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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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

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

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

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

[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. Google Scholar

[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. Google Scholar

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[48]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

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

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

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

[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. Google Scholar

[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. Google Scholar

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[48]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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