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Multiple existence of traveling waves of a free boundary problem describing cell motility
Exponential stability of the traveling fronts for a viscous Fisher-KPP equation
1. | Center for PDE, East China Normal University, Shanghai, 200241, China, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China |
References:
[1] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[2] |
G. Barenblatt, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.
doi: 10.1016/0021-8928(60)90107-6. |
[3] |
M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Traveling Waves, Mem. Amer. Math. Soc., 44 (1983), No. 285, iv+190.
doi: 10.1090/memo/0285. |
[4] |
C. M. Cuesta, Linear stability analysis of travelling waves for a pseudo-parabolic Burgers' equation, Dyn. Partial Differ. Equ, 7 (2010), 77-105.
doi: 10.4310/DPDE.2010.v7.n1.a5. |
[5] |
C. J. van Duijn, L. A. Peletier and I. S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.
doi: 10.1137/05064518X. |
[6] |
C. J. van Duijn, L. A. Peletier and I. S. Pop, Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media, Nonlinear Anal. Real World Appl., 14 (2013), 1361-1383.
doi: 10.1016/j.nonrwa.2012.10.002. |
[7] |
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[8] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[9] |
R. Gardner and C. Jones, Traveling waves of a perturbed diffusion equation arising in a phase field model, Indiana U. Math. J., 39 (1990), 1197-1222.
doi: 10.1512/iumj.1990.39.39054. |
[10] |
F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Differential Equations, 249 (2010), 1726-1745.
doi: 10.1016/j.jde.2010.06.025. |
[11] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[12] |
F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. |
[13] |
C. K. R. T. Jones, Geometric singular perturbation theories, in Dynamical systems Lecture Notes in Mathematics 1609, Springer-Verlag, Berlin, (1995), 44-118.
doi: 10.1007/BFb0095239. |
[14] |
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. État Moscou Sér. Inter. A, 1 (1937), 1-26. |
[15] |
P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation, Math. Models Methods Appl. Sci., 22 (2012), 1250004, 33 pages.
doi: 10.1142/S0218202512500042. |
[16] |
K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differential Equations, 59 (1985), 44-70.
doi: 10.1016/0022-0396(85)90137-8. |
[17] |
C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Ration. Mech. Anal., 194 (2009), 887-925.
doi: 10.1007/s00205-008-0185-6. |
[18] |
V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.
doi: 10.1090/S0002-9947-03-03340-3. |
[19] |
R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.
doi: 10.1007/BF02101705. |
[20] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[21] |
R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math.Anal., 1 (1970), 1-26.
doi: 10.1137/0501001. |
[22] |
A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem, SIAM J. Math.Anal., 43 (2011), 228-252.
doi: 10.1137/090778833. |
[23] |
K. Uchiyama, The behaviour of solutions of some non-linear diffusion equation for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. |
[24] |
L. N. Wang, Y. P. Wu and T. Li, Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion, Physica D., 240 (2011), 971-983.
doi: 10.1016/j.physd.2011.02.003. |
show all references
References:
[1] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[2] |
G. Barenblatt, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.
doi: 10.1016/0021-8928(60)90107-6. |
[3] |
M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Traveling Waves, Mem. Amer. Math. Soc., 44 (1983), No. 285, iv+190.
doi: 10.1090/memo/0285. |
[4] |
C. M. Cuesta, Linear stability analysis of travelling waves for a pseudo-parabolic Burgers' equation, Dyn. Partial Differ. Equ, 7 (2010), 77-105.
doi: 10.4310/DPDE.2010.v7.n1.a5. |
[5] |
C. J. van Duijn, L. A. Peletier and I. S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.
doi: 10.1137/05064518X. |
[6] |
C. J. van Duijn, L. A. Peletier and I. S. Pop, Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media, Nonlinear Anal. Real World Appl., 14 (2013), 1361-1383.
doi: 10.1016/j.nonrwa.2012.10.002. |
[7] |
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[8] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[9] |
R. Gardner and C. Jones, Traveling waves of a perturbed diffusion equation arising in a phase field model, Indiana U. Math. J., 39 (1990), 1197-1222.
doi: 10.1512/iumj.1990.39.39054. |
[10] |
F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Differential Equations, 249 (2010), 1726-1745.
doi: 10.1016/j.jde.2010.06.025. |
[11] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[12] |
F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. |
[13] |
C. K. R. T. Jones, Geometric singular perturbation theories, in Dynamical systems Lecture Notes in Mathematics 1609, Springer-Verlag, Berlin, (1995), 44-118.
doi: 10.1007/BFb0095239. |
[14] |
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. État Moscou Sér. Inter. A, 1 (1937), 1-26. |
[15] |
P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation, Math. Models Methods Appl. Sci., 22 (2012), 1250004, 33 pages.
doi: 10.1142/S0218202512500042. |
[16] |
K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differential Equations, 59 (1985), 44-70.
doi: 10.1016/0022-0396(85)90137-8. |
[17] |
C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Ration. Mech. Anal., 194 (2009), 887-925.
doi: 10.1007/s00205-008-0185-6. |
[18] |
V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.
doi: 10.1090/S0002-9947-03-03340-3. |
[19] |
R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.
doi: 10.1007/BF02101705. |
[20] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[21] |
R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math.Anal., 1 (1970), 1-26.
doi: 10.1137/0501001. |
[22] |
A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem, SIAM J. Math.Anal., 43 (2011), 228-252.
doi: 10.1137/090778833. |
[23] |
K. Uchiyama, The behaviour of solutions of some non-linear diffusion equation for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. |
[24] |
L. N. Wang, Y. P. Wu and T. Li, Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion, Physica D., 240 (2011), 971-983.
doi: 10.1016/j.physd.2011.02.003. |
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