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Analysis of a delayed free boundary problem for tumor growth
Traveling waves for models of phase transitions of solids driven by configurational forces
1. | Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan |
2. | Basque Center for Applied Mathematics, Building 500, Bizkaia Technology Park, E-48160 Derio, Spain |
References:
[1] |
R. Abeyaratne and J. Knowles, On the driving traction acting on a surface of strain discontinuity in a continuum, J. Mech. Phys. Solids, 38 (1990), 345-360.
doi: doi:10.1016/0022-5096(90)90003-M. |
[2] |
H. D. Alber, Evolving microstructure and homogenization, Continuum. Mech. Thermodyn., 12 (2000), 235-287.
doi: doi:10.1007/s001610050137. |
[3] |
H. D. Alber and P. Zhu, Evolution of phase boundaries by configurational forces, Archive Rat. Mech. Anal., 185 (2007), 235-286.
doi: doi:10.1007/s00205-007-0054-8. |
[4] |
H. D. Alber and P. Zhu, Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces, SIAM J. Appl. Math., 66 (2006), 680-699.
doi: doi:10.1137/050629951. |
[5] |
H. D. Alber and P. Zhu, Solutions to a model for interface motion by interface diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 923-955.
doi: doi:10.1017/S0308210507000170. |
[6] |
H. D. Alber and P. Zhu, Interface motion by interface diffusion driven by bulk energy: Justification of a diffusive interface model, preprint, accepted for publication in Conti. Mech. Thermodyna, 2010. |
[7] |
S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Met., 27 (1979), 1084-1095.
doi: doi:10.1016/0001-6160(79)90196-2. |
[8] |
D. Aronson and H. Weiberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: doi:10.1016/0001-8708(78)90130-5. |
[9] |
H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations," Contemp. Mathematics, 446, Amer. Math. Soc., Providence, RI, (2007), 101-123. |
[10] |
T. Blesgen and U. Weikard, Multi-component Allen-Cahn equation for elastically stressed solids, Electronic J. Diff. Eqs., 2005 (2005), 1-17. |
[11] |
J. Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, J. Chem. Phys., 30 (1959), 1121-1124.
doi: doi:10.1063/1.1730145. |
[12] |
J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1959), 258-267.
doi: doi:10.1063/1.1744102. |
[13] |
J. Cahn and J. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699.
doi: doi:10.1063/1.1730447. |
[14] |
J. Cahn and J. Taylor, Surface motion by surface diffusion, Acta Metall. Mater., 42 (1994), 1045-1063.
doi: doi:10.1016/0956-7151(94)90123-6. |
[15] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Diff. Eq., 96 (1992), 116-141. |
[16] |
P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer Verlag, 1979. |
[17] |
P. Fife and J. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rati. Mech. Anal., 65 (1977), 335-361.
doi: doi:10.1007/BF00250432. |
[18] |
M. Gurtin, "Configurational Forces as Basic Concepts of Continuum Physics," Springer Verlag, New York, 2000. |
[19] |
H. Garcke, On the Cahn-Hilliard system with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331.
doi: doi:10.1017/S0308210500002419. |
[20] |
E. Hornbogen and H. Warlimont, "Metallkunde," 4th edition, Springer-Verlag, 2001. |
[21] |
Y. Kanel, On the stabilization of solutions of the Cauchy problem for equations arising in the theory of combustion, Mat. Sbornik, 59 (1962), 245-288. |
[22] |
W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), 333-339.
doi: doi:10.1063/1.1722742. |
[23] |
H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Diff. Eq., 213 (2005), 204-233. |
[24] |
H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Disc. Conti. Dyna. Syst., 15 (2006), 819-832.
doi: doi:10.3934/dcds.2006.15.819. |
[25] |
R. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Soc. Lond. Sect. A, 422 (1989), 261-278.
doi: doi:10.1098/rspa.1989.0027. |
[26] |
J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer Verlag, New York, 1983. |
[27] |
M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: doi:10.1137/060661788. |
[28] |
J. Taylor and J. Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Stat. Phys., 77 (1994), 183-197.
doi: doi:10.1007/BF02186838. |
[29] |
A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, Rhode Island, 1994. |
show all references
References:
[1] |
R. Abeyaratne and J. Knowles, On the driving traction acting on a surface of strain discontinuity in a continuum, J. Mech. Phys. Solids, 38 (1990), 345-360.
doi: doi:10.1016/0022-5096(90)90003-M. |
[2] |
H. D. Alber, Evolving microstructure and homogenization, Continuum. Mech. Thermodyn., 12 (2000), 235-287.
doi: doi:10.1007/s001610050137. |
[3] |
H. D. Alber and P. Zhu, Evolution of phase boundaries by configurational forces, Archive Rat. Mech. Anal., 185 (2007), 235-286.
doi: doi:10.1007/s00205-007-0054-8. |
[4] |
H. D. Alber and P. Zhu, Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces, SIAM J. Appl. Math., 66 (2006), 680-699.
doi: doi:10.1137/050629951. |
[5] |
H. D. Alber and P. Zhu, Solutions to a model for interface motion by interface diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 923-955.
doi: doi:10.1017/S0308210507000170. |
[6] |
H. D. Alber and P. Zhu, Interface motion by interface diffusion driven by bulk energy: Justification of a diffusive interface model, preprint, accepted for publication in Conti. Mech. Thermodyna, 2010. |
[7] |
S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Met., 27 (1979), 1084-1095.
doi: doi:10.1016/0001-6160(79)90196-2. |
[8] |
D. Aronson and H. Weiberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: doi:10.1016/0001-8708(78)90130-5. |
[9] |
H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations," Contemp. Mathematics, 446, Amer. Math. Soc., Providence, RI, (2007), 101-123. |
[10] |
T. Blesgen and U. Weikard, Multi-component Allen-Cahn equation for elastically stressed solids, Electronic J. Diff. Eqs., 2005 (2005), 1-17. |
[11] |
J. Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, J. Chem. Phys., 30 (1959), 1121-1124.
doi: doi:10.1063/1.1730145. |
[12] |
J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1959), 258-267.
doi: doi:10.1063/1.1744102. |
[13] |
J. Cahn and J. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699.
doi: doi:10.1063/1.1730447. |
[14] |
J. Cahn and J. Taylor, Surface motion by surface diffusion, Acta Metall. Mater., 42 (1994), 1045-1063.
doi: doi:10.1016/0956-7151(94)90123-6. |
[15] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Diff. Eq., 96 (1992), 116-141. |
[16] |
P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer Verlag, 1979. |
[17] |
P. Fife and J. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rati. Mech. Anal., 65 (1977), 335-361.
doi: doi:10.1007/BF00250432. |
[18] |
M. Gurtin, "Configurational Forces as Basic Concepts of Continuum Physics," Springer Verlag, New York, 2000. |
[19] |
H. Garcke, On the Cahn-Hilliard system with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331.
doi: doi:10.1017/S0308210500002419. |
[20] |
E. Hornbogen and H. Warlimont, "Metallkunde," 4th edition, Springer-Verlag, 2001. |
[21] |
Y. Kanel, On the stabilization of solutions of the Cauchy problem for equations arising in the theory of combustion, Mat. Sbornik, 59 (1962), 245-288. |
[22] |
W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), 333-339.
doi: doi:10.1063/1.1722742. |
[23] |
H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Diff. Eq., 213 (2005), 204-233. |
[24] |
H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Disc. Conti. Dyna. Syst., 15 (2006), 819-832.
doi: doi:10.3934/dcds.2006.15.819. |
[25] |
R. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Soc. Lond. Sect. A, 422 (1989), 261-278.
doi: doi:10.1098/rspa.1989.0027. |
[26] |
J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer Verlag, New York, 1983. |
[27] |
M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: doi:10.1137/060661788. |
[28] |
J. Taylor and J. Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Stat. Phys., 77 (1994), 183-197.
doi: doi:10.1007/BF02186838. |
[29] |
A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, Rhode Island, 1994. |
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