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
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

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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

show all references

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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