# Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation

• Volume: 15, Issue: 3-4, page 271-280
• ISSN: 1120-6330

top

## Abstract

top
We study the behaviour of the solutions of the Cauchy problem ${u}_{t}={\left({u}^{m}\right)}_{xx}+u\left(1-{u}^{m-1}\right),x\in \mathbb{R},t>0u\left(0,x\right)={u}_{0}\left(x\right),{u}_{0}\left(x\right)\ge 0,$ and prove that if initial data ${u}_{0}\left(x\right)$ decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.

## How to cite

top

Kamin, Shoshana, and Rosenau, Philip. "Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 271-280. <http://eudml.org/doc/252421>.

@article{Kamin2004,
abstract = {We study the behaviour of the solutions of the Cauchy problem $$u\_\{t\} = (u^\{m\})\_\{xx\}+u(1-u^\{m-1\}), \quad x \in \mathbb\{R\}, \quad t > 0 \quad u(0,x)=u\_\{0\}(x), \quad u\_\{0\}(x) \ge 0,$$ and prove that if initial data $u_\{0\}(x)$ decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.},
author = {Kamin, Shoshana, Rosenau, Philip},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Asymptotic behaviour of solutions; nonlinear diffusion; reaction-diffusion equation; traveling waves},
language = {eng},
month = {12},
number = {3-4},
pages = {271-280},
publisher = {Accademia Nazionale dei Lincei},
title = {Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation},
url = {http://eudml.org/doc/252421},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Kamin, Shoshana
AU - Rosenau, Philip
TI - Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/12//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 3-4
SP - 271
EP - 280
AB - We study the behaviour of the solutions of the Cauchy problem $$u_{t} = (u^{m})_{xx}+u(1-u^{m-1}), \quad x \in \mathbb{R}, \quad t > 0 \quad u(0,x)=u_{0}(x), \quad u_{0}(x) \ge 0,$$ and prove that if initial data $u_{0}(x)$ decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.
LA - eng
KW - Asymptotic behaviour of solutions; nonlinear diffusion; reaction-diffusion equation; traveling waves
UR - http://eudml.org/doc/252421
ER -

## References

top
1. ARONSON, D., Density-dependent interaction-diffusion systems. In: W.E. STEWART et al. (eds.), Dynamics and Modeling of Reactive Systems. Academic Press, New York1980, 161-176. MR588018
2. BERTSCH, M. - KERSNER, R. - PELETIER, L.A., Positivity versus localization in degenerate diffusion equation. Nonlinear Anal., 9, 1985, 987-1008. Zbl0596.35073MR804564DOI10.1016/0362-546X(85)90081-1
3. BIRO, Z., Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type. Adv. Nonlin. St., 2, 2002, 357-371. Zbl1021.35053MR1936043
4. XINFU CHEN, - YUANWEI QI, - MINGXIN WANG, , Existence and uniqueness of singular solutions of fast diffusion porous medium equation. Preprint.
5. DI BENEDETTO, E., Continuity of weak solutions to a general porous media equation. Ind. Univ. Math. J., 32, 1983, 83-118. Zbl0526.35042MR684758DOI10.1512/iumj.1983.32.32008
6. FREISTÜHLER, H. - SERRE, D., ${L}^{1}$ stability of shock waves in scalar viscous conservation laws. Commun. Pure Appl. Math., 51, 1998, 291-301. Zbl0907.76046MR1488516DOI10.1002/(SICI)1097-0312(199803)51:3<291::AID-CPA4>3.3.CO;2-S
7. GILDING, B.H. - KERSNER, R., Travelling waves in nonlinear diffusion-advection-reaction. Memorandum no. 1585, June 2001, version available at www.math.utwente.nl/publications/2001/1585.pdf. Zbl1073.35002
8. KALASHNIKOV, A.S., Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russian Math. Surv., 42, 1987, 169-222. Zbl0642.35047MR898624
9. KAMIN, S. - ROSENAU, P., Emergence of waves in a nonlinear convection-reaction-diffusion equation. Advanced Nonlinear Studies, 4(3), 2004, 251. Zbl1058.35126MR2079814
10. KERSNER, R., Some properties of generalized solutions of quasilinear degenerate parabolic equations. Acta Math. Acad. Sc. Hungaricae, 32, 1978, 301-330. Zbl0399.35067MR512406DOI10.1007/BF01902368
11. KOLMOGOROV, A. - PETROVSKY, I. - PISCUNOV, N., Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique. Bulletin Univ. Moscou, Ser. Internationale, Math., Mec., 1, 1937, 1-25 (see English translation in: P. PELCÉ (ed.), Dynamics of Curved Fronts. Academic Press, Boston1988, 105-130; and in: O.A. OLEINIK (ed.), Petrowsky, I.G.Selected Works Part II. Differential Equations and Probability Theory. Gordon and Breach Publishers, 1996, 106-132). Zbl0018.32106
12. MALAGUTI, L. - MARCELLI, C., Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations. Preprint. Zbl1042.34056MR2016820DOI10.1016/j.jde.2003.06.005
13. NEWMAN, W.I., Some exact solutions to a nonlinear diffusion problem in population genetics and combustion. J. Theor. Biol., 85, 1980, 325-334. MR586127DOI10.1016/0022-5193(80)90024-7
14. OLEINIK, O.A. - KALASHNIKOV, A.S. - CHZHOU, Y.-L., The Cauchy problem and boundary problems for equations of the type of nonstationary filtration (in Russian). Izv. Akad. Nauk. SSSR Ser. Mat., 22, 1958, 667-704. Zbl0093.10302MR99834
15. OSHER, S. - RALSON, J., ${L}^{1}$ stability of travelling waves with applications to convective porous media flow. Commun. Pure Appl. Math., 35, 1982, 737-749. Zbl0479.35053MR673828DOI10.1002/cpa.3160350602
16. YUANWEI QI, - MINGXIN WANG, , Singular solutions of doubly singular parabolic equations with absorption. Elect. J. Diff. Eq., 67, 2000, 1-22. Zbl0959.35099MR1800832
17. ROSENAU, P., Reaction and concentration dependent diffusion model. Phys. Rev. Lett., 88, 2002, 194501-4.
18. SÁNCHEZ-GARDUÑO, F. - MAINI, P.K., Travelling wave phenomena in some degenerate reaction-diffusion equations. J. Diff. Eq., 117, 1995, 281-319. Zbl0821.35085MR1325800DOI10.1006/jdeq.1995.1055
19. SERRE, D., Stabilité des ondes de choc de viscosité qui peuvent etre characteristiques. In: D. CIORANESCU - J.-L. LIONS (eds.), Nonlinear Partial Differential Equations and Their Applications. Studies in Mathematics and its Applications, 31, Elsevier, 2002, 647-654. Zbl1034.35076
20. UCHIYAMA, K., The behaviour of solutions of some nonlinear diffusion equations for large time. J. Math. Kyoto Univ., 18, 1978, 453-508. Zbl0408.35053MR509494
21. VÁZQUEZ, J.L., An introduction to the mathematical theory of the porous medium equation. In: Shape Optimization and Free Boundaries. NATO ASI series, series C, Mathematical and physical sciences, v. 380, Kluwer Acad. Publ., Dordrecht1992, 347-389. Zbl0765.76086MR1260981
22. VÁZQUEZ, J.L., A note on the asymptotic behaviour for ${u}_{t}=\mathrm{△}{u}^{m}-{u}^{m}$. Preprint.
23. VOLPERT, A.I., On propagation of waves described by nonlinear parabolic equations. In: O.A. OLEINIK (ed.), I.G. Petrowsky Selected Works, Part II. Differential Equations and Probability Theory. Gordon and Breach Publishers, 1996.
24. VOLPERT, A.I. - VOLPERT, VI.A. - VOLPERT, VL.A., Travelling wave solutions of parabolic systems. American Mathematical Society, Providence, Rhode Island1994, Translation of Mathematical Monographs, vol. 140 (translated by James F. Heyda from an original Russian manuscript). Zbl1001.35060MR1297766

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.