabolic system, Interfaces Free Bound., 6 (2004), 175193. MR 2079602 (2006a:35316)
 12.
 S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the early growth of ductal carcinoma in situ of the breast, J. Math. Biol., 47 (2003), 424452. MR 2029006 (2004j:92030)
 13.
 S. J. H. Franks, H. M. Byrne, J. P. King, J. C. E. Underwood and C. E. Lewis, Modelling the growth of comedo ductal carcinoma in situ, Math. Med. Biol., 20 (2003), 277308.
 14.
 S. J. H. Franks, H. M. Byrne, J. C. E. Underwood and C. E. Lewis, Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast, J. Theoret. Biol., 232 (2005), 523543. MR 2125830
 15.
 S. J. H. Franks and J. P. King, Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties, Math. Med. Biol., 20 (2003), 4789.
 16.
 A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic and Stokes equations modeling tumor growth, Interfaces Free Bound., 8 (2006), 247261. MR 2256843 (2007e:35299)
 17.
 A. Friedman, Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17, suppl. (2007), 17511772. MR 2362763 (2009d:92028)
 18.
 A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174194. MR 2318381 (2008f:35407)
 19.
 A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl., 327 (2007), 643664. MR 2277439 (2008e:35213)
 20.
 A. Friedman and F. Reitich, Analysis of a mathematical model for growth of tumor, J. Math. Biol., 38 (1999), 262284. MR 1684873 (2001f:92011)
 21.
 A. Friedman and F. Reitich, Symmetrybreaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth, Trans. Amer. Math. Soc., 353 (2001), 15871634. MR 1806728 (2002a:35208)
 22.
 H. P. Greenspan, On the growth of cell culture and solid tumors, J. Theoret. Biol., 56 (1976), 229242. MR 0429164 (55:2183)
 23.
 E. L. Hill, The theory of vector spherical harmonics, Amer. J. Phys., 222 (1954), 211214. MR 0061226 (15:792h)
 24.
 J. S. Lowengrub, H. B. Frieboes, F. Jin, et al., Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1R91. MR 2576370
 25.
 A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. MR 1329547 (96e:47039)
 26.
 G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944. MR 0010746 (6:64a)
 27.
 J. Wu and S. Cui, Asymptotic behaviour of solutions of a free boundary problem modelling the growth of tumours in the presence of inhibitors, Nonlinearity, 20 (2007), 23892408. MR 2356116 (2009b:35446)
 28.
 J. Wu and S. Cui, Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with Stokes equations, Discrete Contin. Dyn. Syst., 24 (2009), 625651. MR 2486594 (2010g:35349)
 29.
 J. Wu and S. Cui, Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues, SIAM J. Math. Anal., 41 (2009), 391414. MR 2505864 (2010m:35578)
 30.
 J. Wu and S. Cui, Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth, Discrete Contin. Dyn. Syst., 26 (2010), 737765. MR 2556506 (2011d:35528)
 31.
 F. Zhou and S. Cui, Wellposedness and stability of a multidimentional moving boundary problem modeling the growth of tumor cord, Discrete Contin. Dyn. Syst., 21 (2008), 929943. MR 2399443 (2009c:35487)
 32.
 F. Zhou and J. Wu, Regularity of solutions to a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl., 377 (2011), 540556. MR 2769156
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Additional Information
Junde Wu
Affiliation:
Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, People’s Republic of China
Email:
wjdmath@yahoo.com.cn
Fujun Zhou
Affiliation:
Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, People’s Republic of China
Email:
zhoufujunht@yahoo.com.cn
DOI:
http://dx.doi.org/10.1090/S000299472013057790
PII:
S 00029947(2013)057790
Keywords:
Free boundary problem,
tumor growth,
Stokes equation,
radial stationary solution,
asymptotic stability
Received by editor(s):
May 23, 2011
Received by editor(s) in revised form:
November 18, 2011
Published electronically:
January 28, 2013
Additional Notes:
This work was supported by the National Natural Science Foundation of China under the grant numbers 10901057 and 11001192, the Doctoral Foundation of Education Ministry of China under the grant numbers 200805611027 and 20103201120017, the Natural Science Fund for Colleges and Universities in Jiangsu Province under the grant number 10KJB110008, and the Fundamental Research Funds for the Central Universities of SCUT under the grant number 2012ZZ0072.
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
