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The global existence and numerical method for the free boundary problem of ductal carcinoma in situ. (English) Zbl 1527.35505


MSC:

35R35 Free boundary problems for PDEs
35K55 Nonlinear parabolic equations
65N06 Finite difference methods for boundary value problems involving PDEs
92C50 Medical applications (general)
Full Text: DOI

References:

[1] ByrneHM, ChaplainMAJ. Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math Bios. 1995;130:151‐181. · Zbl 0836.92011
[2] FriedmanA. Free boundary problems in biology. Phil Trans R Soc A. 2015;373:20140368. · Zbl 1353.35291
[3] LiuK, XuY, XuD. Numerical algorithms for a free boundary problem model of DCIS and a related inverse problem. Appl Anal. 2020;99:1181‐1194. · Zbl 1442.65231
[4] WardJP, KingJP. Mathematical modelling of avascular‐tumor growth. Math Medi Biol. 1997;14:39‐70. · Zbl 0866.92011
[5] XuY. A free boundary problem model of ductal carcinoma in situ. Disc Cont Dyna Syst Ser B. 2004;4:337‐348. · Zbl 1050.35158
[6] XuY. A mathematical model of ductal carcinoma in situ and its characteristic stationary solutions. In: BegehrH (ed.) et al., eds. Advances in Analysis. World Scientific; 2005.
[7] XuY, GilbertR. Some inverse problems raised from a mathematical model of ductal carcinoma in situ. Math Comput Mod. 2009;49:814‐828. · Zbl 1165.65388
[8] FriedmanA, ReitichF. Analysis of a mathematical model for growth of tumors. J Math Biol. 1999;38:262‐284. · Zbl 0944.92018
[9] FriedmanA, ReitichF. Symmetry‐breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth. Trans Amer Math Soc. 2001;353:1587‐1634. · Zbl 0983.35019
[10] PettetGJ, PettetGJ, PleaseCP, TindallM, McElwainD. The migration of cells in multicell tumor spheroids. Bull Math Biol. 2001;63:231‐257. · Zbl 1323.92037
[11] CuiXCS, FriedmanA. A hyperbolic free boundary problem modeling tumor growth: asymptotic behavior. Trans Amer Math Soc. 2005;357:4771‐4804. · Zbl 1082.35166
[12] ChenX, FriedmanA. Free boundary problem for an elliptic‐hyperbolic system: an application to tumor growth. SIAM Math Anal. 2003;35:974‐986. · Zbl 1054.35144
[13] LiebermanGM. Second Order Parabolic Differential Equations. River Edge NJ: World Scientific Publishing Co., Inc.; 1996. · Zbl 0884.35001
[14] GidasB, NiW, NirenbergL. Symmetry and related properties via the maximum principle. Commun Math Phys. 1979;68:209‐243. · Zbl 0425.35020
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