Non-sharp travelling waves for a dual porous medium equation

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

Non-sharp travelling waves for a dual porous medium equation

Jing Li ^1, , Yifu Wang ^2, and Jingxue Yin ^3,

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

Abstract
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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.

Keywords: Travelling wave, dual porous medium equation..

Mathematics Subject Classification: Primary: 35K65, 35K4.

Citation: Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure and Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623

References:

[1]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.
[2]	Ph. Bénilan and K. S. Ha, Equation d'évolution du type $(du/dt) +\beta\delta\Phi_\varepsilon(u) \ni 0$ dans $L^\infty(\Omega)$, Comptes Rendus Acad. Sci. Paris, A, 281 (1975), 947-950.
[3]	G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.
[4]	R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 353-369.
[5]	V. A. Galaktionov, Geometric sturmian theory of nonlinear parabolic equations with applications, Chapman $&$ Hall, 2005. doi: 10.1201/9780203998069.
[6]	K. S. Ha, Sur des semigroups non linéaires dans les espaces $L^\infty(\Omega)$, J. Math. Soc. Japan, 31 (1979), 593-622. doi: 10.2969/jmsj/03140593.
[7]	S. L. Kamenomostskaya (Kamin), On the Stefan Problem, Mat. Sbornik, 53 (1961), 489-514.
[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, Bull. Univ. Moskov Ser. Internat. Sec. Math., 1 (1937), 1-25.
[9]	Y. Konishi, On the nonlinear semi-groups associated with $u_t=\Delta\beta(u)$ and $\Phi_\varepsilon(u_t)=\Delta u$, J. Math. Soc. Japan, 25 (1973), 622-628.
[10]	P. L. Lions, Some problems related to the Bellman-Dirichlet equation for two operators, Comm. Partial Differential Equations, 5 (1980), 753-771. doi: 10.1080/03605308008820153.
[11]	L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equations, 195 (2003), 471-496. doi: 10.1016/j.jde.2003.06.005.
[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, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026.
[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, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020.
[14]	O. A. Oleinik, A. S. Kalashnikov and Chzhou Yui-Lin, The Cauchy problem and boundary-value problems for equations of unsteady filtration type, Izv. Akad. Nauk SSSR, Ser. Mat., 22 (1958), 667-704.
[15]	A. D. Pablo and A. Sánchez, Global travelling waves in reaction-convection-diffusion equations, J. Differential Equations, 165 (2000), 377-413. doi: 10.1006/jdeq.2000.3781.
[16]	A. D. Pablo and J. L. Vazquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93 (1991), 19-61. doi: 10.1016/0022-0396(91)90021-Z.
[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, Proc. R. Soc. Lond. Ser. A, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789.
[18]	J. W.-H. So and X. Zou, Traveling waves for the diffusive Nicholson's blowflies equation, Appl. Math. Comput., 22 (2001), 385-392. doi: 10.1016/S0096-3003(00)00055-2.
[19]	W. Strauss, Evolution equations non-linear in the time-derivative, Jour. Math. Mech., 15 (1966), 49-82.
[20]	C. P. Wang and J. X. Yin, Travelling wave fronts of a degenerate parabolic equation with non-divergence form, J. PDEs, 16 (2003), 62-74.
[21]	J. X. Yin and C. H. Jin, Travelling wavefronts for a non-divergent degenerate and singular parabolic equation with changing sign sources, Proceedings of the Royal Society of Edinburgh, 139A (2009), 1179-1207. doi: 10.1017/S0308210508000231.
[22]	J. X. Yin, J. Li and C. H. Jin, Classical solutions for a class of fully nonlinear degenerate parabolic equations, J. Math. Anal. Appl., 360 (2009), 119-129. doi: 10.1016/j.jmaa.2009.06.038.

show all references

References:

[1]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.
[2]	Ph. Bénilan and K. S. Ha, Equation d'évolution du type $(du/dt) +\beta\delta\Phi_\varepsilon(u) \ni 0$ dans $L^\infty(\Omega)$, Comptes Rendus Acad. Sci. Paris, A, 281 (1975), 947-950.
[3]	G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.
[4]	R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 353-369.
[5]	V. A. Galaktionov, Geometric sturmian theory of nonlinear parabolic equations with applications, Chapman $&$ Hall, 2005. doi: 10.1201/9780203998069.
[6]	K. S. Ha, Sur des semigroups non linéaires dans les espaces $L^\infty(\Omega)$, J. Math. Soc. Japan, 31 (1979), 593-622. doi: 10.2969/jmsj/03140593.
[7]	S. L. Kamenomostskaya (Kamin), On the Stefan Problem, Mat. Sbornik, 53 (1961), 489-514.
[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, Bull. Univ. Moskov Ser. Internat. Sec. Math., 1 (1937), 1-25.
[9]	Y. Konishi, On the nonlinear semi-groups associated with $u_t=\Delta\beta(u)$ and $\Phi_\varepsilon(u_t)=\Delta u$, J. Math. Soc. Japan, 25 (1973), 622-628.
[10]	P. L. Lions, Some problems related to the Bellman-Dirichlet equation for two operators, Comm. Partial Differential Equations, 5 (1980), 753-771. doi: 10.1080/03605308008820153.
[11]	L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equations, 195 (2003), 471-496. doi: 10.1016/j.jde.2003.06.005.
[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, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026.
[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, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020.
[14]	O. A. Oleinik, A. S. Kalashnikov and Chzhou Yui-Lin, The Cauchy problem and boundary-value problems for equations of unsteady filtration type, Izv. Akad. Nauk SSSR, Ser. Mat., 22 (1958), 667-704.
[15]	A. D. Pablo and A. Sánchez, Global travelling waves in reaction-convection-diffusion equations, J. Differential Equations, 165 (2000), 377-413. doi: 10.1006/jdeq.2000.3781.
[16]	A. D. Pablo and J. L. Vazquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93 (1991), 19-61. doi: 10.1016/0022-0396(91)90021-Z.
[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, Proc. R. Soc. Lond. Ser. A, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789.
[18]	J. W.-H. So and X. Zou, Traveling waves for the diffusive Nicholson's blowflies equation, Appl. Math. Comput., 22 (2001), 385-392. doi: 10.1016/S0096-3003(00)00055-2.
[19]	W. Strauss, Evolution equations non-linear in the time-derivative, Jour. Math. Mech., 15 (1966), 49-82.
[20]	C. P. Wang and J. X. Yin, Travelling wave fronts of a degenerate parabolic equation with non-divergence form, J. PDEs, 16 (2003), 62-74.
[21]	J. X. Yin and C. H. Jin, Travelling wavefronts for a non-divergent degenerate and singular parabolic equation with changing sign sources, Proceedings of the Royal Society of Edinburgh, 139A (2009), 1179-1207. doi: 10.1017/S0308210508000231.
[22]	J. X. Yin, J. Li and C. H. Jin, Classical solutions for a class of fully nonlinear degenerate parabolic equations, J. Math. Anal. Appl., 360 (2009), 119-129. doi: 10.1016/j.jmaa.2009.06.038.

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