doi: 10.3934/dcdsb.2022114
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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

* Corresponding author: Yu Ichida

Received  October 2021 Revised  March 2022 Early access June 2022

Fund Project: The author was partially supported by JSPS KAKENHI Grant Number JP21J20035

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.

Citation: Yu Ichida. Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022114
References:
[1]

M. J. ÁlvarezA. 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. AnadaI. 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. IchidaK. 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. ÁlvarezA. 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. AnadaI. 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. IchidaK. 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.

Figure 1.  Schematic picture of the traveling wave solutions obtained in Theorems. Here it should be noted that the position of the singularity point $ \xi_{*} $ is not determined in our studies, however, they are shown in the figures for the convenience. [Top left: The weak traveling wave solution with singularity in Theorem 2.3.] [Top right: The weak traveling wave solution with singularity in Theorem 2.4 in the case that $ D<0 $.] [Lower center: The traveling wave solution on $ \xi\in \mathbb{R} $ obtained in Theorem 2.5 in the case that $ D>0 $.]
Figure 2.  Schematic pictures of the dynamics on the Poincaré disk in the case that $ \delta = 0 $ or $ 1 $, $ 1<p\in \mathbb{R} $. [Left: Case $ \delta = 0 $.] [Right: Case $ \delta = 1 $.]
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