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February  2017, 37(2): 743-756. doi: 10.3934/dcds.2017031

Transition fronts and stretching phenomena for a general class of reaction-dispersion equations

1. 

Université Savoie Mont-Blanc, LAMA, F-73000 Chambéry, France

2. 

CNRS, LAMA, F-73000 Chambéry, France

3. 

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

4. 

BioSP, INRA, 84914, Avignon, France

* Corresponding author: François Hamel

Received  April 2015 Revised  September 2015 Published  November 2016

Fund Project: This work has been carried out in the framework of Archimède LabEx (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR). The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n.321186 -ReaDi -Reaction-Diffusion Equations, Propagation and Modelling, and from the ANR project NONLOCAL (ANR-14-CE25-0013).

We consider a general form of reaction-dispersion equations with non-local or nonlinear dispersal operators and local reaction terms. Under some general conditions, we prove the non-existence of transition fronts, as well as some stretching properties at large time for the solutions of the Cauchy problem. These conditions are satisfied in particular when the reaction is monostable and when the dispersal operator is either the fractional Laplacian, a convolution operator with a fat-tailed kernel or a nonlinear fast diffusion operator.

Citation: Jimmy Garnier, FranÇois Hamel, Lionel Roques. Transition fronts and stretching phenomena for a general class of reaction-dispersion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 743-756. doi: 10.3934/dcds.2017031
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]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in Perspectives in Nonlinear Partial Differential Equations. In honor of H. Brezis, Amer. Math. Soc. , Contemp. Math. , 446 (2007), 101-123.

[4]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.

[5]

X. CabréA.-C. Coulon and J.-M. Roquejoffre, Propagation in Fisher-KPP type equations with fractional diffusion in periodic media, C. R. Math. Acad. Sci. Paris, 350 (2012), 885-890.  doi: 10.1016/j.crma.2012.10.007.

[6]

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.

[7]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Am. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.

[8]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Diff. Equations, 2 (1997), 125-160. 

[9]

A. Chmaj, Existence of traveling waves in the fractional bistable equation, Arch. Math., 100 (2013), 473-480.  doi: 10.1007/s00013-013-0511-6.

[10]

J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Amer. Natur., 152 (1998), 204-224.  doi: 10.1086/286162.

[11]

A. -C. Coulon, Propagation in Reaction-Diffusion Equations with Fractional Diffusion Ph. D thesis, Université Paul Sabatier and Universitat Politécnica de Catalunya, 2014.

[12]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.  doi: 10.1007/s10231-005-0163-7.

[13]

J. CovilleJ. Dávila and S. Martínez, Non-local anisotropic dispersal with monostable nonlinearity, J. Diff. Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.

[14]

J. CovilleJ. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.

[15]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.

[16]

J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.

[17]

D. del-Castillo-Negrete, Truncation effects in superdiffusive front propagation with Lévy flights, Phys. Rev., 79 (2009), 031120.  doi: 10.1103/PhysRevE.79.031120.

[18]

D. del-Castillo-NegreteB. A. Carreras and V. Lynch, Front propagation and segregation in a reaction-diffusion model with cross-diffusion, Phys. D, 168/169 (2002), 45-60.  doi: 10.1016/S0167-2789(02)00494-3.

[19] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin, New York, 1979. 
[20]

P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.

[21]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369. 

[22]

J. Garnier, Fast propagation in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.

[23]

C. Gui and T. Huan, Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc. Var. Part. Diff. Equations, 54 (2015), 251-273.  doi: 10.1007/s00526-014-0785-y.

[24]

C. Gui and M. Zhao, Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 32 (2015), 785-812.  doi: 10.1016/j.anihpc.2014.03.005.

[25]

F. Hamel and N. Nadirashvili, Travelling waves and entire solutions of the Fisher-KPP equation in $\mathbb{R}^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.

[26]

F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Diff. Equations, 249 (2010), 1726-1745.  doi: 10.1016/j.jde.2010.06.025.

[27]

F. Hamel and L. Rossi, Transition fronts for the Fisher-KPP equation, Trans. Amer. Math. Soc., 368 (2016), 8675-8713.  doi: 10.1090/tran/6609.

[28]

F. Hamel and L. Rossi, Admissible speeds of transition fronts for non-autonomous monostable equations, SIAM J. Math. Anal., 47 (2015), 3342-3392.  doi: 10.1137/140995519.

[29]

J. R. King and P. M. McCabe, On the Fisher-KPP equation with fast nonlinear diffusion, Proc. Royal Soc. A, 459 (2003), 2529-2546.  doi: 10.1098/rspa.2003.1134.

[30]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique, Bull. Univ. Etat Moscou, Série Intern. A, 1 (1937), 1-26. 

[31]

M. KotM. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.  doi: 10.2307/2265698.

[32]

R. MancinelliD. Vergni and A. Vulpiani, Front propagation in reactive systems with anomalous diffusion, Phys. D, 185 (2003), 175-195.  doi: 10.1016/S0167-2789(03)00235-5.

[33]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.

[34]

A. MelletJ. NolenJ.-M. Roquejoffre and L. Ryzhik, Stability of generalized transition fronts, Comm. Part. Diff. Equations, 34 (2009), 521-552.  doi: 10.1080/03605300902768677.

[35]

A. MelletJ.-M. Roquejoffre and Y. Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Disc. Cont. Dyn. Syst. A, 26 (2010), 303-312.  doi: 10.3934/dcds.2010.26.303.

[36]

A. MelletJ.-M. Roquejoffre and Y. Sire, Existence and asymptotics of fronts in non local combustion models, Comm. Math. Sci., 12 (2014), 1-11.  doi: 10.4310/CMS.2014.v12.n1.a1.

[37]

D. Mollison, Spatial contact models for ecological and epidemic spread, J. Royal Stat. Ser. B Stat. Methodol, 39 (1977), 283-326. 

[38]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653.  doi: 10.1016/j.matpur.2012.05.005.

[39]

J. NolenJ.-M. RoquejoffreL. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217-246.  doi: 10.1007/s00205-011-0449-4.

[40]

J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 26 (2009), 1021-1047.  doi: 10.1016/j.anihpc.2009.02.003.

[41]

J. -M. Roquejoffre and A. Tarfulea, Gradient estimates and symmetrization for Fisher-KPP front propagation with fractional diffusion, arXiv: 1502.06304.

[42]

D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Equations, 25 (1977), 130-144.  doi: 10.1016/0022-0396(77)90185-1.

[43]

K. Schumacher, Travelling-front solutions for integro-differential equations. Ⅰ, J. Reine Angew. Math., 316 (1980), 54-70.  doi: 10.1515/crll.1980.316.54.

[44]

W. Shen, Traveling waves in diffusive random media, J. Dyn. Diff. Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.

[45]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Diff. Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.

[46]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101. 

[47]

D. Stan and J. L. Vázquez, The Fisher-KPP equation with nonlinear fractional diffusion, SIAM J. Math. Anal., 46 (2014), 3241-3276.  doi: 10.1137/130918289.

[48]

K. Uchiyama, The behavior of solutions of some semilinear diffusion equation for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. 

[49]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. RIMS Kyoto Univ., 45 (2009), 925-953.  doi: 10.2977/prims/1260476648.

[50]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 89-102.  doi: 10.1016/j.matpur.2011.11.007.

[51]

A. Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability, Arch. Ration. Mech. Anal., 208 (2013), 447-480.  doi: 10.1007/s00205-012-0600-x.

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]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in Perspectives in Nonlinear Partial Differential Equations. In honor of H. Brezis, Amer. Math. Soc. , Contemp. Math. , 446 (2007), 101-123.

[4]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.

[5]

X. CabréA.-C. Coulon and J.-M. Roquejoffre, Propagation in Fisher-KPP type equations with fractional diffusion in periodic media, C. R. Math. Acad. Sci. Paris, 350 (2012), 885-890.  doi: 10.1016/j.crma.2012.10.007.

[6]

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.

[7]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Am. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.

[8]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Diff. Equations, 2 (1997), 125-160. 

[9]

A. Chmaj, Existence of traveling waves in the fractional bistable equation, Arch. Math., 100 (2013), 473-480.  doi: 10.1007/s00013-013-0511-6.

[10]

J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Amer. Natur., 152 (1998), 204-224.  doi: 10.1086/286162.

[11]

A. -C. Coulon, Propagation in Reaction-Diffusion Equations with Fractional Diffusion Ph. D thesis, Université Paul Sabatier and Universitat Politécnica de Catalunya, 2014.

[12]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.  doi: 10.1007/s10231-005-0163-7.

[13]

J. CovilleJ. Dávila and S. Martínez, Non-local anisotropic dispersal with monostable nonlinearity, J. Diff. Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.

[14]

J. CovilleJ. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.

[15]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.

[16]

J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.

[17]

D. del-Castillo-Negrete, Truncation effects in superdiffusive front propagation with Lévy flights, Phys. Rev., 79 (2009), 031120.  doi: 10.1103/PhysRevE.79.031120.

[18]

D. del-Castillo-NegreteB. A. Carreras and V. Lynch, Front propagation and segregation in a reaction-diffusion model with cross-diffusion, Phys. D, 168/169 (2002), 45-60.  doi: 10.1016/S0167-2789(02)00494-3.

[19] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin, New York, 1979. 
[20]

P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.

[21]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369. 

[22]

J. Garnier, Fast propagation in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.

[23]

C. Gui and T. Huan, Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc. Var. Part. Diff. Equations, 54 (2015), 251-273.  doi: 10.1007/s00526-014-0785-y.

[24]

C. Gui and M. Zhao, Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 32 (2015), 785-812.  doi: 10.1016/j.anihpc.2014.03.005.

[25]

F. Hamel and N. Nadirashvili, Travelling waves and entire solutions of the Fisher-KPP equation in $\mathbb{R}^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.

[26]

F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Diff. Equations, 249 (2010), 1726-1745.  doi: 10.1016/j.jde.2010.06.025.

[27]

F. Hamel and L. Rossi, Transition fronts for the Fisher-KPP equation, Trans. Amer. Math. Soc., 368 (2016), 8675-8713.  doi: 10.1090/tran/6609.

[28]

F. Hamel and L. Rossi, Admissible speeds of transition fronts for non-autonomous monostable equations, SIAM J. Math. Anal., 47 (2015), 3342-3392.  doi: 10.1137/140995519.

[29]

J. R. King and P. M. McCabe, On the Fisher-KPP equation with fast nonlinear diffusion, Proc. Royal Soc. A, 459 (2003), 2529-2546.  doi: 10.1098/rspa.2003.1134.

[30]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique, Bull. Univ. Etat Moscou, Série Intern. A, 1 (1937), 1-26. 

[31]

M. KotM. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.  doi: 10.2307/2265698.

[32]

R. MancinelliD. Vergni and A. Vulpiani, Front propagation in reactive systems with anomalous diffusion, Phys. D, 185 (2003), 175-195.  doi: 10.1016/S0167-2789(03)00235-5.

[33]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.

[34]

A. MelletJ. NolenJ.-M. Roquejoffre and L. Ryzhik, Stability of generalized transition fronts, Comm. Part. Diff. Equations, 34 (2009), 521-552.  doi: 10.1080/03605300902768677.

[35]

A. MelletJ.-M. Roquejoffre and Y. Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Disc. Cont. Dyn. Syst. A, 26 (2010), 303-312.  doi: 10.3934/dcds.2010.26.303.

[36]

A. MelletJ.-M. Roquejoffre and Y. Sire, Existence and asymptotics of fronts in non local combustion models, Comm. Math. Sci., 12 (2014), 1-11.  doi: 10.4310/CMS.2014.v12.n1.a1.

[37]

D. Mollison, Spatial contact models for ecological and epidemic spread, J. Royal Stat. Ser. B Stat. Methodol, 39 (1977), 283-326. 

[38]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653.  doi: 10.1016/j.matpur.2012.05.005.

[39]

J. NolenJ.-M. RoquejoffreL. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217-246.  doi: 10.1007/s00205-011-0449-4.

[40]

J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 26 (2009), 1021-1047.  doi: 10.1016/j.anihpc.2009.02.003.

[41]

J. -M. Roquejoffre and A. Tarfulea, Gradient estimates and symmetrization for Fisher-KPP front propagation with fractional diffusion, arXiv: 1502.06304.

[42]

D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Equations, 25 (1977), 130-144.  doi: 10.1016/0022-0396(77)90185-1.

[43]

K. Schumacher, Travelling-front solutions for integro-differential equations. Ⅰ, J. Reine Angew. Math., 316 (1980), 54-70.  doi: 10.1515/crll.1980.316.54.

[44]

W. Shen, Traveling waves in diffusive random media, J. Dyn. Diff. Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.

[45]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Diff. Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.

[46]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101. 

[47]

D. Stan and J. L. Vázquez, The Fisher-KPP equation with nonlinear fractional diffusion, SIAM J. Math. Anal., 46 (2014), 3241-3276.  doi: 10.1137/130918289.

[48]

K. Uchiyama, The behavior of solutions of some semilinear diffusion equation for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. 

[49]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. RIMS Kyoto Univ., 45 (2009), 925-953.  doi: 10.2977/prims/1260476648.

[50]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 89-102.  doi: 10.1016/j.matpur.2011.11.007.

[51]

A. Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability, Arch. Ration. Mech. Anal., 208 (2013), 447-480.  doi: 10.1007/s00205-012-0600-x.

Figure 1.  Propagation to the right of the solution of the Cauchy problem (2), at successive times $t=0,1,\ldots,20,$ with (a) $\mathcal D u = -(-\Delta_x)^\alpha u$ with $\alpha=0.9$; (b) $\mathcal D u = J*u-u$ with $J(x)=\exp(-\sqrt{|x|})/4$; (c) $\mathcal D u=(u^\gamma)_{xx}$ with $\gamma=1/2$; and (d) $\mathcal D u = u_{xx}$. In all cases, the initial condition was $u_0(x)=\exp(-x^2/100)$ and the function $f$ was of the KPP type $f(u)=u\, (1-u)$.
Figure 2.  Distance $x_{0.4}(t)-x_{0.6}(t)$ between two level sets of the solution of the Cauchy problem (2), for $t\in(0,20)$, with: (solid line) $\mathcal D u = -(-\Delta_x)^\alpha u$ with $\alpha=0.9$; (dashed line) $\mathcal D u = J*u-u$ with $J(x)=\exp(-\sqrt{|x|})/4$; (dash-dot line) $\mathcal D u=(u^\gamma)_{xx}$ with $\gamma=1/2$; and (circles) $\mathcal D u = u_{xx}$. The assumptions on $f$ and $u_0$ are the same as in Fig. 1
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