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Stability and dynamics of self-similarity in evolution equations. (English) Zbl 1194.35091
Summary: A methodology for studying the linear stability of self-similar solutions is discussed. These fundamental ideas are illustrated on three prototype problems: a simple ODE with finite-time blow-up, a second-order semilinear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, the use of dimensional analysis to derive scaling invariant similarity variables is discussed, as well as the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions it is demonstrated that the symmetries give rise to positive eigenvalues associated with the symmetries, and it is shown how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

##### MSC:
 35C06 Self-similar solutions to PDEs 35A30 Geometric theory, characteristics, transformations in context of PDEs 35B44 Blow-up in context of PDEs 35B35 Stability in context of PDEs 35B06 Symmetries, invariants, etc. in context of PDEs
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##### References:
 [1] Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press, Cambridge, p 386 · Zbl 0907.76002 [2] Barenblatt GI (2003) Scaling. Cambridge University Press, Cambridge, p 171 · Zbl 1094.00006 [3] Kadanoff LP (1997) Singularities and blowups. Phys Today 50: 11–13 [4] Eggers J, Fontelos MA (2009) The role of self-similarity in singularities of partial differential equations. Nonlinearity 22: R1–R44 · Zbl 1152.35300 [5] Tanner LH (1979) The spreading of silicone oil drops on horizontal surfaces. J Phys D 12: 1473–1484 [6] Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69: 931–980 [7] Witelski TP, Bernoff AJ (1998) Self-similar asymptotics for linear and nonlinear diffusion equations. Stud Appl Math 100: 153–193 · Zbl 1001.35056 [8] Zel’dovich YB, Raizer YP (2002) Physics of shock waves and high temperature hydrodynamic phenomena. Dover, New York, p 944 [9] Kleinstein G, Ting L (1971) Optimum one-term solutions for heat conduction problems. Z Angew Math Mech 51: 1–16 · Zbl 0241.35036 [10] Kloosterziel RC (1990) On the large-time asymptotics of the diffusion equation on infinite domains. J Eng Math 24: 213–236 · Zbl 0726.35019 [11] Bernoff AJ, Lingevitch JF (1994) Rapid relaxation of an axisymmetric vortex. Phys Fluids 6: 3717–3723 · Zbl 0838.76024 [12] Gallay T, Wayne CE (2002) Invariant manifolds and the long-time asymptotics of the Navier–Stokes and vorticity equations on R 2. Arch Ration Mech Anal 163: 209–258 · Zbl 1042.37058 [13] Bebernes J, Eberly D (1989) Mathematical problems from combustion theory vol 83 of Applied Mathematical Sciences. Springer, New York, p 177 · Zbl 0692.35001 [14] Stevens A, Othmer HG (1997) Aggregation, blowup, and collapse. SIAM J Appl Math 57: 1044–1081 · Zbl 0990.35128 [15] Levine HA (1989) Quenching, nonquenching, and beyond quenching for solution of some parabolic equations. Ann Math Pura Appl 155: 243–260 · Zbl 0743.35010 [16] Galaktionov VA, Vázquez JL (2002) The problem of blow-up in nonlinear parabolic equations. Discret Contin Dyn Syst 8: 399–433 · Zbl 1010.35057 [17] Guo YJ, Pan ZG, Ward MJ (2005) Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties. SIAM J Appl Math 66: 309–338 · Zbl 1103.35042 [18] Flores G, Mercado G, Pelesko JA, Smyth N (2007) Analysis of the dynamics and touchdown in a model of electrostatic MEMS. SIAM J Appl Math 67: 434–446 · Zbl 1132.35045 [19] Brenner MP, Lister JR, Stone HA (1996) Pinching threads, singularities and the number 0.0304.... Phys Fluids 8: 2827–2836 · Zbl 1027.76510 [20] Eggers J, Villermaux E (2008) Physics of liquid jets. Rep Prog Phys 71: 1–79 [21] Bernoff AJ, Bertozzi AL, Witelski TP (1998) Axisymmetric surface diffusion: dynamics and stability of self-similar pinchoff. J Stat Phys 93: 725–776 · Zbl 0951.74007 [22] Zhang WW, Lister JR (1999) Similarity solutions for van der Waals rupture of a thin film on a solid substrate. Phys Fluids 11: 2454–2462 · Zbl 1149.76597 [23] Vaynblat D, Lister JR, Witelski TP (2001) Rupture of thin viscous films by van der Waals forces I: evolution and self-similarity. Phys Fluids 13: 1130–1140 · Zbl 1184.76571 [24] Vaynblat D, Lister JR, Witelski TP (2001) Symmetry and self-similarity in rupture and pinchoff: a geometric bifurcation. Eur J Appl Math 12: 209–232 · Zbl 1011.76068 [25] Witelski TP, Bernoff AJ (1999) Stability of self-similar solutions for van der Waals driven thin film rupture. Phys Fluids 11: 2443–2445 · Zbl 1149.76588 [26] Witelski TP, Bernoff AJ (2000) Dynamics of three-dimensional thin film rupture. Physica D 147: 155–176 · Zbl 0992.76013 [27] Levine HA (1990) The role of critical exponents in blowup theorems. SIAM Rev 32: 262–288 · Zbl 0706.35008 [28] Bandle C, Brunner H (1998) Blowup in diffusion equations: a survey. J Comput Appl Math 97: 3–22 · Zbl 0932.65098 [29] Samarskii AA, Galaktionov VA, Kurdyumov SP, Mikhailov AP (1995) Blow-up in quasilinear parabolic equations. Walter de Gruyter & Co, Berlin, p 535 [30] Hydon PE (2000) Symmetry methods for differential equations. Cambridge University Press, Cambridge, p 213 · Zbl 0951.34001 [31] Bluman GW, Anco SC (2002) Symmetry and integration methods for differential equations. Springer, New York, p 419 · Zbl 1013.34004 [32] Dresner L (1999) Applications of Lie’s theory of ordinary and partial differential equations. Institute of Physics Publishing, Bristol, p 225 · Zbl 0914.34002 [33] Olver PJ (1993) Applications of Lie groups to differential equations. Springer, New York, p 513 [34] Merle F, Zaag H (2002) O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discret Contin Dyn Syst 8: 435–450 · Zbl 1009.35039 [35] Berger M, Kohn RV (1988) A rescaling algorithm for the numerical calculation of blowing-up solutions. Commun Pure Appl Math 41: 841–863 · Zbl 0652.65070 [36] Giga Y, Kohn RV (1985) Asymptotically self-similar blow-up of semilinear heat equations. Commun Pure Appl Math 38: 297–319 · Zbl 0585.35051 [37] Giga Y, Kohn RV (1987) Characterizing blowup using similarity variables. Indiana Univ Math J 36: 1–40 · Zbl 0601.35052 [38] Velazquez JJ, Galaktionov VA, Herrero MA (1991) The space structure near a blow-up point for semilinear heat equations: a formal approach. Comput Math Math Phys 31: 46–55 · Zbl 0747.35014 [39] Galaktionov VA, Kurdyumov SP, Samarskiĭ AA (1985) Asymptotic ”eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation. Mat Sb (N.S.) 126(168):435–472, 592 [40] Kamin S, Peletier LA (1985) Singular solutions of the heat equation with absorption. Proc Am Math Soc 95: 205–210 · Zbl 0607.35046 [41] Brezis H, Peletier LA, Terman D (1986) A very singular solution of the heat equation with absorption. Arch Ration Mech Anal 95: 185–209 · Zbl 0627.35046 [42] Wayne CE (1997) Invariant manifolds for parabolic partial differential equations on unbounded domains. Arch Ration Mech Anal 138: 279–306 · Zbl 0882.35061 [43] Bricmont J, Kupiainen A, Lin G (1994) Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Commun Pure Appl Math 47: 893–922 · Zbl 0806.35067 [44] Bricmont J, Kupiainen A (1996) Stable non-Gaussian diffusive profiles. Nonlinear Anal 26: 583–593 · Zbl 0841.35045 [45] Filippas S, Kohn RV (1992) Refined asymptotics for the blowup of u t u = u p . Commun Pure Appl Math 45: 821–869 · Zbl 0784.35010 [46] Bebernes J, Bricher S (1992) Final time blowup profiles for semilinear parabolic equations via center manifold theory. SIAM J Math Anal 23: 852–869 · Zbl 0754.35055 [47] Galaktionov VA, Williams JF (2004) On very singular similarity solutions of a higher-order semilinear parabolic equation. Nonlinearity 17: 1075–1099 · Zbl 1063.35077 [48] Galaktionov VA, Vázquez JL (2004) A stability technique for evolution partial differential equations: a dynamical systems approach. Birkhäuser, Boston, MA, p 377 [49] Sarocka DC, Bernoff AJ, Rossi LF (1999) Large-amplitude solutions to the Sivashinsky and Riley–Davis equations for directional solidification. Physica D 127: 146–176 · Zbl 0949.76088 [50] Sivashinsky GI (1983) On cellular instability in the solidification of a dilute binary alloy. Physica D 8: 243–248 [51] Childress S, Spiegel EA (2004) Pattern formation in a suspension of swimming microorganisms: nonlinear aspects. In: Givoli D, Grote MJ, Papanicolaou GC (eds) A celebration of mathematical modeling. Kluwer, Dordrecht, pp 33–52 [52] Novick-Cohen A (1990) On Cahn-Hilliard type equations. Nonlinear Anal 15: 797–814 · Zbl 0731.35057 [53] Novick-Cohen A (1992) Blow up and growth in the directional solidification of dilute binary alloys. Appl Anal 47: 241–257 · Zbl 0727.35012 [54] Straughan B (1998) Explosive instabilities in mechanics. Springer, Berlin, p 196 · Zbl 0911.35002 [55] Bernoff AJ, Bertozzi AL (1995) Singularities in a modified Kuramoto–Sivashinsky equation describing interface motion for phase transition. Physica D 85: 375–404 · Zbl 0899.76190 [56] Bertozzi AL, Pugh MC (1998) Long-wave instabilities and saturation in thin film equations. Commun Pure Appl Math 51: 625–661 · Zbl 0916.35008 [57] Bertozzi AL, Pugh MC (2000) Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ Math J 49: 1323–1366 · Zbl 0978.35007 [58] Evans JD, Galaktionov VA, King JR (2007) Source-type solutions of the fourth-order unstable thin film equation. Eur J Appl Math 18: 273–321 · Zbl 1156.35387 [59] Witelski TP, Bernoff AJ, Bertozzi AL (2004) Blowup and dissipation in a critical-case unstable thin film equation. Eur J Appl Math 15: 223–256 · Zbl 1062.76005 [60] Hocherman T, Rosenau P (1993) On KS-type equations the evolution and rupture of a liquid interface. Physica D 67: 113–125 · Zbl 0787.76092 [61] Gandarias ML, Ibragimov NH (2008) Equivalence group of a fourth-order evolution equations unifying various non-linear models. Commun Nonlinear Sci Numer Simul 13: 259–268 · Zbl 1155.35410 [62] Levine HA (1973) Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $${Pu_{t}=-Au+{\mathcal F}(u)}$$ . Arch Ration Mech Anal 51: 371–386 · Zbl 0278.35052 [63] Evans JD, Galaktionov VA, Williams JF (2006) Blow-up and global asymptotics of the limit unstable Cahn–Hilliard equation. SIAM J Math Anal 38: 64–102 · Zbl 1110.35023 [64] Witelski TP (2002) Computing finite-time singularities in interfacial flows. In: Bourlioux A, Gander MJ (eds) Modern methods in scientific computing and applications. Kluwer, Dordrecht, pp 451–487 · Zbl 1053.76026 [65] Budd CJ, Piggott MD (2001) The geometric integration of scale-invariant ordinary and partial differential equations. J Comput Appl Math 128: 399–422 · Zbl 0974.65095 [66] Budd CJ, Piggott MD (2003) Geometric integration and applications. In: Cucker F (eds) Handbook of numerical analysis, vol XI. Elsevier/North Holland, Amsterdam, pp 35–139 [67] Bernis F, Peletier LA (1996) Two problems from draining flows involving third-order ordinary differential equations. SIAM J Math Anal 27: 515–527 · Zbl 0845.34033 [68] Boatto S, Kadanoff LP, Olla P (1993) Traveling-wave solutions to thin-film equations. Phys Rev E 48: 4423–4431 [69] Carr J (1981) Applications of centre manifold theory. Springer, New York, p 142 · Zbl 0464.58001 [70] Guckenheimer J, Holmes P (1990) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York, p 459 · Zbl 0515.34001 [71] Evans JD, Galaktionov VA, King JR (2007) Unstable sixth-order thin film equation. I. Blow-up similarity solutions. Nonlinearity 20: 1799–1841 · Zbl 1173.35562 [72] Budd CJ, Galaktionov VA, Williams JF (2004) Self-similar blow-up in higher-order semilinear parabolic equations. SIAM J Appl Math 64: 1775–1809 · Zbl 1112.35095 [73] Budd CJ, Rottschäfer V, Williams JF (2005) Multibump, blow-up, self-similar solutions of the complex Ginzburg-Landau equation. SIAM J Appl Dyn Syst 4: 649–678 · Zbl 1170.35530 [74] Palais B (1988) Blowup for nonlinear equations using a comparison principle in Fourier space. Commun Pure Appl Math 41: 165–196 · Zbl 0674.35045 [75] Chapman SJ, King JR, Adams KL (1998) Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc R Soc Lond Ser A 454: 2733–2755 · Zbl 0916.34017 [76] Ascher UM, Mattheij RMM, Russell RD (1995) Numerical solution of boundary value problems for ordinary differential equations. Society for Industrial and Applied Mathematics, Philadelphia, PA, p 595 [77] Burke J, Knobloch E (2007) Homoclinic snaking:structure and stability. Chaos 17: 037102 · Zbl 1163.37317 [78] Beck M, Knobloch J, Lloyd D, Sandstede B, Wagenknecht T (2009) Snakes, ladders, and isolas of localized patterns. SIAM J Math Anal 41: 936–972 · Zbl 1200.37015 [79] Rottschäfer V, Kaper TJ (2003) Geometric theory for multi-bump, self-similar, blowup solutions of the cubic nonlinear Schrödinger equation. Nonlinearity 16: 929–961 · Zbl 1033.35118
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