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

Kamin, Shoshana ; Rosenau, Philip

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004), p. 271-280 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

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.

Détail

Publié le : 2004-12-01

MR 2148885 | Zbl 1113.35094

Bibliographie

[1] Aronson, D., Density-dependent interaction-diffusion systems. In: W.E. Stewart et al. (eds.), Dynamics and Modeling of Reactive Systems. Academic Press, New York 1980, 161-176. | MR 588018

[2] Bertsch, M. - Kersner, R. - Peletier, L.A., Positivity versus localization in degenerate diffusion equation. Nonlinear Anal., 9, 1985, 987-1008. | MR 804564 | Zbl 0596.35073

[3] Biro, Z., Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type. Adv. Nonlin. St., 2, 2002, 357-371. | MR 1936043 | Zbl 1021.35053

[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. | MR 684758 | Zbl 0526.35042

[6] Freistühler, H. - Serre, D., $L^{1}$ stability of shock waves in scalar viscous conservation laws. Commun. Pure Appl. Math., 51, 1998, 291-301. | MR 1488516 | Zbl 0907.76046

[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. | Zbl 1073.35002

[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. | MR 898624 | Zbl 0642.35047

[9] Kamin, S. - Rosenau, P., Emergence of waves in a nonlinear convection-reaction-diffusion equation. Advanced Nonlinear Studies, 4(3), 2004, 251. | MR 2079814 | Zbl 1058.35126

[10] Kersner, R., Some properties of generalized solutions of quasilinear degenerate parabolic equations. Acta Math. Acad. Sc. Hungaricae, 32, 1978, 301-330. | MR 512406 | Zbl 0399.35067

[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, Boston 1988, 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). | Zbl 0018.32106

[12] Malaguti, L. - Marcelli, C., Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations. Preprint. | MR 2016820 | Zbl 1042.34056

[13] Newman, W.I., Some exact solutions to a nonlinear diffusion problem in population genetics and combustion. J. Theor. Biol., 85, 1980, 325-334. | MR 586127

[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. | MR 99834 | Zbl 0093.10302

[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. | MR 673828 | Zbl 0479.35053

[16] Yuanwei Qi, - Mingxin Wang, , Singular solutions of doubly singular parabolic equations with absorption. Elect. J. Diff. Eq., 67, 2000, 1-22. | MR 1800832 | Zbl 0959.35099

[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. | MR 1325800 | Zbl 0821.35085

[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. | Zbl 1034.35076

[20] Uchiyama, K., The behaviour of solutions of some nonlinear diffusion equations for large time. J. Math. Kyoto Univ., 18, 1978, 453-508. | MR 509494 | Zbl 0408.35053

[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. | MR 1260981 | Zbl 0765.76086

[22] Vázquez, J.L., A note on the asymptotic behaviour for $u_{t} = \triangle u^{m} - u^{m}$. 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 Island 1994, Translation of Mathematical Monographs, vol. 140 (translated by James F. Heyda from an original Russian manuscript). | MR 1297766 | Zbl 1001.35060