Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

A. V. Gasnikov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1458-1487

Résumé

The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition.

Bibliographie

[1] Oleinik O. A., Venttsel T. D., “Pervaya kraevaya zadacha i zadacha Koshi dlya kvazilineinykh uravnenii parabolicheskogo tipa”, Matem. sb., 41(83):1 (1957), 105–128 | MR | Zbl

[2] Oleinik O. A., “Razryvnye resheniya nelineinykh differentsialnykh uravnenii”, Uspekhi matem. nauk, 12:3(75) (1957), 3–73 | MR

[3] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva H. H., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[4] Fridman A., Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, M., 1968

[5] Kruzhkov S. N., “Nelineinye parabolicheskie uravneniya s dvumya nezavisimymi peremennymi”, Tr. Mosk. matem. ob-va, 16, 1967, 329–346 | Zbl

[6] Kruzhkov S. N., “Kvazilineinye uravneniya pervogo poryadka so mnogimi nezavisimymi peremennymi”, Matem. sb., 81(123):2 (1970), 228–255 | Zbl

[7] Hoff D., Smoller J., “Solutions in the large for certain nonlinear parabolic systems”, Ann. Inst. H. Poincaré, Analys. Non Linéaire, 2 (1985), 213–235 | MR | Zbl

[8] Serre D., “$L^1$ stability of shock waves in scalar conservation laws”, Evolutionary Equations. Handbook of Differential Equations, v. 1, North-Holland, Amsterdam, 2004, 473–553 | MR | Zbl

[9] Uizem Dzh., Lineinye i nelineinye volny, Mir, M., 1977 | MR

[10] Peletier L. A., “The porous media equation”, Applic. Nonlinear Analys. Phys. Sci., Proc. Consr. on Bifurcations Theory, 1981, 229–241 | MR | Zbl

[11] Harten A., Hyman J. M., Lax P. D., “On finite-difference approximations and entropy conditions for shocks”, Communs. Pure Appl. Math., 29 (1976), 297–322 | DOI | MR | Zbl

[12] Henkin G. M., Polterovich V. M., “Shumpetrian dynamics as non-linear wave theory”, J. Math. Econom., 20 (1991), 551–590 | DOI | MR | Zbl

[13] Henkin G. M., Polterovich V. M., “A difference-differential analogue of the Burgers equation and some models of economic development”, Discrete and Continuous Dynamic Systems, 5:4 (1999), 697–728 | DOI | MR | Zbl

[14] Mejai M., Volpert Vit., “Convergence to systems of waves for viscous scalar conservation laws”, Asymptotic Analys., 20 (1999), 351–366 | MR | Zbl

[15] Gasnikov A. B., “O promezhutochnoi asimptotike resheniya zadachi Koshi dlya kvazilineinogo uravneniya parabolicheskogo tipa s monotonnym nachalnym usloviem”, Izv. RAN. Teoriya i sistemy upravleniya, 2008, no. 3 (to appear) | MR

[16] Gelfand I. M., “Nekotorye zadachi teorii kvazilineinykh uravnenii”, Uspekhi matem. nauk, 14:2(86) (1959), 87–158 | MR

[17] Ilin A. M., Oleinik O. A., “Asimptoticheskoe povedenie reshenii zadachi Koshi dlya nekotorykh kvazilineinykh uravnenii pri bolshikh znacheniyakh vremeni”, Matem. sb., 51(93):2 (1960), 191–216 | MR

[18] Rozhdestvenskii B. L.,Yanenko H. H., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR

[19] Weinberger H. F., “Long-time behavior for regularized scalar conservation law in absence of genuine nonlinearity”, Ann. Inst. H. Poincare, Anal. Non Linéaire, 7 (1990), 407–425 | MR | Zbl

[20] Gasnikov A. B., “Asimptoticheskoe po vremeni povedenie resheniya kvazilineinogo uravneniya parabolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 46:12 (2006), 2235–2253 | MR

[21] Volpert A. I., Volpert Vit. A., Volpert Vl. A., Traveling waves solutions of parabolic system, Transl. Math. Monographs., 140, 2000, 455 pp. | MR

[22] Engeiberg S., Schocket S., “Nonintegrable perturbation of scalar viscous shock profiles”, Asymptotic Analys., 48 (2006), 121–140 | MR

[23] Gasnikov A. B., Asimptoticheskoe po vremeni povedenie resheniya nachalnoi zadachi Koshi dlya zakona sokhraneniya s nelineinoi divergentnoi vyazkostyu, Dis. $\dots$ kand. fiz.-matem. nauk, RUDN, M., 2007, 64 pp.

[24] Freistühler H., Serre D., “$L^1$ stability of shock waves in scalar viscous conservation laws”, Communs. Pure Appl. Math., 51 (1998), 291–301 | 3.0.CO;2-5 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[25] Dafermos C. M., Hyperbolic conservation laws in continuum physics, Springer, Berlin et al., 2005 | MR

[26] Liu T. P., Admissible solutions of hyperbolic conservation laws, Mem. Amer. Math. Soc., 30, no. 240, 1981, 78 pp. | MR

[27] Cheng K.-S., “Asymptotic behaviour of solution of a conservation law without convexity condition”, J. Different. Equat., 40:3 (1981), 343–376 | DOI | MR | Zbl

[28] Petrosyan H. S., “Ob asimptotike resheniya zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka s nevypukloi funktsiei sostoyaniya”, Uspekhi matem. nauk, 38:2(230) (1983), 213–214 | MR | Zbl

[29] Kruzhkov C. H., Petrosyan N. S., “Asimptoticheskoe povedenie reshenii zadachi Koshi dlya nelineinykh uravnenii pervogo pryadka”, Uspekhi matem. nauk, 42:5(257) (1987), 3–40 | MR | Zbl

[30] Petrosyan N. S., “Ob asimptoticheskikh svoistvakh resheniya zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka”, Differents. ur-niya, 20:3 (1984), 502–508 | MR | Zbl

[31] Osher S., Ralston J., “$L^1$ stability of traveling waves with application to convective porous media flow”, Communs. Pure Appl. Math., 35 (1982), 737–749 | DOI | MR | Zbl

[32] Jones C. K. R. T., Gardner R., Kapitula T., “Stability of traveling waves for nonconvex scalar conservation laws”, Communs. Pure Appl. Math., 46 (1993), 505–526 | DOI | MR | Zbl

[33] Henkin G. M., Shananin A. A., Tumanov A. E., “Estimates for solution of Burgers type equations and some applications”, J. Math. Purés Appl., 84 (2005), 717–752 | MR | Zbl

[34] Henkin G. M., Shananin A. A., “Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations”, J. Math. Purés Appl., 83 (2004), 1457–1500 | MR | Zbl

[35] Henkin G. M., “Asymptotic structure for solutions of the Cauchy problem for Burgers type equations”, J. Fixed Point Theory Appl., 1:2, July (2007), 239–291 | DOI | MR | Zbl

[36] Kolmogorov A. H., Petrovskii I. G., Piskunov N. S., “Issledovanie uravneniya diffuzii, soedinennoi s vozrastaniem kolichestva veschestva i ego primenenie k odnoi biologicheskoi probleme”, Byul. MGU. Matem. i mekhan., 6:1 (1937), 1–26

[37] Uchiyama K., “The behaviour of solutions of some nonlinear diffusion equation for large time”, J. Math. Kyoto Univ., 18 (1978), 453–508 | MR | Zbl

[38] Khaikin B. I., Filonenko A. K., Khudyaev S. I., “Flame propagation in the presence of two successive gas-phase reactions”, Combust., Expl. and Shock Waves, 4 (1968), 343–349 | DOI

[39] Merzhanov A. G., Rumanov E. N., Khaikin B. I., “Multizonnoe gorenie kondensirovannykh sistem”, Prikl. mekhan. i tekhn. fiz., 1972, no. 6, 99–105

[40] Fife P. C., McLeod J. B., “The approach of solutions of nonlinear diffustion equations to travelling front solutions”, Arch. Ration. Mech. Analyse, 65 (1977), 335–361 | MR | Zbl

[41] Kumei S., Bluman G. W., “When nonlinear differential equations are equivalent to linear differential equations”, SIAM J. Appl. Math., 42:5 (1982), 1157–1173 | DOI | MR | Zbl

[42] Meirmanov A. M., Pukhnachov V. V., Shmarev S. I., Evolution equations and Lagrangian coordinates, Walter de Gruyter Co., Berlin, 1997 | MR

[43] Hopf E., “The partial differential equation $u_t+uu_x=\mu u_{xx}$”, Communs. Pure Appl. Math., 3:3 (1950), 201–230 | DOI | MR | Zbl

[44] Magaril-Ilyaev G. G., Tikhomirov V. M., Vypuklyi analiz i ego prilozheniya, URSS, M., 2003

[45] Protter M. H., Weinberger H. F., Maximum principles in differential equations, Springer, New York, 1986 | Zbl