# American Institute of Mathematical Sciences

March  2016, 15(2): 623-636. doi: 10.3934/cpaa.2016.15.623

## Non-sharp travelling waves for a dual porous medium equation

 1 College of Science, Minzu University of China, Beijing, 100081, China 2 Department of Mathematics, Beijing Institute of Technology, Beijing 100081 3 Department of Mathematics, South China Normal University, Guangzhou, Guangdong, 510631

Received  February 2015 Revised  October 2015 Published  January 2016

We discuss non-sharp travelling waves of a dual porous medium equation with monostable source and bistable source respectively. We show the existence of non-sharp travelling waves and find that though the equation is degenerate, the travelling waves are classical ones. Furthermore, for the monostable source, we show that the non-sharp travelling waves are infinite, while for the bistable source, the non-sharp travelling waves are semi-finite, which is in contrast with the case of the heat equation.
Citation: Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure & Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623
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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, \emph{Adv. Math.}, 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5. Google Scholar [2] Ph. Bénilan and K. S. Ha, Equation d'évolution du type $(du/dt) +\beta\delta\Phi_\varepsilon(u) \ni 0$ dans $L^\infty(\Omega)$,, \emph{Comptes Rendus Acad. Sci. Paris, 281 (1975), 947. Google Scholar [3] G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique,, Dunod, (1972). Google Scholar [4] R. A. Fisher, The wave of advance of advantageous genes,, \emph{Annals of Eugenics}, 7 (1937), 353. Google Scholar [5] V. A. Galaktionov, Geometric sturmian theory of nonlinear parabolic equations with applications,, Chapman $&$ Hall, (2005). doi: 10.1201/9780203998069. Google Scholar [6] K. S. Ha, Sur des semigroups non linéaires dans les espaces $L^\infty(\Omega)$,, \emph{J. Math. Soc. Japan}, 31 (1979), 593. doi: 10.2969/jmsj/03140593. Google Scholar [7] S. L. Kamenomostskaya (Kamin), On the Stefan Problem,, \emph{Mat. Sbornik}, 53 (1961), 489. Google Scholar [8] A. Kolmogorov, I. Petrovsky and N. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probleme biologique,, \emph{Bull. Univ. Moskov Ser. Internat. Sec. Math.}, 1 (1937), 1. Google Scholar [9] Y. Konishi, On the nonlinear semi-groups associated with $u_t=\Delta\beta(u)$ and $\Phi_\varepsilon(u_t)=\Delta u$,, \emph{J. Math. Soc. Japan}, 25 (1973), 622. Google Scholar [10] P. L. Lions, Some problems related to the Bellman-Dirichlet equation for two operators,, \emph{Comm. Partial Differential Equations}, 5 (1980), 753. doi: 10.1080/03605308008820153. Google Scholar [11] L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations,, \emph{J. Differential Equations}, 195 (2003), 471. doi: 10.1016/j.jde.2003.06.005. Google Scholar [12] M. Mei, C. -K. Lin, C. -T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (I) local nonlinearity,, \emph{J. Differential Equations}, 247 (2009), 495. doi: 10.1016/j.jde.2008.12.026. Google Scholar [13] M. Mei, C. -K. Lin, C. -T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (II) nonlocal nonlinearity,, \emph{J. Differential Equations}, 247 (2009), 511. doi: 10.1016/j.jde.2008.12.020. Google Scholar [14] O. A. Oleinik, A. S. Kalashnikov and Chzhou Yui-Lin, The Cauchy problem and boundary-value problems for equations of unsteady filtration type,, \emph{Izv. Akad. Nauk SSSR, 22 (1958), 667. Google Scholar [15] A. D. Pablo and A. Sánchez, Global travelling waves in reaction-convection-diffusion equations,, \emph{J. Differential Equations}, 165 (2000), 377. doi: 10.1006/jdeq.2000.3781. Google Scholar [16] A. D. Pablo and J. L. Vazquez, Travelling waves and finite propagation in a reaction-diffusion equation,, \emph{J. Differential Equations}, 93 (1991), 19. doi: 10.1016/0022-0396(91)90021-Z. Google Scholar [17] J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure:(I) Traveling wavefronts on unbounded domains,, \emph{Proc. R. Soc. Lond. Ser. A}, 457 (2001), 1841. doi: 10.1098/rspa.2001.0789. Google Scholar [18] J. W.-H. So and X. Zou, Traveling waves for the diffusive Nicholson's blowflies equation,, \emph{Appl. Math. Comput.}, 22 (2001), 385. doi: 10.1016/S0096-3003(00)00055-2. Google Scholar [19] W. Strauss, Evolution equations non-linear in the time-derivative,, \emph{Jour. Math. Mech.}, 15 (1966), 49. Google Scholar [20] C. P. Wang and J. X. Yin, Travelling wave fronts of a degenerate parabolic equation with non-divergence form,, \emph{J. PDEs}, 16 (2003), 62. Google Scholar [21] J. X. Yin and C. H. Jin, Travelling wavefronts for a non-divergent degenerate and singular parabolic equation with changing sign sources,, \emph{Proceedings of the Royal Society of Edinburgh}, 139A (2009), 1179. doi: 10.1017/S0308210508000231. Google Scholar [22] J. X. Yin, J. Li and C. H. Jin, Classical solutions for a class of fully nonlinear degenerate parabolic equations,, \emph{J. Math. Anal. Appl.}, 360 (2009), 119. doi: 10.1016/j.jmaa.2009.06.038. Google Scholar
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