June  2019, 39(6): 2977-3015. doi: 10.3934/dcds.2019124

Bistable reaction equations with doubly nonlinear diffusion

Dipartimento di Matematica "Giuseppe Luigi Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italia

Dedicated to Professor Juan Luis Vázquez

Received  January 2018 Revised  November 2018 Published  February 2019

Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves that describe the asymptotic behaviour of a larger class of solutions $ 0\leq u(x, t)\leq 1 $ of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In the present setting we consider bistable reaction terms, which present interesting differences w.r.t. the Fisher-KPP framework recently studied in [5]. We find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call "pseudo-linear", i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the "pseudo-linear" case, the travelling waves of the "slow" diffusion setting exhibit free boundaries.

Finally, as a complement of [5], we study the asymptotic behaviour of more general solutions in the presence of a "heterozygote superior" reaction function and doubly nonlinear diffusion ("slow" and "pseudo-linear").

Citation: Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124
References:
[1]

F. AndreuV. Caselles and J. M. Mazón, A Fisher-Kolmogorov equation with finite speed of propagation, J. Differential Equations, 248 (2010), 2528-2561. doi: 10.1016/j.jde.2010.01.005. Google Scholar

[2]

D. G. Aronson, The porous medium equation, in Some Problems in Nonlinear Diffusion (A. Fasano and M. Primicerio, Eds.), Lecture Notes in Mathematics, Springer-Verlag, New York/Berlin, 1224 (1986), 1–46. doi: 10.1007/BFb0072687. Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, in Partial Differential Equations and Related Topic, Lecture Notes in Mathematics, Pub., New York, 446 (1975), 5–49. Google Scholar

[4]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar

[5]

A. Audrito and J. L. Vázquez, The Fisher-KPP problem with doubly nonlinear diffusion, J. Differential Equations, 263 (2017), 7647-7708. doi: 10.1016/j.jde.2017.08.025. Google Scholar

[6]

A. Audrito and J. L. Vázquez, The Fisher-KPP problem with doubly nonlinear "fast" diffusion, Nonlinear Anal., 157 (2017), 212-248. doi: 10.1016/j.na.2017.03.015. Google Scholar

[7]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022. Google Scholar

[8]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅰ: Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26. Google Scholar

[9]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅱ: General domains, J. Amer. Math. Soc., 23 (2010), 1-34. doi: 10.1090/S0894-0347-09-00633-X. Google Scholar

[10]

H. Berestycki and L. Nirenberg, Traveling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 9 (1992), 497-572. doi: 10.1016/S0294-1449(16)30229-3. Google Scholar

[11]

X. Cabré and J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5. Google Scholar

[12]

J. CalvoJ. CamposV. CasellesO. Sánchez and J. Soler, Pattern formation in a flux limited reaction-diffusion of porous media type, Invent. Math., 206 (2016), 57-108. doi: 10.1007/s00222-016-0649-5. Google Scholar

[13]

N. CalvoJ. I. DíazJ. DuranyE. Schiavi and C. Vázquez, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics, SIAM J. Appl. Math., 63 (2002), 683-707. doi: 10.1137/S0036139901385345. Google Scholar

[14]

J. CamposP. GuerreroO. Sánchez and J. Soler, On the analysis of travelling waves to a nonlinear flux limited reaction–diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 141-155. doi: 10.1016/j.anihpc.2012.07.001. Google Scholar

[15]

J. Campos and J. Soler, Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation, Nonlinear Anal., 137 (2016), 266-290. doi: 10.1016/j.na.2015.12.021. Google Scholar

[16]

A. De Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equation, 93 (1991), 19-61. doi: 10.1016/0022-0396(91)90021-Z. Google Scholar

[17]

E. DiBenedetto, Degenerate Parabolic Equations, Springer Verlag, New York, Series Universitext, 1993. doi: 10.1007/978-1-4612-0895-2. Google Scholar

[18]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224. doi: 10.1090/S0002-9947-1989-0962278-5. Google Scholar

[19]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312. doi: 10.4171/JEMS/198. Google Scholar

[20]

R. EnguicaA. Gavioli and L. Sanchez, A class of singular first order differential equations with applications in reaction-diffusion, Discrete Contin. Dyn. Syst., 33 (2013), 173-191. doi: 10.3934/dcds.2013.33.173. Google Scholar

[21]

J. R. Esteban and J. L. Vázquez, Homogeneous diffusion in $\mathbb{R}$ with power-like nonlinear diffusivity, Arch. Rational Mech. Anal., 103 (1988), 39-80. doi: 10.1007/BF00292920. Google Scholar

[22]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432. Google Scholar

[23]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. Google Scholar

[24]

M. Garrione and M. Strani, Monotone wave fronts for $ (p,q) $-Laplacian driven reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 91-103. doi: 10.3934/dcdss.2019006. Google Scholar

[25]

A. Gárriz, Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour, preprint, arXiv: 1805.10955.Google Scholar

[26]

C. JinJ. Yin and S. Zheng, Traveling waves for a time delayed Newtonian filtration equation, J. Differential Equations, 254 (2013), 1-29. doi: 10.1016/j.jde.2012.08.034. Google Scholar

[27]

A. S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys, 42 (1987), 135-176. Google Scholar

[28]

A. N. KolmogorovI. Petrovsky and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Bull. Univ. Moskou Ser. Internat. Sec. A, 1 (1937), 1-25. Google Scholar

[29]

T. KuusiJ. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J., 61 (2012), 399-430. doi: 10.1512/iumj.2012.61.4513. Google Scholar

[30]

R. LaisterA. T. Peplow and R. E. Beardmore, Finite time extinction in nonlinear diffusion equations, Appl. Math. Lett., 17 (2004), 561-567. doi: 10.1016/S0893-9659(04)90126-7. Google Scholar

[31]

L. S. Leibenzon, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk SSSR, Geography and Geophysics, 9 (1945), 7–10 (Russian). Google Scholar

[32]

J. Li, Cauchy problem and initial trace for a doubly degenerate parabolic equation with strongly nonlinear sources, J. Math. Anal. Appl., 264 (2001), 49-67. doi: 10.1006/jmaa.2001.7553. Google Scholar

[33]

P. Lindqvist, Notes on the $ p $-Laplace equation, University of Jyväskylä, Jyväskylä, 2006. Google Scholar

[34]

H. MatanoF. Punzo and A. Tesei, Front propagation for nonlinear diffusion equations on the hyperbolic space, J. Eur. Math. Soc., 17 (2015), 1199-1227. doi: 10.4171/JEMS/529. Google Scholar

[35]

H. P. McKean, Nagumo's Equation, Adv. Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar

[36]

C. B. Muratov and X. Zhong, Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations, Discrete Contin. Dyn. Syst., 37 (2017), 915-944. doi: 10.3934/dcds.2017038. Google Scholar

[37]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. Google Scholar

[38]

P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on $ \mathbb{R}^N $, Arch. Ration. Mech. Anal., 199 (2011), 69-97. doi: 10.1007/s00205-010-0316-8. Google Scholar

[39]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. Google Scholar

[40]

J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. Henri Poincaré, 14 (1997), 499-552. doi: 10.1016/S0294-1449(97)80137-0. Google Scholar

[41]

F. Sanchez-Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178. Google Scholar

[42] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001. Google Scholar
[43] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007. Google Scholar
[44]

V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math., 75 (1992), 65-80. doi: 10.1007/BF02567072. Google Scholar

[45]

Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791. Google Scholar

show all references

References:
[1]

F. AndreuV. Caselles and J. M. Mazón, A Fisher-Kolmogorov equation with finite speed of propagation, J. Differential Equations, 248 (2010), 2528-2561. doi: 10.1016/j.jde.2010.01.005. Google Scholar

[2]

D. G. Aronson, The porous medium equation, in Some Problems in Nonlinear Diffusion (A. Fasano and M. Primicerio, Eds.), Lecture Notes in Mathematics, Springer-Verlag, New York/Berlin, 1224 (1986), 1–46. doi: 10.1007/BFb0072687. Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, in Partial Differential Equations and Related Topic, Lecture Notes in Mathematics, Pub., New York, 446 (1975), 5–49. Google Scholar

[4]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar

[5]

A. Audrito and J. L. Vázquez, The Fisher-KPP problem with doubly nonlinear diffusion, J. Differential Equations, 263 (2017), 7647-7708. doi: 10.1016/j.jde.2017.08.025. Google Scholar

[6]

A. Audrito and J. L. Vázquez, The Fisher-KPP problem with doubly nonlinear "fast" diffusion, Nonlinear Anal., 157 (2017), 212-248. doi: 10.1016/j.na.2017.03.015. Google Scholar

[7]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022. Google Scholar

[8]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅰ: Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26. Google Scholar

[9]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅱ: General domains, J. Amer. Math. Soc., 23 (2010), 1-34. doi: 10.1090/S0894-0347-09-00633-X. Google Scholar

[10]

H. Berestycki and L. Nirenberg, Traveling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 9 (1992), 497-572. doi: 10.1016/S0294-1449(16)30229-3. Google Scholar

[11]

X. Cabré and J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5. Google Scholar

[12]

J. CalvoJ. CamposV. CasellesO. Sánchez and J. Soler, Pattern formation in a flux limited reaction-diffusion of porous media type, Invent. Math., 206 (2016), 57-108. doi: 10.1007/s00222-016-0649-5. Google Scholar

[13]

N. CalvoJ. I. DíazJ. DuranyE. Schiavi and C. Vázquez, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics, SIAM J. Appl. Math., 63 (2002), 683-707. doi: 10.1137/S0036139901385345. Google Scholar

[14]

J. CamposP. GuerreroO. Sánchez and J. Soler, On the analysis of travelling waves to a nonlinear flux limited reaction–diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 141-155. doi: 10.1016/j.anihpc.2012.07.001. Google Scholar

[15]

J. Campos and J. Soler, Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation, Nonlinear Anal., 137 (2016), 266-290. doi: 10.1016/j.na.2015.12.021. Google Scholar

[16]

A. De Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equation, 93 (1991), 19-61. doi: 10.1016/0022-0396(91)90021-Z. Google Scholar

[17]

E. DiBenedetto, Degenerate Parabolic Equations, Springer Verlag, New York, Series Universitext, 1993. doi: 10.1007/978-1-4612-0895-2. Google Scholar

[18]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224. doi: 10.1090/S0002-9947-1989-0962278-5. Google Scholar

[19]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312. doi: 10.4171/JEMS/198. Google Scholar

[20]

R. EnguicaA. Gavioli and L. Sanchez, A class of singular first order differential equations with applications in reaction-diffusion, Discrete Contin. Dyn. Syst., 33 (2013), 173-191. doi: 10.3934/dcds.2013.33.173. Google Scholar

[21]

J. R. Esteban and J. L. Vázquez, Homogeneous diffusion in $\mathbb{R}$ with power-like nonlinear diffusivity, Arch. Rational Mech. Anal., 103 (1988), 39-80. doi: 10.1007/BF00292920. Google Scholar

[22]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432. Google Scholar

[23]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. Google Scholar

[24]

M. Garrione and M. Strani, Monotone wave fronts for $ (p,q) $-Laplacian driven reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 91-103. doi: 10.3934/dcdss.2019006. Google Scholar

[25]

A. Gárriz, Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour, preprint, arXiv: 1805.10955.Google Scholar

[26]

C. JinJ. Yin and S. Zheng, Traveling waves for a time delayed Newtonian filtration equation, J. Differential Equations, 254 (2013), 1-29. doi: 10.1016/j.jde.2012.08.034. Google Scholar

[27]

A. S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys, 42 (1987), 135-176. Google Scholar

[28]

A. N. KolmogorovI. Petrovsky and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Bull. Univ. Moskou Ser. Internat. Sec. A, 1 (1937), 1-25. Google Scholar

[29]

T. KuusiJ. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J., 61 (2012), 399-430. doi: 10.1512/iumj.2012.61.4513. Google Scholar

[30]

R. LaisterA. T. Peplow and R. E. Beardmore, Finite time extinction in nonlinear diffusion equations, Appl. Math. Lett., 17 (2004), 561-567. doi: 10.1016/S0893-9659(04)90126-7. Google Scholar

[31]

L. S. Leibenzon, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk SSSR, Geography and Geophysics, 9 (1945), 7–10 (Russian). Google Scholar

[32]

J. Li, Cauchy problem and initial trace for a doubly degenerate parabolic equation with strongly nonlinear sources, J. Math. Anal. Appl., 264 (2001), 49-67. doi: 10.1006/jmaa.2001.7553. Google Scholar

[33]

P. Lindqvist, Notes on the $ p $-Laplace equation, University of Jyväskylä, Jyväskylä, 2006. Google Scholar

[34]

H. MatanoF. Punzo and A. Tesei, Front propagation for nonlinear diffusion equations on the hyperbolic space, J. Eur. Math. Soc., 17 (2015), 1199-1227. doi: 10.4171/JEMS/529. Google Scholar

[35]

H. P. McKean, Nagumo's Equation, Adv. Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar

[36]

C. B. Muratov and X. Zhong, Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations, Discrete Contin. Dyn. Syst., 37 (2017), 915-944. doi: 10.3934/dcds.2017038. Google Scholar

[37]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. Google Scholar

[38]

P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on $ \mathbb{R}^N $, Arch. Ration. Mech. Anal., 199 (2011), 69-97. doi: 10.1007/s00205-010-0316-8. Google Scholar

[39]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. Google Scholar

[40]

J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. Henri Poincaré, 14 (1997), 499-552. doi: 10.1016/S0294-1449(97)80137-0. Google Scholar

[41]

F. Sanchez-Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178. Google Scholar

[42] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001. Google Scholar
[43] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007. Google Scholar
[44]

V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math., 75 (1992), 65-80. doi: 10.1007/BF02567072. Google Scholar

[45]

Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791. Google Scholar

Figure 1.  The "slow diffusion" area and the "pseudo-linear" line in $ (m, p-1) $-plane. The yellow and orange area are called "fast diffusion" and "very fast diffusion" range, respectively, and they will not be studied in this paper
Figure 2.  Qualitative representation of the reactions of type C and type C', respectively.
Figure 3.  Examples of admissible TWs: Finite and Positive types
Figure 4.  Reactions of type C, range $ \gamma > 0 $, case $ c = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3, 0.7 $. The second case is excluded by the assumption $ \int_0^1u^{m-1}f(u)du > 0 $
Figure 6.  Reactions of type C, range $ \gamma > 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $. The first two pictures show the case $ 0< c < c_{\ast} $, while the others the cases $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 5.  Reactions of type C, range $ \gamma > 0 $. Null isoclines in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $, in the cases $ 0< c < c_0 $ and $ c > c_0 $, respectively
Figure 7.  Reactions of type C, range $ \gamma = 0$ , case $ c = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $.
Figure 8.  Reactions of type C, range $ \gamma = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $. The first picture shows the case $ 0< c < c_{\ast} $, while the others the cases $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 9.  Reactions of type C', range $ \gamma > 0 $. Null isoclines in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the cases $ 0< c < c_0 $ and $ c > c_0 $, respectively
Figure 10.  Reactions of type C', range $ \gamma > 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the ranges $ 0< c < c_{\ast} $, $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 11.  Reactions of type C', range $ \gamma = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the ranges $ 0< c < c_{\ast} $, $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
[1]

A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373

[2]

A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1185-1198. doi: 10.3934/dcds.2008.21.1185

[3]

Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775

[4]

Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15

[5]

Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141

[6]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767

[7]

Xinmin Xiang. The long-time behaviour for nonlinear Schrödinger equation and its rational pseudospectral approximation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 469-488. doi: 10.3934/dcdsb.2005.5.469

[8]

H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119

[9]

Yuguo Lin, Daqing Jiang. Long-time behaviour of a perturbed SIR model by white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873

[10]

Lingbing He, Claude Le Bris, Tony Lelièvre. Periodic long-time behaviour for an approximate model of nematic polymers. Kinetic & Related Models, 2012, 5 (2) : 357-382. doi: 10.3934/krm.2012.5.357

[11]

Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 273-309. doi: 10.3934/dcdss.2011.4.273

[12]

Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2551-2576. doi: 10.3934/dcdsb.2018265

[13]

Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102

[14]

Amjad Khan, Dmitry E. Pelinovsky. Long-time stability of small FPU solitary waves. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2065-2075. doi: 10.3934/dcds.2017088

[15]

Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920

[16]

Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557

[17]

Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*. Evolution Equations & Control Theory, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023

[18]

Igor Chueshov, Stanislav Kolbasin. Long-time dynamics in plate models with strong nonlinear damping. Communications on Pure & Applied Analysis, 2012, 11 (2) : 659-674. doi: 10.3934/cpaa.2012.11.659

[19]

Eduard Feireisl, Françoise Issard-Roch, Hana Petzeltová. Long-time behaviour and convergence towards equilibria for a conserved phase field model. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 239-252. doi: 10.3934/dcds.2004.10.239

[20]

Tamara Fastovska. Long-time behaviour of a radially symmetric fluid-shell interaction system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1315-1348. doi: 10.3934/dcds.2018054

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (144)
  • HTML views (122)
  • Cited by (0)

Other articles
by authors

[Back to Top]