American Institute of Mathematical Sciences

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February  2018, 38(2): 791-821. doi: 10.3934/dcds.2018034

N-barrier maximum principle for degenerate elliptic systems and its application

Chiun-Chuan Chen 1, , Li-Chang Hung 2,, and  Hsiao-Feng Liu 3,

1. 

Department of Mathematics, National Taiwan University, National Center for Theoretical Sciences, Taipei, Taiwan
2. 

College of Engineering, National Taiwan University of Science and Technology, Department of Mathematics, National Taiwan University, Taipei, Taiwan
3. 

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan

* Corresponding author

Received  November 2016 Revised  September 2017 Published  February 2018

Fund Project: The research of C.-C. Chen is partly supported by the grant 102-2115-M-002-011-MY3 of Ministry of Science and Technology, Taiwan. The research of L.-C. Hung is partly supported by the grant 104EFA0101550 of Ministry of Science and Technology, Taiwan.

In this paper, we prove the N-barrier maximum principle, which extends the result in C.-C. Chen and L.-C. Hung (2016) from linear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems of porous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of the solutions to the above-mentioned degenerate nonlinear diffusion equations including the Shigesada-Kawasaki-Teramoto model as a special case. We also apply the N-barrier maximum principle to a coexistence problem in ecology, where we show the nonexistence of traveling waves in a three-species degenerate elliptic system.

Keywords: Maximum principles, traveling wave solutions, degenerate elliptic systems, reaction-diffusion equations, Lotka-Volterra.
Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.
Citation: Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034
References:
[1]

M. W. Adamson and A. Y. Morozov, Revising the role of species mobility in maintaining biodiversity in communities with cyclic competition, Bull. Math. Biol., 74 (2012), 2004-2031.  doi: 10.1007/s11538-012-9743-z.

[2]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.

[3]

A. J. Baczkowski, D. N. Joanes and G. M. Shamia, Range of validity of $α$ and $β$ for a generalized diversity index $H(α,β)$ due to Good, Math. Biosci., 148 (1998), 115-128.  doi: 10.1016/S0025-5564(97)10013-X.

[4]

R. S. Cantrell and J. R., Jr. Ward, On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.  doi: 10.1137/S0036139995292367.

[5]

C.-C. Chen and L.-C. Hung, A maximum principle for diffusive lotka-volterra systems of two competing species, J. Differential Equations, 261 (2016), 4573-4592.  doi: 10.1016/j.jde.2016.07.001.

[6]

C.-C. Chen and L.-C. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl. Anal., 15 (2016), 1451-1469.  doi: 10.3934/cpaa.2016.15.1451.

[7]

C. -C. Chen, L. -C. Hung and C. -C. Lai, An N-barrier maximum principle for autonomous systems of n species and its application to problems arising from population dynamics, submitted.

[8]

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653-2669.  doi: 10.3934/dcdsb.2012.17.2653.

[9]

P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234.  doi: 10.1137/S0036139995294767.

[10]

S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, Interfaces Free Bound, 1 (1999), 57-80.  doi: 10.4171/IFB/4.

[11]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction Progress in Nonlinear Differential Equations and their Applications, 60, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.

[12]

I. J. Good, The population frequencies of species and the estimation of population parameters, Biometrika, 40 (1953), 237-264.  doi: 10.1093/biomet/40.3-4.237.

[13]

S. Grossberg, Decisions, patterns, and oscillations in nonlinear competitve systems with applications to Volterra-Lotka systems, J. Theoret. Biol., 73 (1978), 101-130.  doi: 10.1016/0022-5193(78)90182-0.

[14]

M. Guedda, R. Kersner, M. Klincsik and E. Logak, Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1589-1600.  doi: 10.3934/dcdsb.2014.19.1589.

[15]

W. Gurney and R. Nisbet, The regulation of inhomogeneous populations, Journal of Theoretical Biology, 52 (1975), 441-457.  doi: 10.1016/0022-5193(75)90011-9.

[16]

W. Gurney and R. Nisbet, A note on non-linear population transport, Journal of theoretical biology, 56 (1976), 249-251.  doi: 10.1016/S0022-5193(76)80056-2.

[17]

M. Gyllenberg and P. Yan, On a conjecture for three-dimensional competitive Lotka-Volterra systems with a heteroclinic cycle, Differ. Equ. Appl., 1 (2009), 473-490.  doi: 10.7153/dea-01-26.

[18]

T. G. Hallam, L. J. Svoboda and T. C. Gard, Persistence and extinction in three species Lotka-Volterra competitive systems, Math. Biosci., 46 (1979), 117-124.  doi: 10.1016/0025-5564(79)90018-X.

[19]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.  doi: 10.1137/070700784.

[20]

S. B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.

[21]

L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251.  doi: 10.1007/s13160-012-0056-2.

[22]

S. R.-J. Jang, Competitive exclusion and coexistence in a Leslie-Gower competition model with Allee effects, Appl. Anal., 92 (2013), 1527-1540.  doi: 10.1080/00036811.2012.692365.

[23]

J. Kastendiek, Competitor-mediated coexistence: interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210.  doi: 10.1016/0022-0981(82)90201-5.

[24]

W. Ko, K. Ryu and I. Ahn, Coexistence of three competing species with non-negative cross-diffusion rate, J. Dyn. Control Syst., 20 (2014), 229-240.  doi: 10.1007/s10883-014-9219-6.

[25]

R. S. Maier, The integration of three-dimensional Lotka-Volterra systems Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 469 (2013), 20120693, 27pp. doi: 10.1098/rspa.2012.0693.

[26]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion, J. Differential Equations, 23 (1977), 30-52.  doi: 10.1016/0022-0396(77)90135-8.

[27]

M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition--diffusion system, Ecological Complexity, 21 (2015), 215-232.  doi: 10.1016/j.ecocom.2014.05.004.

[28]

J. D. Murray, Mathematical Biology Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.

[29]

A. de Pablo and A. Sánchez, Travelling wave behaviour for a porous-Fisher equation, European J. Appl. Math., 9 (1998), 285-304.  doi: 10.1017/S0956792598003465.

[30]

A. de Pablo and J. L. Vázquez, 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.

[31]

S. Petrovskii, K. Kawasaki, F. Takasu and N. Shigesada, Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species, Japan J. Indust. Appl. Math., 18 (2001), 459-481.  doi: 10.1007/BF03168586.

[32]

H. Ramezani and S. Holm, Sample based estimation of landscape metrics; accuracy of line intersect sampling for estimating edge density and Shannon's diversity index, Environ. Ecol. Stat., 18 (2011), 109-130.  doi: 10.1007/s10651-009-0123-2.

[33]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270. 

[34]

M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657-696.  doi: 10.1007/BF03167410.

[35]

F. Sánchez-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.

[36]

F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319.  doi: 10.1006/jdeq.1995.1055.

[37]

J. A. Sherratt and B. P. Marchant, Nonsharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion, Appl. Math. Lett., 9 (1996), 33-38.  doi: 10.1016/0893-9659(96)00069-9.

[38]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[39]

E. H. Simpson, Measurement of diversity Nature, 163 (1949), p688. doi: 10.1038/163688a0.

[40]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.

[41]

T. P. Witelski, Merging traveling waves for the porous-Fisher's equation, Appl. Math. Lett., 8 (1995), 57-62.  doi: 10.1016/0893-9659(95)00047-T.

[42]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.  doi: 10.1080/02681119308806158.

[43]

J. B. Zeldovič, G. I. Barenblatt, V. B. Librovič and G. M. Mahviladze, Matematicheskaya Teoriya Goreniya I Vzryva (Mathematical Theory of Combustion and Explosion) "Nauka", Moscow, 1980.

[44]

Y. B. Zeldovich, Theory of Flame Propagation 1951.

show all references

References:
[1]

M. W. Adamson and A. Y. Morozov, Revising the role of species mobility in maintaining biodiversity in communities with cyclic competition, Bull. Math. Biol., 74 (2012), 2004-2031.  doi: 10.1007/s11538-012-9743-z.

[2]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.

[3]

A. J. Baczkowski, D. N. Joanes and G. M. Shamia, Range of validity of $α$ and $β$ for a generalized diversity index $H(α,β)$ due to Good, Math. Biosci., 148 (1998), 115-128.  doi: 10.1016/S0025-5564(97)10013-X.

[4]

R. S. Cantrell and J. R., Jr. Ward, On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.  doi: 10.1137/S0036139995292367.

[5]

C.-C. Chen and L.-C. Hung, A maximum principle for diffusive lotka-volterra systems of two competing species, J. Differential Equations, 261 (2016), 4573-4592.  doi: 10.1016/j.jde.2016.07.001.

[6]

C.-C. Chen and L.-C. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl. Anal., 15 (2016), 1451-1469.  doi: 10.3934/cpaa.2016.15.1451.

[7]

C. -C. Chen, L. -C. Hung and C. -C. Lai, An N-barrier maximum principle for autonomous systems of n species and its application to problems arising from population dynamics, submitted.

[8]

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653-2669.  doi: 10.3934/dcdsb.2012.17.2653.

[9]

P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234.  doi: 10.1137/S0036139995294767.

[10]

S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, Interfaces Free Bound, 1 (1999), 57-80.  doi: 10.4171/IFB/4.

[11]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction Progress in Nonlinear Differential Equations and their Applications, 60, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.

[12]

I. J. Good, The population frequencies of species and the estimation of population parameters, Biometrika, 40 (1953), 237-264.  doi: 10.1093/biomet/40.3-4.237.

[13]

S. Grossberg, Decisions, patterns, and oscillations in nonlinear competitve systems with applications to Volterra-Lotka systems, J. Theoret. Biol., 73 (1978), 101-130.  doi: 10.1016/0022-5193(78)90182-0.

[14]

M. Guedda, R. Kersner, M. Klincsik and E. Logak, Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1589-1600.  doi: 10.3934/dcdsb.2014.19.1589.

[15]

W. Gurney and R. Nisbet, The regulation of inhomogeneous populations, Journal of Theoretical Biology, 52 (1975), 441-457.  doi: 10.1016/0022-5193(75)90011-9.

[16]

W. Gurney and R. Nisbet, A note on non-linear population transport, Journal of theoretical biology, 56 (1976), 249-251.  doi: 10.1016/S0022-5193(76)80056-2.

[17]

M. Gyllenberg and P. Yan, On a conjecture for three-dimensional competitive Lotka-Volterra systems with a heteroclinic cycle, Differ. Equ. Appl., 1 (2009), 473-490.  doi: 10.7153/dea-01-26.

[18]

T. G. Hallam, L. J. Svoboda and T. C. Gard, Persistence and extinction in three species Lotka-Volterra competitive systems, Math. Biosci., 46 (1979), 117-124.  doi: 10.1016/0025-5564(79)90018-X.

[19]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.  doi: 10.1137/070700784.

[20]

S. B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.

[21]

L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251.  doi: 10.1007/s13160-012-0056-2.

[22]

S. R.-J. Jang, Competitive exclusion and coexistence in a Leslie-Gower competition model with Allee effects, Appl. Anal., 92 (2013), 1527-1540.  doi: 10.1080/00036811.2012.692365.

[23]

J. Kastendiek, Competitor-mediated coexistence: interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210.  doi: 10.1016/0022-0981(82)90201-5.

[24]

W. Ko, K. Ryu and I. Ahn, Coexistence of three competing species with non-negative cross-diffusion rate, J. Dyn. Control Syst., 20 (2014), 229-240.  doi: 10.1007/s10883-014-9219-6.

[25]

R. S. Maier, The integration of three-dimensional Lotka-Volterra systems Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 469 (2013), 20120693, 27pp. doi: 10.1098/rspa.2012.0693.

[26]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion, J. Differential Equations, 23 (1977), 30-52.  doi: 10.1016/0022-0396(77)90135-8.

[27]

M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition--diffusion system, Ecological Complexity, 21 (2015), 215-232.  doi: 10.1016/j.ecocom.2014.05.004.

[28]

J. D. Murray, Mathematical Biology Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.

[29]

A. de Pablo and A. Sánchez, Travelling wave behaviour for a porous-Fisher equation, European J. Appl. Math., 9 (1998), 285-304.  doi: 10.1017/S0956792598003465.

[30]

A. de Pablo and J. L. Vázquez, 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.

[31]

S. Petrovskii, K. Kawasaki, F. Takasu and N. Shigesada, Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species, Japan J. Indust. Appl. Math., 18 (2001), 459-481.  doi: 10.1007/BF03168586.

[32]

H. Ramezani and S. Holm, Sample based estimation of landscape metrics; accuracy of line intersect sampling for estimating edge density and Shannon's diversity index, Environ. Ecol. Stat., 18 (2011), 109-130.  doi: 10.1007/s10651-009-0123-2.

[33]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270. 

[34]

M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657-696.  doi: 10.1007/BF03167410.

[35]

F. Sánchez-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.

[36]

F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319.  doi: 10.1006/jdeq.1995.1055.

[37]

J. A. Sherratt and B. P. Marchant, Nonsharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion, Appl. Math. Lett., 9 (1996), 33-38.  doi: 10.1016/0893-9659(96)00069-9.

[38]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[39]

E. H. Simpson, Measurement of diversity Nature, 163 (1949), p688. doi: 10.1038/163688a0.

[40]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.

[41]

T. P. Witelski, Merging traveling waves for the porous-Fisher's equation, Appl. Math. Lett., 8 (1995), 57-62.  doi: 10.1016/0893-9659(95)00047-T.

[42]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.  doi: 10.1080/02681119308806158.

[43]

J. B. Zeldovič, G. I. Barenblatt, V. B. Librovič and G. M. Mahviladze, Matematicheskaya Teoriya Goreniya I Vzryva (Mathematical Theory of Combustion and Explosion) "Nauka", Moscow, 1980.

[44]

Y. B. Zeldovich, Theory of Flame Propagation 1951.

Figure 1.  Red line: $1-u-a_1\, v=0$; blue line: $1-a_2\, u-v=0$; green curve: $F(u, v):=\alpha\, u\, (1-u-a_1\, v)+\beta\, v\, (1-a_2\, u-v)=0$; brown line: $\displaystyle\frac{u}{\underline{u}}+\frac{v}{\underline{v}}=1$, where $\underline{u}$ and $\underline{v}$ are given by (55) and (56); magenta ellipse (above): $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_1$, where $\lambda_1$ is given by (61); magenta ellipse (below): $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_2$, where $\lambda_2$ is given by (70); yellow line (above): $\alpha\, u+\beta\, v=\eta_1$, where $\eta_1$ is given by (62); yellow line (below): $\alpha\, u+\beta\, v=\eta_2$, where $\eta_2$ is given by (71); dashed orange curve: the solution $(u(x), v(x))$; dotted line (above): $\displaystyle\frac{u}{\sqrt{\frac{\lambda_1}{\alpha\, d_1}}}+\displaystyle\frac{v}{\sqrt{\frac{\lambda_1}{\beta\, d_2}}}=1$; dotted line (below): $\displaystyle\frac{u}{\sqrt{\frac{\lambda_2}{\alpha\, d_1}}}+\displaystyle\frac{v}{\sqrt{\frac{\lambda_2}{\beta\, d_2}}}=1$
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Figure 2.  Red line: $1-u-a_1\, v=0$; blue line: $1-a_2\, u-v=0$; green curve: $F(u, v)=0$; brown line: $\displaystyle\frac{u}{\overline{u}}+\frac{v}{\overline{v}}=1$, where $\overline{u}$ and $\overline{v}$ are given by (55) and (56); magenta ellipses : $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_1, \lambda_2$, where $\lambda_1$ (below) is given by (96) and $\lambda_2$ (above) by (103); yellow lines: $\alpha\, u+\beta\, v=\eta_1, \eta_2$, where $\eta_1$ (below) is given by (102) and $\eta_2$ (above) by (109); dashed orange curve: the solution $(u(x), v(x))$; dotted lines: $\displaystyle\sqrt{\alpha\, d_1}\, u+\sqrt{\beta\, d_2}\, v=\sqrt{\lambda_1}$ (below), $\displaystyle\sqrt{\lambda_2}$ (above); $\overline{u}=\overline{v}=1$; $d_1=3$, $a_1=2$, $a_2=3$, $\alpha=1$
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Figure 3.  Red: $u(x) =60\, \big(1-\tanh x\big)^2$; green: $v(x) =8\, \big(1+\tanh x\big)$
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Figure 4.  Red: $u(x) =\displaystyle\frac{1}{10}\big(1-\cos\, (2\, x)\big)$; green: $v(x) =\displaystyle\frac{1}{11}\big(1+\cos\, (2\, x)\big)$; blue: $w(x) =\displaystyle\frac{1}{12}\big(1+\cos\, (2\, x)\big)$
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