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

Kamin, Shoshana ; Rosenau, Philip

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 271-280.

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.

Bibliographie

[1] D. Aronson, 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

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

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

[4] Xinfu Chen - Yuanwei Qi - Mingxin Wang, Existence and uniqueness of singular solutions of fast diffusion porous medium equation. Preprint.

[5] E. Di Benedetto, Continuity of weak solutions to a general porous media equation. Ind. Univ. Math. J., 32, 1983, 83-118. | DOI | MR | Zbl

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

[7] B.H. Gilding - R. Kersner, Travelling waves in nonlinear diffusion-advection-reaction. Memorandum no. 1585, June 2001, version available at www.math.utwente.nl/publications/2001/1585.pdf. | Zbl

[8] A.S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russian Math. Surv., 42, 1987, 169-222. | MR | Zbl

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

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

[11] A. Kolmogorov - I. Petrovsky - N. Piscunov, 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.), I.G. Petrowsky Selected Works Part II. Differential Equations and Probability Theory. Gordon and Breach Publishers, 1996, 106-132). | Zbl

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

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

[14] O.A. Oleinik - A.S. Kalashnikov - Y.-L. Chzhou, 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 | Zbl

[15] S. Osher - J. Ralson, $L^{1}$ stability of travelling waves with applications to convective porous media flow. Commun. Pure Appl. Math., 35, 1982, 737-749. | DOI | MR | Zbl

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

[17] P. Rosenau, Reaction and concentration dependent diffusion model. Phys. Rev. Lett., 88, 2002, 194501-4.

[18] F. Sánchez-Garduño - P.K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations. J. Diff. Eq., 117, 1995, 281-319. | DOI | MR | Zbl

[19] D. Serre, 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

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

[21] J.L. Vázquez, 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., Dordrecht 1992, 347-389. | MR | Zbl

[22] J.L. Vázquez, A note on the asymptotic behaviour for $u_{t} = \triangle u^{m} - u^{m}$. Preprint.

[23] A.I. Volpert, 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] A.I. Volpert - Vi.A. Volpert - Vl.A. Volpert, 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 | Zbl