-
Previous Article
Asymptotic analysis of a spatially and size-structured population model with delayed birth process
- CPAA Home
- This Issue
-
Next Article
Existence and concentration of semiclassical solutions for Hamiltonian elliptic system
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 |
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. |
[1] |
Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks and Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337 |
[2] |
Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 |
[3] |
Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783 |
[4] |
Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1761-1774. doi: 10.3934/dcdsb.2012.17.1761 |
[5] |
Sofía Nieto, Guillermo Reyes. Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1123-1139. doi: 10.3934/cpaa.2013.12.1123 |
[6] |
Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927 |
[7] |
Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure and Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183 |
[8] |
Zhilei Liang. On the critical exponents for porous medium equation with a localized reaction in high dimensions. Communications on Pure and Applied Analysis, 2012, 11 (2) : 649-658. doi: 10.3934/cpaa.2012.11.649 |
[9] |
Edoardo Mainini. On the signed porous medium flow. Networks and Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525 |
[10] |
A. Ducrot. Travelling wave solutions for a scalar age-structured equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 251-273. doi: 10.3934/dcdsb.2007.7.251 |
[11] |
Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925 |
[12] |
Liangwei Wang, Jingxue Yin, Chunhua Jin. $\omega$-limit sets for porous medium equation with initial data in some weighted spaces. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 223-236. doi: 10.3934/dcdsb.2013.18.223 |
[13] |
Ansgar Jüngel, Stefan Schuchnigg. A discrete Bakry-Emery method and its application to the porous-medium equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5541-5560. doi: 10.3934/dcds.2017241 |
[14] |
Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107 |
[15] |
Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665 |
[16] |
Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4031-4053. doi: 10.3934/dcdsb.2019049 |
[17] |
Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011 |
[18] |
Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093 |
[19] |
Nikolaos Roidos, Yuanzhen Shao. Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation. Evolution Equations and Control Theory, 2022, 11 (3) : 793-825. doi: 10.3934/eect.2021026 |
[20] |
Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]