The KPP-problem and $\log t$-front shift for higher-order semilinear parabolic equations

V. A. Galaktionov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 49-79.

Résumé

The seminal paper by Kolmogorov, Petrovskii, and Piskunov (KPP) of 1937 on the travelling wave propagation in the reaction–diffusion equation $u_t=u_{xx}+u(1-u)$ in $\mathbb R\times\mathbb R_+$ with $u_0(x)=H(-x)\equiv1$ for $x0$ and $0$ for $x\ge0$ (here $H(\cdot)$ is the Heaviside function) opened a new era in the general theory of nonlinear PDEs and various applications. This paper became an encyclopedia of deep mathematical techniques and tools for nonlinear parabolic equations, which, in the last seventy years, were further developed in hundreds of papers and in dozens of monographs. The KPP paper established the fundamental fact that, in the above equation, there occurs a travelling wave $f(x-\lambda _0t)$, with the minimal speed $\lambda_0=2$, and, in the moving frame with the front shift $x_f(t)$ ($u(x_f(t),t)\equiv1/2$), there is uniform convergence $u(x_f(t)+y,t)\to f(y)$ as $t\to+\infty$, where $x_f(t)=2t(1+o(1))$. In 1983, by a probabilistic approach, Bramson proved that there exists an unbounded $\log t$-shift of the wave front in the indicated PDE problem and $x_f(t)=2t-(3/2)\log t(1+o(1))$ as $t\to+\infty$. Our goal is to reveal some aspects of KPP-type problems for higher-order semilinear parabolic PDEs, including the bi-harmonic equation and the tri-harmonic one, $u_t=-u_{xxxx}+u(1-u)$ and $u_t=u_{xxxxxx}+u(1-u)$. Two main questions to study are (i) existence of travelling waves via any analytical/numerical methods and (ii) a formal derivation of the $\log t$-shifting of moving fronts.

Bibliographie

[1] Barenblatt G.I., Similarity, self-similarity, and intermediate asymptotics, Consultants Bureau, New York, 1979 | MR | Zbl

[2] Bertozzi A.L., Münch A., Shearer M., Zumbrun K., “Stability of compressive and undercompressive thin film travelling waves”, Eur. J. Appl. Math., 12 (2001), 253–291 | DOI | MR | Zbl

[3] Bramson M., Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. AMS, no. 285, Amer. Math. Soc., Providence, RI, 1983 | MR | Zbl

[4] Brezis H., Peletier L.A., Terman D., “A very singular solution of the heat equation with absorption”, Arch. Ration. Mech. Anal., 95 (1986), 185–209 | DOI | MR | Zbl

[5] Budd C.J., Galaktionov V.A., Williams J.F., “Self-similar blow-up in higher-order semilinear parabolic equations”, SIAM J. Appl. Math., 64 (2004), 1775–1809 | DOI | MR | Zbl

[6] Busca J., Jendoubi M.A., Poláčik P., “Convergence to equilibrium for semilinear parabolic problems in $\mathbb R^N$”, Commun. Partial Diff. Eqns., 27 (2002), 1793–1814 | DOI | MR | Zbl

[7] Chill R., “On the Łojasiewicz–Simon gradient inequality”, J. Funct. Anal., 201 (2003), 572–601 | DOI | MR | Zbl

[8] Coddington E.A., Levinson N., Theory of ordinary differential equations, McGraw-Hill, New York, 1955 | MR | Zbl

[9] Collet P., Eckmann J.-P., Instabilities and fronts in extended systems, Princeton Univ. Press, Princeton, NJ, 1990 | MR | Zbl

[10] Coulon A.-C., Roquejoffre J.-M., “Transition between linear and exponential propagation in Fisher–KPP type reaction–diffusion equations”, Commun. Partial Diff. Eqns., 37 (2012), 2029–2049, arXiv: 1111.0408 [math.AP] | DOI | MR | Zbl

[11] Ebert U., van Saarloos W., “Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts”, Physica D, 146 (2000), 1–99 | DOI | MR | Zbl

[12] Egorov Yu.V., Galaktionov V.A., Kondratiev V.A., Pohozaev S.I., “On the necessary conditions of existence to a quasilinear inequality in the half-space”, C. r. Acad. sci. Paris. Sér. 1: Math., 330 (2000), 93–98 | DOI | MR | Zbl

[13] Egorov Yu.V., Galaktionov V.A., Kondratiev V.A., Pohozaev S.I., “Global solutions of higher-order semilinear parabolic equations in the supercritical range”, Adv. Diff. Eqns., 9 (2004), 1009–1038 | MR | Zbl

[14] Eidelman S.D., Parabolic systems, North-Holland, Amsterdam, 1969 | MR | Zbl

[15] Evans J.D., Galaktionov V.A., King J.R., “Source-type solutions of the fourth-order unstable thin film equation”, Eur. J. Appl. Math., 18 (2007), 273–321 | DOI | MR | Zbl

[16] Evans J.D., Galaktionov V.A., Williams J.F., “Blow-up and global asymptotics of the limit unstable Cahn–Hilliard equation”, SIAM J. Math. Anal., 38 (2006), 64–102 | DOI | MR | Zbl

[17] Feireisl E., Issard-Roch F., Petzeltová H., “A non-smooth version of the Lojasiewicz–Simon theorem with applications to non-local phase-field systems”, J. Diff. Eqns., 199 (2004), 1–21 | DOI | MR | Zbl

[18] Friedman A., “On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations”, J. Math. Mech., 7 (1958), 43–59 | MR | Zbl

[19] Galaktionov V.A., Comments on the magic exponent $\frac 32$ in Kolmogorov–Petrovskii–Piskunov problem, Discussion proposal, Dept. Math. Sci. Univ. Bath, Bath, Dec. 1995

[20] Galaktionov V.A., “On a spectrum of blow-up patterns for a higher-order semilinear parabolic equation”, Proc. R. Soc. London A: Math. Phys. Eng. Sci., 457:2011 (2001), 1623–1643 | DOI | MR | Zbl

[21] Galaktionov V.A., “Critical global asymptotics in higher-order semilinear parabolic equations”, Int. J. Math. Math. Sci., 2003, no. 60, 3809–3825 | DOI | MR | Zbl

[22] Galaktionov V.A., Geometric Sturmian theory of nonlinear parabolic equations and applications, Chapman Hall/CRC, Boca Raton, FL, 2004 | MR | Zbl

[23] Galaktionov V.A., “On interfaces and oscillatory solutions of higher-order semilinear parabolic equations with non-Lipschitz nonlinearities”, Stud. Appl. Math., 117 (2006), 353–389 | DOI | MR | Zbl

[24] Galaktionov V.A., Towards the KPP-problem and $\log t$-front shift for higher-order nonlinear PDEs. II: Quasilinear bi- and tri-harmonic equations, E-print, 2012, arXiv: 1210.5063 [math.AP]

[25] Galaktionov V.A., Towards the KPP-problem and $\log t$-front shift for higher-order nonlinear PDEs. III: Dispersion and hyperbolic equations, E-print, 2012, arXiv: 1210.5084 [math.AP]

[26] Galaktionov V.A., Kurdyumov S.P., Samarskii A.A., “On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation”, Math. USSR. Sb., 54 (1986), 421–455 | DOI | MR | Zbl

[27] Galaktionov V.A., Mitidieri E., Pokhozhaev S.I., “Variational approach to complicated similarity solutions of higher order nonlinear evolution partial differential equations”, Sobolev spaces in mathematics. V. 2: Applications in analysis and partial differential equations, Int. Math. Ser., 9, eds. V. Maz'ya, Springer, New York, 2009, 147–197, arXiv: 0902.1425 [math.AP] | DOI | MR | Zbl

[28] Galaktionov V.A., Mitidieri E., Pohozaev S.I., “Variational approach to complicated similarity solutions of higher-order nonlinear PDEs. II”, Nonlinear Anal., Real World Appl., 12 (2011), 2435–2466, arXiv: 1103.2643 [math.AP] | DOI | MR | Zbl

[29] Galaktionov V.A., Pohozaev S.I., “Existence and blow-up for higher-order semilinear parabolic equations: Majorizing order-preserving operators”, Indiana Univ. Math. J., 51 (2002), 1321–1338 | DOI | MR | Zbl

[30] Galaktionov V.A., Pohozaev S.I., Shishkov A.E., “Convergence in gradient systems with branching of equilibria”, Sb. Math., 198 (2007), 817–838, arXiv: 0902.0286 [math.AP] | DOI | DOI | MR | Zbl

[31] Galaktionov V.A., Vazquez J.L., “Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach”, J. Funct. Anal., 100 (1991), 435–462 | DOI | MR | Zbl

[32] Galaktionov V.A., Vazquez J.L., “Extinction for a quasilinear heat equation with absorption. II: A dynamical systems approach”, Commun. Partial Diff. Eqns., 19 (1994), 1107–1137 | DOI | MR | Zbl

[33] Galaktionov V.A., Vázquez J.L., A stability technique for evolution partial differential equations: A dynamical systems approach, Birkhäuser, Boston, MA, 2004 | MR | Zbl

[34] Gao H., Liu C., “Instability of traveling waves of the convective–diffusive Cahn–Hilliard equation”, Chaos Solitons Fractals, 20 (2004), 253–258 | DOI | MR | Zbl

[35] Gärtner J., “Location of wave fronts for the multi-dimensional K-P-P equation and Brownian first exit densities”, Math. Nachr., 105 (1982), 317–351 | DOI | MR

[36] Gmira A., Veron L., “Large time behaviour of the solutions of a semilinear parabolic equation in $\mathbb R^N$”, J. Diff. Eqns., 53 (1984), 258–276 | DOI | MR | Zbl

[37] Hale J.K., Raugel G., “Convergence in gradient-like systems with applications to PDE”, Z. angew. Math. Phys., 43 (1992), 63–124 | DOI | MR | Zbl

[38] Hamel F., Roques L., “Fast propagation for KPP equations with slowly decaying initial conditions”, J. Diff. Eqns., 249 (2010), 1726–1745, arXiv: 0906.3164 [math.AP] | DOI | MR | Zbl

[39] Haraux A., Jendoubi M.A., “Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity”, Asymptotic Anal., 26 (2001), 21–36 | MR | Zbl

[40] Haraux A., Jendoubi M.A., “The Łojasiewicz gradient inequality in the infinite-dimensional Hilbert space framework”, J. Funct. Anal., 260 (2011), 2826–2842 | DOI | MR | Zbl

[41] Kalies W.D., Kwapisz J., VandenBerg J.B., VanderVorst R.C.A.M., “Homotopy classes for stable periodic and chaotic patterns in fourth-order Hamiltonian systems”, Commun. Math. Phys., 214 (2000), 573–592 | DOI | MR | Zbl

[42] Kamenomostskaya S., “The asymptotic behaviour of the solution of the filtration equation”, Isr. J. Math., 14 (1973), 76–87 | DOI | MR | Zbl

[43] Kamin S., Peletier L.A., “Large time behaviour of solutions of the heat equation with absorption”, Ann. Sc. Norm. Pisa. Cl. Sci. Ser. 4, 12 (1985), 393–408 | MR | Zbl

[44] Kolmogorov A.N., “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers”, Proc. R. Soc. London A, 434:1890 (1991), 9–13 | DOI | MR | Zbl

[45] Kolmogoroff A., Petrovsky I., Piscounoff N., “Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem”, Dynamics of curved fronts, ed. P. Pelcé, Academic, Boston, MA, 1988, 105–130 | DOI | MR

[46] Li Z., Liu C., “On the nonlinear instability of traveling waves for a sixth-order parabolic equation”, Abstr. Appl. Anal., 2012, 739156 | MR | Zbl

[47] Lunardi A., Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser, Basel, 1995 | MR | Zbl

[48] Naimark M.A., Linear differential operators, Part 1, Frederick Ungar Publ., New York, 1967 | MR | Zbl

[49] Obukhov A.M., “O raspredelenii energii v spektre turbulentnogo potoka”, DAN SSSR, 32:1 (1941), 22–24 | MR

[50] Peletier L.A., Troy W.C., Spatial patterns: Higher order models in physics and mechanics, Birkhäuser, Boston, MA, 2001 | MR

[51] Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P., Blow-up in quasilinear parabolic equations, W. de Gruyter, Berlin, 1995 | MR | Zbl

[52] Strauss W., Wang G., “Instability of traveling waves of the Kuramoto–Sivashinsky equation”, Frontiers in mathematical analysis and numerical methods, World Sci., River Edge, NJ, 2004, 253–266 | DOI | MR

[53] van den Berg J.B., Vandervorst R.C., “Stable patterns for fourth-order parabolic equations”, Duke Math. J., 115 (2002), 513–558 | DOI | MR | Zbl

[54] Vergara V., “Convergence to steady states of solutions to nonlinear integral evolution equations”, Calc. Var. Partial Diff. Eqns., 40 (2011), 319–334 | DOI | MR | Zbl

[55] Yanagida E., “Irregular behavior of solutions for Fisher's equation”, J. Dyn. Diff. Eqns., 19 (2007), 895–914 | DOI | MR | Zbl

[56] Yassine H., “Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order”, Nonlinear Anal., Theory Methods Appl., 74 (2011), 2309–2326 | DOI | MR | Zbl

[57] Zelenyak T.I., “Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable”, Diff. Eqns., 4 (1972), 17–22 | Zbl | Zbl