February 2017, 37(2): 915-944. doi: 10.3934/dcds.2017038

Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA

1The author is deceased.

Received  June 2015 Revised  January 2016 Published  November 2016

Fund Project: This work was supported, in part, by NSF via grants DMS-0908279, DMS-1119724 and DMS-1313687. CBM wishes to express his gratitude to V. Moroz for many valuable discussions

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in $\mathbb{R}^N$ with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For $L^2$ initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.

Citation: Cyrill B. Muratov, Xing Zhong. Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 915-944. doi: 10.3934/dcds.2017038
References:
[1]

S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal., 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2.

[2]

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

[3]

R. BamónI. Flores and M. del Pino, Ground states of semilinear elliptic equations: A geometric approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 551-581. doi: 10.1016/S0294-1449(00)00126-8.

[4]

P.W. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264. doi: 10.1016/S0022-1236(02)00013-7.

[5]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[6]

H. BerestyckiP.-L. Lions and L.A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb R^n$, Indiana Univ. Math. J., 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012.

[7]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on ${\bf R}^n$ or ${\bf R}^n_{+}$ through the method of moving planes, Comm. Partial Differential Equations, 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.

[8]

J. BuscaM.A. Jendoubi and P. Poláčik, Convergence to equilibrium for semilinear parabolic problems in $\mathbb{R}^n$, Comm. Partial Differential Equations, 27 (2002), 1793-1814. doi: 10.1081/PDE-120016128.

[9]

X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $\mathbb{R}^n$, C. R. Math. Acad. Sci. Paris, 338 (2004), 769-774. doi: 10.1016/j.crma.2004.03.013.

[10]

X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.

[11]

L.A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.

[12]

A. Capella-Kort, Stable Solutions of Nonlinear Elliptic Equations: Qualitative and Regularity Properties, PhD thesis, Universitat Politècnica de Catalunya, 2005.

[13]

E.N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., 131 (2003), 1891-1899. doi: 10.1090/S0002-9939-02-06733-3.

[14]

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.

[15]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[16]

E. Fašangová, Asymptotic analysis for a nonlinear parabolic equation on $\mathbb R$, Comment. Math. Univ. Carolinae, 39 (1998), 525--544.

[17]

E. Feireisl, On the long time behaviour of solutions to nonlinear diffusion equations on Rn, Nonlin. Diff. Eq. Appl., 4 (1997), 43-60. doi: 10.1007/PL00001410.

[18]

E. Feireisl and H. Petzeltová, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential Integral Equations, 10 (1997), 181-196.

[19]

P.C. Fife, Long time behavior of solutions of bistable nonlinear diffusion equations, Arch. Rational Mech. Anal., 70 (1979), 31-46. doi: 10.1007/BF00276380.

[20]

J. Földes and P. Poláčik, Convergence to a steady state for asymptotically autonomous semilinear heat equations on $\mathbb{R}^n$, J. Differential Equations, 251 (2011), 1903-1922. doi: 10.1016/j.jde.2011.04.002.

[21]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc. , Englewood Cliffs, NJ, 1964.

[22]

V.A. GalaktionovS.I. Pokhozhaev and A.E. Shishkov, On convergence in gradient systems with a degenerate equilibrium position, Mat. Sb., 198 (2007), 65-88. doi: 10.1070/SM2007v198n06ABEH003862.

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.

[24]

C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in Rn, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906.

[25]

C.K. R.T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364. doi: 10.1216/RMJ-1983-13-2-355.

[26]

C.K. R.T. Jones, Spherically symmetric solutions of a reaction-diffusion equation, J. Diff. Equations, 49 (1983), 142-169. doi: 10.1016/0022-0396(83)90023-2.

[27]

Y.I. Kanel', On the stabilization of solutions of the Cauchy problem for the equations arising in the theory of combusion, Mat. Sbornik, 59 (1962), 245-288.

[28]

B. S. Kerner and V. V. Osipov, Autosolitons, Kluwer, Dordrecht, 1994.

[29]

E. H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 1997.

[30]

C.S. Lin and W.-M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Math. Soc., 102 (1988), 271-277. doi: 10.1090/S0002-9939-1988-0920985-9.

[31]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Basel, 1995.

[32]

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

[33]

A.G. Merzhanov and E.N. Rumanov, Physics of reaction waves, Rev. Mod. Phys., 71 (1999), 1173-1210. doi: 10.1103/RevModPhys.71.1173.

[34]

A. S. Mikhailov, Foundations of Synergetics, Springer-Verlag, Berlin, 1990.

[35]

C.B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 867-892. doi: 10.3934/dcdsb.2004.4.867.

[36]

C.B. Muratov and M. Novaga, Front propagation in infinite cylinders. I. A variational approach, Comm. Math. Sci., 6 (2008), 799-826. doi: 10.4310/CMS.2008.v6.n4.a1.

[37]

C.B. Muratov and M. Novaga, Global stability and exponential convergence to variational traveling waves in cylinders, SIAM J. Math. Anal., 44 (2012), 293-315. doi: 10.1137/110833269.

[38]

C.B. Muratov and X. Zhong, Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations, Nonlin. Diff. Eq. Appl., 20 (2013), 1519-1552. doi: 10.1007/s00030-013-0220-7.

[39]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.

[40]

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

[41]

P. Poláčik, Morse indices and bifurcations of positive solutions of $Δ u+f(u)=0$ on $\mathbb{R}^n$, Indiana Univ. Math. J., 50 (2001), 1407-1432. doi: 10.1512/iumj.2001.50.1909.

[42]

P. Poláčik and K.P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations, 124 (1996), 472-494. doi: 10.1006/jdeq.1996.0020.

[43]

P. Poláčik and E. Yanagida, Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics, SIAM J. Math. Anal., 46 (2014), 3481-3496. doi: 10.1137/140958566.

[44]

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.

[45]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. , Birkhäuser Verlag, Basel, Switzerland, 2007.

[46]

V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 341-379. doi: 10.1016/S0294-1449(03)00042-8.

[47]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893.

[48]

J. Shi and X. Wang, Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations, J. Differential Equations, 231 (2006), 235-251. doi: 10.1016/j.jde.2006.06.008.

[49]

L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals Math., 118 (1983), 525-571. doi: 10.2307/2006981.

[50]

M. Tang, Existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations, J. Differential Equations, 164 (2000), 155-179. doi: 10.1006/jdeq.1999.3752.

[51]

K. Uchiyama, Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients, Arch. Rational Mech. Anal., 90 (1985), 291-311. doi: 10.1007/BF00276293.

[52]

J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296.

[53]

A. Zlatoš, Sharp transition between extinction and propagation of reaction, J. Amer. Math. Soc., 19 (2006), 251-263. doi: 10.1090/S0894-0347-05-00504-7.

show all references

References:
[1]

S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal., 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2.

[2]

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

[3]

R. BamónI. Flores and M. del Pino, Ground states of semilinear elliptic equations: A geometric approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 551-581. doi: 10.1016/S0294-1449(00)00126-8.

[4]

P.W. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264. doi: 10.1016/S0022-1236(02)00013-7.

[5]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[6]

H. BerestyckiP.-L. Lions and L.A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb R^n$, Indiana Univ. Math. J., 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012.

[7]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on ${\bf R}^n$ or ${\bf R}^n_{+}$ through the method of moving planes, Comm. Partial Differential Equations, 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.

[8]

J. BuscaM.A. Jendoubi and P. Poláčik, Convergence to equilibrium for semilinear parabolic problems in $\mathbb{R}^n$, Comm. Partial Differential Equations, 27 (2002), 1793-1814. doi: 10.1081/PDE-120016128.

[9]

X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $\mathbb{R}^n$, C. R. Math. Acad. Sci. Paris, 338 (2004), 769-774. doi: 10.1016/j.crma.2004.03.013.

[10]

X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.

[11]

L.A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.

[12]

A. Capella-Kort, Stable Solutions of Nonlinear Elliptic Equations: Qualitative and Regularity Properties, PhD thesis, Universitat Politècnica de Catalunya, 2005.

[13]

E.N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., 131 (2003), 1891-1899. doi: 10.1090/S0002-9939-02-06733-3.

[14]

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.

[15]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[16]

E. Fašangová, Asymptotic analysis for a nonlinear parabolic equation on $\mathbb R$, Comment. Math. Univ. Carolinae, 39 (1998), 525--544.

[17]

E. Feireisl, On the long time behaviour of solutions to nonlinear diffusion equations on Rn, Nonlin. Diff. Eq. Appl., 4 (1997), 43-60. doi: 10.1007/PL00001410.

[18]

E. Feireisl and H. Petzeltová, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential Integral Equations, 10 (1997), 181-196.

[19]

P.C. Fife, Long time behavior of solutions of bistable nonlinear diffusion equations, Arch. Rational Mech. Anal., 70 (1979), 31-46. doi: 10.1007/BF00276380.

[20]

J. Földes and P. Poláčik, Convergence to a steady state for asymptotically autonomous semilinear heat equations on $\mathbb{R}^n$, J. Differential Equations, 251 (2011), 1903-1922. doi: 10.1016/j.jde.2011.04.002.

[21]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc. , Englewood Cliffs, NJ, 1964.

[22]

V.A. GalaktionovS.I. Pokhozhaev and A.E. Shishkov, On convergence in gradient systems with a degenerate equilibrium position, Mat. Sb., 198 (2007), 65-88. doi: 10.1070/SM2007v198n06ABEH003862.

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.

[24]

C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in Rn, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906.

[25]

C.K. R.T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364. doi: 10.1216/RMJ-1983-13-2-355.

[26]

C.K. R.T. Jones, Spherically symmetric solutions of a reaction-diffusion equation, J. Diff. Equations, 49 (1983), 142-169. doi: 10.1016/0022-0396(83)90023-2.

[27]

Y.I. Kanel', On the stabilization of solutions of the Cauchy problem for the equations arising in the theory of combusion, Mat. Sbornik, 59 (1962), 245-288.

[28]

B. S. Kerner and V. V. Osipov, Autosolitons, Kluwer, Dordrecht, 1994.

[29]

E. H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 1997.

[30]

C.S. Lin and W.-M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Math. Soc., 102 (1988), 271-277. doi: 10.1090/S0002-9939-1988-0920985-9.

[31]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Basel, 1995.

[32]

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

[33]

A.G. Merzhanov and E.N. Rumanov, Physics of reaction waves, Rev. Mod. Phys., 71 (1999), 1173-1210. doi: 10.1103/RevModPhys.71.1173.

[34]

A. S. Mikhailov, Foundations of Synergetics, Springer-Verlag, Berlin, 1990.

[35]

C.B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 867-892. doi: 10.3934/dcdsb.2004.4.867.

[36]

C.B. Muratov and M. Novaga, Front propagation in infinite cylinders. I. A variational approach, Comm. Math. Sci., 6 (2008), 799-826. doi: 10.4310/CMS.2008.v6.n4.a1.

[37]

C.B. Muratov and M. Novaga, Global stability and exponential convergence to variational traveling waves in cylinders, SIAM J. Math. Anal., 44 (2012), 293-315. doi: 10.1137/110833269.

[38]

C.B. Muratov and X. Zhong, Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations, Nonlin. Diff. Eq. Appl., 20 (2013), 1519-1552. doi: 10.1007/s00030-013-0220-7.

[39]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.

[40]

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

[41]

P. Poláčik, Morse indices and bifurcations of positive solutions of $Δ u+f(u)=0$ on $\mathbb{R}^n$, Indiana Univ. Math. J., 50 (2001), 1407-1432. doi: 10.1512/iumj.2001.50.1909.

[42]

P. Poláčik and K.P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations, 124 (1996), 472-494. doi: 10.1006/jdeq.1996.0020.

[43]

P. Poláčik and E. Yanagida, Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics, SIAM J. Math. Anal., 46 (2014), 3481-3496. doi: 10.1137/140958566.

[44]

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.

[45]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. , Birkhäuser Verlag, Basel, Switzerland, 2007.

[46]

V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 341-379. doi: 10.1016/S0294-1449(03)00042-8.

[47]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893.

[48]

J. Shi and X. Wang, Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations, J. Differential Equations, 231 (2006), 235-251. doi: 10.1016/j.jde.2006.06.008.

[49]

L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals Math., 118 (1983), 525-571. doi: 10.2307/2006981.

[50]

M. Tang, Existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations, J. Differential Equations, 164 (2000), 155-179. doi: 10.1006/jdeq.1999.3752.

[51]

K. Uchiyama, Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients, Arch. Rational Mech. Anal., 90 (1985), 291-311. doi: 10.1007/BF00276293.

[52]

J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296.

[53]

A. Zlatoš, Sharp transition between extinction and propagation of reaction, J. Amer. Math. Soc., 19 (2006), 251-263. doi: 10.1090/S0894-0347-05-00504-7.

Table 1.  List of critical exponents.
Name Exponent Validity $N = 3$
Fujita $p_F = (N + 2)/N$ $N \geq 1$ 5/3
Serrin $p_{sg} = N / (N - 2)$ $N \geq 3$ 3
Sobolev $p_S = (N + 2) / (N - 2)$ $N \geq 3$ 5
Joseph-Lundgren $p_{JL} = 1 + 4/ \left( N - 4 - 2 \sqrt{N - 1} \, \right)$ $N \geq 11$ -
Name Exponent Validity $N = 3$
Fujita $p_F = (N + 2)/N$ $N \geq 1$ 5/3
Serrin $p_{sg} = N / (N - 2)$ $N \geq 3$ 3
Sobolev $p_S = (N + 2) / (N - 2)$ $N \geq 3$ 5
Joseph-Lundgren $p_{JL} = 1 + 4/ \left( N - 4 - 2 \sqrt{N - 1} \, \right)$ $N \geq 11$ -
[1]

Wen Shen, Karim Shikh-Khalil. Traveling waves for a microscopic model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2571-2589. doi: 10.3934/dcds.2018108

[2]

Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231

[3]

Wenzhang Huang. Weakly coupled traveling waves for a model of growth and competition in a flow reactor. Mathematical Biosciences & Engineering, 2006, 3 (1) : 79-87. doi: 10.3934/mbe.2006.3.79

[4]

Yanxia Wu, Yaping Wu. Existence of traveling waves with transition layers for some degenerate cross-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 911-934. doi: 10.3934/cpaa.2012.11.911

[5]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i

[6]

Wenzhang Huang. Co-existence of traveling waves for a model of microbial growth and competition in a flow reactor. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 883-896. doi: 10.3934/dcds.2009.24.883

[7]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[8]

Jonatan Lenells. Traveling waves in compressible elastic rods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 151-167. doi: 10.3934/dcdsb.2006.6.151

[9]

Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure & Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623

[10]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[11]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69

[12]

Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935

[13]

Da-Peng Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2601-2614. doi: 10.3934/dcdsb.2016063

[14]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141

[15]

Guangyu Zhao. Multidimensional periodic traveling waves in infinite cylinders. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1025-1045. doi: 10.3934/dcds.2009.24.1025

[16]

Alejandro B. Aceves, Luis A. Cisneros-Ake, Antonmaria A. Minzoni. Asymptotics for supersonic traveling waves in the Morse lattice. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 975-994. doi: 10.3934/dcdss.2011.4.975

[17]

Thuc Manh Le, Nguyen Van Minh. Monotone traveling waves in a general discrete model for populations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3221-3234. doi: 10.3934/dcdsb.2017171

[18]

Chuncheng Wang, Rongsong Liu, Junping Shi, Carlos Martinez del Rio. Traveling waves of a mutualistic model of mistletoes and birds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1743-1765. doi: 10.3934/dcds.2015.35.1743

[19]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[20]

Zhiting Xu. Traveling waves for a diffusive SEIR epidemic model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 871-892. doi: 10.3934/cpaa.2016.15.871

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]