[6] M. BERTSCH, C. B. MURATOV, and I. PRIMI. "Traveling wave solutions of harmonic heat flow". In: *Calc. Var. Partial Differ. Equ.* 26.4 (2006), pp. 489–509. DOI: 10.1007/s00526-006-0016-2 (cit. on p. 18).

[7] J. BOUHOURS and G. NADIN. "A variational approach to reaction-diffusion equations with forced speed in dimension 1". In: *Discret. Contin. Dyn. Syst.* - A 35.5 (2015), pp. 1843–1872. DOI: 10.3934/dcds.2015.35.1843. arXiv: 1310.3689 (cit. on p. 18).

[8] C. N. CHEN, C. C. CHEN, and C. C. HUANG. "Traveling waves for the FitzHugh-Nagumo system on an infinite channel". In: *J. Differ. Equ.* 261.6 (2016), pp. 3010–3041. DOI: 10.1016/j.jde.2016.05.014 (cit. on p. 18).

[9] C. N. CHEN and Y. S. CHOI. "Traveling pulse solutions to FitzHugh-Nagumo equations". In: *Calc. Var. Partial Differ. Equ.* 54.1 (2015), pp. 1–45. DOI: 10.1007/s00526-014-0776-z (cit. on p. 18).

[10] C. C. CHEN, H. Y. CHIEN, and C. C. HUANG. "A variational approach to three-phase traveling waves for a gradient system". In: *Discret. Contin. Dyn. Syst. Ser. A* 41.10 (2021), pp. 4737–4765. DOI: 10.3934/dcds.2021055 (cit. on p. 18).

[11] M. C. CROSS and P. C. HOHENBERG. "Pattern formation outside of equilibrium". In: *Rev. Mod. Phys.* 65.3 (1993), pp. 851–1112. DOI: 10.1103/RevModPhys.65.851 (cit. on p. 6).

[12] P. C. FIFE and J. B. McLEOD. "The approach of solutions of nonlinear diffusion equations to travelling front solutions". In: *Arch. Ration. Mech. Anal.* 65.4 (1977), pp. 335–361. DOI: 10.1007/BF00250432 (cit. on pp. 16, 17).

[13] P. C. FIFE and J. B. McLEOD. "A phase plane discussion of convergence to travelling fronts for nonlinear diffusion". In: *Arch. Ration. Mech. Anal.* 75.4 (1981), pp. 281–314. DOI: 10.1007/BF00256381 (cit. on p. 16).

[14] R. A. FISHER. "The wave of advance of advantageous genes". In: *Ann. Eugen.* 7.4 (1937), pp. 355–369. DOI: 10.1111/j.1469-1809.1937.tb02153.x (cit. on p. 73).

[15] T. GALLAY and R. JOLY. "Global stability of travelling fronts for a damped wave equation with bistable nonlinearity". In: *Ann. Sci. l'École Norm. Supérieure* 42.1 (2009), pp. 103–140. DOI: 10.24033/asens.2091. arXiv: 0710.0794 (cit. on pp. 18, 24, 26, 32, 35, 40, 43, 45, 46, 48, 50, 52–54, 56).

[16] T. GALLAY and E. RISLER. "A variational proof of global stability for bistable travelling waves". In: *Differ. Integr. Equations* 20.8 (2007), pp. 901–926. arXiv: math/0612684 (cit. on pp. 17, 24, 26, 32, 35, 39, 42, 43).

[17] T. GALLAY and S. SLIJEPČEVIĆ. "Energy Flow in Formally Gradient Partial Differential Equations on Unbounded Domains". In: *J. Dyn. Differ. Equations* 13.4 (2001), pp. 757–789. DOI: 10.1023/A:1016624010828 (cit. on p. 18).

[18] J. GARNIER, T. GILETTI, F. HAMEL, and L. ROQUES. "Inside dynamics of pulled and pushed fronts". In: *J. Math. Pures Appl.* 98.4 (2012), pp. 428–449. DOI: 10.1016/j.matpur.2012.02.005. arXiv: 1110.1761 (cit. on p. 17).
[19] K. P. HADELER and F. ROTHE. "Travelling fronts in nonlinear diffusion equations". In: *J. Math. Biol.* 2.3 (1975), pp. 251-263. DOI: 10.1007/BF00277154 (cit. on pp. 17, 73).

[20] A. HARAUX and M. A. JENDOUBI. "Convergence of Solutions of Second-Order Gradient-Like Systems with Analytic Nonlinearities". In: *J. Differ. Equ.* 144 (1998), pp. 313-320. DOI: 10.1006/jdeq.1997.3393 (cit. on p. 2).

[21] S. HEINZE. *A variational approach to travelling waves*. Tech. rep. Leipzig: Technical Report 85, Max Planck Institute for Mathematical Sciences, 2001 (cit. on p. 17).

[22] D. B. HENRY. *Geometric Theory of Semilinear Parabolic Equations*. Vol. 840. Lecture Notes in Mathematics. Berlin, New-York: Springer Berlin Heidelberg, 1981. DOI: 10.1007/BFb0089647 (cit. on p. 19).

[23] M. A. JENDOUBI and P. POLÁČIK. "Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping". In: *R. Soc. Edinburgh - Proc. A* 133.5 (2003), pp. 1137-1153. DOI: 10.1017/s0308210500002845 (cit. on p. 3).

[24] R. JOLY, R. OLIVER-BONAFOUX, and E. RISLER. "Generic transversality of pushed and pulled travelling fronts for parabolic gradient systems (work in progress)" (cit. on pp. 7, 8, 15, 63).

[25] R. JOLY and E. RISLER. "Generic transversality of travelling fronts, standing fronts, and standing pulses for parabolic gradient systems". In: *arXiv* (2023). arXiv: 2301.02095 (cit. on pp. 7, 18).

[26] Y. I. KANEL'. "Some problems involving burning-theory equations". In: *Sov. Math. Dokl.* 2 (1961), pp. 48-51 (cit. on p. 16).

[27] Y. I. KANEL'. "Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory". In: *Mat. Sb.* 59 (101).supplementary (1962), pp. 245-288 (cit. on p. 16).

[28] A. N. KOLMOGOROV, I. PETROVSKII, and N. PISKUNOV. "Study of the Diffusion Equation with Growth of the Quantity of Matter and its Application to a Biology Problem". In: *Moscow Univ. Math. Bull.* 1.6 (1937), pp. 1-25 (cit. on p. 16).

[29] M. LUCIA, C. B. MURATOV, and M. NOVAGA. "Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium". In: *Commun. Pure Appl. Math.* 57.5 (2004), pp. 616-636. DOI: 10.1002/cpa.20014 (cit. on pp. 15, 17, 35).

[30] M. LUCIA, C. B. MURATOV, and M. NOVAGA. "Existence of Traveling Waves of Invasion for Ginzburg-Landau-type Problems in Infinite Cylinders". In: *Arch. Ration. Mech. Anal.* 188.3 (2008), pp. 475-508. DOI: 10.1007/s00205-007-0097-x (cit. on pp. 13, 15, 17).

[31] A. LUNARDI. *Analytic Semigroups and Optimal Regularity in Parabolic Problems*. Basel: Birkhäuser Basel, 1995. DOI: 10.1007/978-3-0348-9234-6 (cit. on p. 20).
[32] C. LUO. "Global stability of travelling fronts for a damped wave equation". In: *J. Math. Anal. Appl.* 399 (2013), pp. 260-278. DOI: 10.1016/j.jmaa.2012.05.089 (cit. on pp. 18, 26, 32, 35, 43, 45, 46, 52).

[33] H. MATANO and P. POLÁČIK. "An entire solution of a bistable parabolic equation on R with two colliding pulses". In: *J. Funct. Anal.* 272.5 (2017), pp. 1956-1979. DOI: 10.1016/j.jfa.2016.11.006 (cit. on p. 19).

[34] A. MONTEIL and F. SANTAMBROGIO. "Metric methods for heteroclinic connections in infinite dimensional spaces". In: *Indiana Univ. Math. J.* 69.4 (2020), pp. 1445-1503. DOI: 10.1512/iumj.2020.69.7869 (cit. on p. 18).

[35] C. B. MURATOV. "A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type". In: *Discret. Contin. Dyn. Syst. - Ser. B* 4.4 (2004), pp. 867-892. DOI: 10.3934/dcdsb.2004.4.867 (cit. on pp. 13, 17, 20, 22, 24, 26, 28, 35, 75).

[36] C. B. MURATOV and M. NOVAGA. "Front propagation in infinite cylinders. I. A variational approach". In: *Commun. Math. Sci.* 6.4 (2008), pp. 799-826. DOI: 10.4310/CMS.2008.v6.n4.a1 (cit. on pp. 15, 17, 20).

[37] C. B. MURATOV and M. NOVAGA. "Global exponential convergence to variational traveling waves in cylinders". In: *SIAM J. Math. Anal.* 44.1 (2012), pp. 293-315. DOI: 10.1137/110833269. arXiv: 1105.1190 (cit. on pp. 17, 20).

[38] C. B. MURATOV and X. ZHONG. "Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations". In: *Nonlinear Differ. Equations Appl. NoDEA* 20.4 (2013), pp. 1519-1552. DOI: 10.1007/s00030-013-0220-7. arXiv: 1203.4623 (cit. on p. 17).

[39] C. B. MURATOV and X. ZHONG. "Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations". In: *Discret. Contin. Dyn. Syst. Ser. A* 37.2 (2017), pp. 915-944. DOI: 10.3934/dcds.2017038. arXiv: 1506.09146 (cit. on p. 17).

[40] R. OLIVER-BONAFOUX. "Heteroclinic traveling waves of 1D parabolic systems with degenerate stable states". In: *arXiv* (2021), pp. 1-37. arXiv: 2111.12546 (cit. on pp. 15, 18).

[41] R. OLIVER-BONAFOUX. "Heteroclinic traveling waves of 2D parabolic Allen-Cahn systems". In: *arXiv* (2021), pp. 1-67. arXiv: 2106.09441 (cit. on p. 18).

[42] R. OLIVER-BONAFOUX and E. RISLER. "Existence of a pulled or pushed travelling front invading a critical point for parabolic gradient systems". In: *arXiv* (2023). arXiv: 2312.10427 (cit. on p. 16).

[43] R. OLIVER-BONAFOUX and E. RISLER. "Travelling fronts invading a degenerate minimum set for parabolic gradient systems (work in progress)" (cit. on p. 18).
[44] E. RISLER. "Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure". In: *Ann. l'Institut Henri Poincare. Ann. Anal. Non Lineaire/Nonlinear Anal.* 25.2 (2008), pp. 381-424. DOI: 10.1016/j.anihpc.2006.12.005 (cit. on pp. 1, 9, 14-18, 29, 66, 68).

[45] E. RISLER. "Generic transversality of radially symmetric stationary solutions stable at infinity for parabolic gradient systems". In: *arXiv* (2023). arXiv: 2301.02605 (cit. on p. 18).

[46] E. RISLER. "Global behaviour of bistable solutions for gradient systems in one unbounded spatial dimension". In: *arXiv* (2023). arXiv: 1604.02002 (cit. on pp. 1, 10, 18, 66).

[47] E. RISLER. "Global behaviour of bistable solutions for hyperbolic gradient systems in one unbounded spatial dimension". In: *arXiv* (2023). arXiv: 1703.01221 (cit. on p. 18).

[48] E. RISLER. "Global behaviour of radially symmetric solutions stable at infinity for gradient systems". In: *arXiv* (2023). arXiv: 1703.02134 (cit. on p. 18).

[49] E. RISLER. "Global behaviour of solutions stable at infinity for gradient systems in higher space dimension: the no invasion case". In: *arXiv* (2023). arXiv: 2206.06288 (cit. on p. 18).

[50] E. RISLER. "Global relaxation of bistable solutions for gradient systems in one unbounded spatial dimension". In: *arXiv* (2023). arXiv: 1604.00804 (cit. on pp. 18, 19).

[51] J.-M. ROQUEJOFFRE. "Convergence to Travelling Waves for Solutions of a Class of Semilinear Parabolic Equations". In: *J. Differ. Equ.* 108.2 (1994), pp. 262-295. DOI: 10.1006/jdeq.1994.1035 (cit. on p. 17).

[52] J.-M. ROQUEJOFFRE. "Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders". In: *Ann. l'Institut Henri Poincare Non Linear Anal.* 14.4 (1997), pp. 499-552. DOI: 10.1016/S0294-1449(97)80137-0 (cit. on p. 16).

[53] F. ROTHE. "Convergence to pushed fronts". In: *Rocky Mt. J. Math.* 11.4 (1981), pp. 617-633. DOI: 10.1216/RMJ-1981-11-4-617 (cit. on pp. 6, 15-17).

[54] W. van SAARLOOS. "Front propagation into unstable states". In: *Phys. Rep.* 386.2-6 (2003), pp. 29-222. DOI: 10.1016/j.physrep.2003.08.001. arXiv: cond-mat/0308540 (cit. on pp. 4, 6, 17).

[55] P. SMYRNELIS. "Connecting orbits in hilbert spaces and applications to P.D.E." In: *Commun. Pure Appl. Anal.* 19.5 (2020), pp. 2797-2818. DOI: 10.3934/CPAA.2020122. arXiv: 1903.09473 (cit. on p. 18).

[56] A. N. STOKES. "On Two Types of Moving Front in Quasilinear Diffusion". In: *Math. Biosci.* 31.3-4 (1976), pp. 307-315. DOI: 10.1016/0025-5564(76)90087-0 (cit. on p. 16).
Ramon OLIVER-BONAFOUX

Dipartimento di Informatica, Università di Verona,
Strada Le Grazie 15, 37134 Verona, Italy
ramon.oliverbonafoux@univr.it

Emmanuel RISLER

Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan,
F-69621 Villeurbanne, France.
emmanuel.risler@insa-lyon.fr