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Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations
Graduate School of Science and Technology, Meiji University / JSPS Research Fellow, 1-1-1, Higashimita Tama-ku Kawasaki Kanagawa 214-8571, Japan |
Traveling wave solutions for the one-dimensional degenerate parabolic equations are considered. The purpose of this paper is to classify the nonnegative traveling wave solutions including sense of weak solutions of these equations and to present their existence, information about their shape and asymptotic behavior. These are studied by applying the framework that combines Poincaré compactification and classical dynamical systems theory. We also aim to use these results to generalize the results of our previous studies. The key to this is the introduction of a transformation, which overcomes the generalization difficulties faced by these studies.
References:
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M. J. Álvarez, A. Ferragut and X. Jarque,
A survey on the blow up technique, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3103-3118.
doi: 10.1142/S0218127411030416. |
[2] |
K. Anada, I. Fukuda and M. Tsutsumi,
Regional blow-up and decay of solutions to the initial-boundary value problem for $u_{t} = uu_xx-\gamma (u_{x})^{2}+ku^{2}$, Funkcial. Ekvac., 39 (1996), 367-387.
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K. Anada and T. Ishiwata,
Asymptotic behavior of blow-up solutions to a degenerate parabolic equation, J. Math-for-Ind., 2011 (2011), 1-8.
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[4] |
K. Anada and T. Ishiwata,
Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation, J. Differential Equations, 262 (2017), 181-271.
doi: 10.1016/j.jde.2016.09.023. |
[5] |
S. B. Angenent and J. J. L. Velázquez,
Asymptotic shape of cusp singularities in curve shortening, Duke Math. J., 77 (1995), 71-110.
doi: 10.1215/S0012-7094-95-07704-7. |
[6] |
M. Brunella and M. Miari,
Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra, J. Differential Equations, 85 (1990), 338-366.
doi: 10.1016/0022-0396(90)90120-E. |
[7] |
J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981. |
[8] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin Heidelberg, 2006. |
[9] |
M. Gage and R. S. Hamilton,
The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.
|
[10] |
M. A. Grayson,
The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.
|
[11] |
Y. Ichida, Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity, submitted. |
[12] |
Y. Ichida, K. Matsue and T. O. Sakamoto,
A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations, JSIAM Lett., 12 (2020), 65-68.
doi: 10.14495/jsiaml.12.65. |
[13] |
Y. Ichida and T. O. Sakamoto,
Quasi traveling waves with quenching in a reaction-diffusion equation in the presence of negative powers nonlinearity, Proc. Japan Acad. Ser. A Math. Sci., 96 (2020), 1-6.
doi: 10.3792/pjaa.96.001. |
[14] |
Y. Ichida and T. O. Sakamoto,
Radial symmetric stationary solutions for a MEMS type reaction-diffusion equation with spatially dependent nonlinearity, Jpn. J. Ind. Appl. Math., 38 (2021), 297-322.
doi: 10.1007/s13160-020-00438-8. |
[15] |
Y. Ichida and T. O. Sakamoto,
Traveling wave solutions for degenerate nonlinear parabolic equations, J. Elliptic Parabol. Equ., 6 (2020), 795-832.
doi: 10.1007/s41808-020-00080-y. |
[16] |
Y. Ichida and T. O. Sakamoto, Singular stationary solutions for a MEMS type reaction-diffusion equation with fringing field, submitted. |
[17] |
C. Kuehn, Multiple Time Scale Dynamics, Springer International Publishing, 2015.
doi: 10.1007/978-3-319-12316-5. |
[18] |
B. C. Low,
Resitive diffusion of force-fee magnetic fields in a passive medium, Astrophys. J., 181 (1973), 209-226.
|
[19] |
B. C. Low,
Nonlinear classical diffusion in a contained plasma, Phys. Fluids, 25 (1982), 402-407.
doi: 10.1063/1.863749. |
[20] |
K. Matsue,
On blow-up solutions of differential equations with Poincaré type compactificaions, SIAM J. Appl. Dyn. Syst., 17 (2018), 2249-2288.
doi: 10.1137/17M1124498. |
[21] |
K. Matsue,
Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs, J. Differential Equations, 267 (2019), 7313-7368.
doi: 10.1016/j.jde.2019.07.022. |
[22] |
C.-C. Poon,
Blow-up of a degenerate non-linear heat equation, Taiwanese J. Math., 15 (2011), 1201-1225.
doi: 10.11650/twjm/1500406295. |
[23] |
C.-C. Poon,
Blowup rate of solutions of a degenerate nonlinear parabolic equation, Discrete Contin. Dyn. System. Ser. B, 24 (2019), 5317-5336.
doi: 10.3934/dcdsb.2019060. |
show all references
References:
[1] |
M. J. Álvarez, A. Ferragut and X. Jarque,
A survey on the blow up technique, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3103-3118.
doi: 10.1142/S0218127411030416. |
[2] |
K. Anada, I. Fukuda and M. Tsutsumi,
Regional blow-up and decay of solutions to the initial-boundary value problem for $u_{t} = uu_xx-\gamma (u_{x})^{2}+ku^{2}$, Funkcial. Ekvac., 39 (1996), 367-387.
|
[3] |
K. Anada and T. Ishiwata,
Asymptotic behavior of blow-up solutions to a degenerate parabolic equation, J. Math-for-Ind., 2011 (2011), 1-8.
|
[4] |
K. Anada and T. Ishiwata,
Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation, J. Differential Equations, 262 (2017), 181-271.
doi: 10.1016/j.jde.2016.09.023. |
[5] |
S. B. Angenent and J. J. L. Velázquez,
Asymptotic shape of cusp singularities in curve shortening, Duke Math. J., 77 (1995), 71-110.
doi: 10.1215/S0012-7094-95-07704-7. |
[6] |
M. Brunella and M. Miari,
Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra, J. Differential Equations, 85 (1990), 338-366.
doi: 10.1016/0022-0396(90)90120-E. |
[7] |
J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981. |
[8] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin Heidelberg, 2006. |
[9] |
M. Gage and R. S. Hamilton,
The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.
|
[10] |
M. A. Grayson,
The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.
|
[11] |
Y. Ichida, Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity, submitted. |
[12] |
Y. Ichida, K. Matsue and T. O. Sakamoto,
A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations, JSIAM Lett., 12 (2020), 65-68.
doi: 10.14495/jsiaml.12.65. |
[13] |
Y. Ichida and T. O. Sakamoto,
Quasi traveling waves with quenching in a reaction-diffusion equation in the presence of negative powers nonlinearity, Proc. Japan Acad. Ser. A Math. Sci., 96 (2020), 1-6.
doi: 10.3792/pjaa.96.001. |
[14] |
Y. Ichida and T. O. Sakamoto,
Radial symmetric stationary solutions for a MEMS type reaction-diffusion equation with spatially dependent nonlinearity, Jpn. J. Ind. Appl. Math., 38 (2021), 297-322.
doi: 10.1007/s13160-020-00438-8. |
[15] |
Y. Ichida and T. O. Sakamoto,
Traveling wave solutions for degenerate nonlinear parabolic equations, J. Elliptic Parabol. Equ., 6 (2020), 795-832.
doi: 10.1007/s41808-020-00080-y. |
[16] |
Y. Ichida and T. O. Sakamoto, Singular stationary solutions for a MEMS type reaction-diffusion equation with fringing field, submitted. |
[17] |
C. Kuehn, Multiple Time Scale Dynamics, Springer International Publishing, 2015.
doi: 10.1007/978-3-319-12316-5. |
[18] |
B. C. Low,
Resitive diffusion of force-fee magnetic fields in a passive medium, Astrophys. J., 181 (1973), 209-226.
|
[19] |
B. C. Low,
Nonlinear classical diffusion in a contained plasma, Phys. Fluids, 25 (1982), 402-407.
doi: 10.1063/1.863749. |
[20] |
K. Matsue,
On blow-up solutions of differential equations with Poincaré type compactificaions, SIAM J. Appl. Dyn. Syst., 17 (2018), 2249-2288.
doi: 10.1137/17M1124498. |
[21] |
K. Matsue,
Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs, J. Differential Equations, 267 (2019), 7313-7368.
doi: 10.1016/j.jde.2019.07.022. |
[22] |
C.-C. Poon,
Blow-up of a degenerate non-linear heat equation, Taiwanese J. Math., 15 (2011), 1201-1225.
doi: 10.11650/twjm/1500406295. |
[23] |
C.-C. Poon,
Blowup rate of solutions of a degenerate nonlinear parabolic equation, Discrete Contin. Dyn. System. Ser. B, 24 (2019), 5317-5336.
doi: 10.3934/dcdsb.2019060. |


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