On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation

V. A. Galaktionov   ; S. P. Kurdyumov   ; A. A. Samarskii

Sbornik. Mathematics, Tome 54 (1986) no. 2, pp. 421-455

Résumé

The asymptotic ($t\to+\infty$) behavior of solutions of the Cauchy problem is studied for the semilinear parabolic equation $$ u_t=\Delta u-u^\beta,\quad t>0,\ x\in R^N;\qquad u(0,x)=u_0(x)\geqslant0,\quad x\in R^N, $$ where $\beta=\mathrm{const}>1$ and $u_0(x)\to0$ as $|x|\to+\infty$. The existence is established of an infinite collection (a continuum) of distinct self-similar solutions of the form $u_A(t,x)=(T+t)^{-1/(\beta-1)}\theta_A(\xi)$, $\xi=|x|/(T+t)^{1/2}$, where the function $\theta_A>0$ satisfies an ordinary differential equation. Conditions for the asymptotic stability of these solutions are established. It is shown that for $\beta\geqslant1+2/N$ there exist solutions of the problem whose behavior as $t\to+\infty$ is described by approximate self-similar solutions (ap.s.-s.s.'s) $u_a(t,x)$ which in the case $\beta>1+2/N$ coincide with a family of self-similar solutions of the heat equation $(u_a)_t=\Delta u_a$, while for $\beta=1+2/N$ and $u_0\in L^1(R^N)$ the ap.s.-s.s. has the form $u_a=[(T+t)\ln(T+t)]^{-N/2}c_N\exp(-|x|^2/4(T+t))$, where $c_N=(N/2)^{N/2}(1+2/N)^{N^2/4}$. Figures: 2. Bibliography: 78 titles.

Bibliographie

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

[2] Kurdyumov S. P., “Sobstvennye funktsii goreniya nelineinoi sredy i konstruktivnye zakony postroeniya ee organizatsii”, Sovremennye problemy vychislitelnoi matem. i matem. fiz., Nauka, M., 1982, 217–243 | MR

[3] Samarskii A. A., Elenin G. G., Zmitrenko P. V., Kurdyumov S. P., Mikhailov A. P., “Gorenie nelineinoi sredy v vide slozhnykh struktur”, DAN SSSR, 237:6 (1977), 1330–1333 | MR

[4] Kurdyumov S. P., Malinetskii G. G., Sinergetika – teoriya samoorganizatsii. Idei, metody, perspektivy, Matematika, kibernetika, Znanie, M., 1983

[5] Galaktionov V. A., Kurdyumov S. P., Samarskii A. A., “Ob odnoi parabolicheskoi sisteme kvazilineinykh uravnenii. I”, Differents. uravneniya, 19:12 (1983), 2123–2140 | MR | Zbl

[6] Galaktionov V. A., Samarskii A. A., “Metody postroeniya priblizhennykh avtomodelnykh reshenii nelineinykh uravnenii teploprovodnosti. I”, Matem. sb., 118(160) (1982), 291–322 | MR | Zbl

[7] Galaktionov V. A., Kurdyumov S. P., Samarskii A. A., “O priblizhennykh avtomodelnykh resheniyakh odnogo klassa kvazilineinykh uravnenii teploprovodnosti s istochnikom”, Matem. sb., 124(166) (1984), 163–188 | MR

[8] Galaktionov V. A., Kurdyumov S. P., Samarskii A. A., “Ob asimptoticheskoi ustoichivosti invariantnykh reshenii nelineinykh uravnenii teploprovodnosti s istochnikom”, Differents. uravneniya, 20:4 (1984), 614–632 | MR

[9] Samarskii A. A., “O novykh metodakh issledovaniya asimptoticheskikh svoistv parabolicheskikh uravnenii”, Tr. MIAN, 158 (1981), 153–162 | MR | Zbl

[10] Galaktionov V. A., Samarskii A. A., “Metody postroeniya priblizhennykh avtomodelnykh reshenii nelineinykh uravnenii teploprovodnosti. II”, Matem. sb., 118(160) (1982), 435–455 | MR | Zbl

[11] Galaktionov V. A., Samarskii A. A., “Metody postroeniya priblizhennykh avtomodelnykh reshenii nelineinykh uravnenii teploprovodnosti. III”, Matem. sb., 120(162) (1983), 3–21 | MR | Zbl

[12] Galaktionov V. A., Samarskii A. A., “Metody postroeniya priblizhennykh avtomodelnykh reshenii nelineinykh uravnenii teploprovodnosti. IV”, Matem. sb., 121(163) (1983), 131–155 | MR

[13] Gidas B., Wei-Ming Ni, Nirenberg L., “Symmetry and related properties via the maximum principle”, Commun. in Math. Phys., 68 (1979), 209–243 | DOI | MR | Zbl

[14] Gmira A., Véron L., “Comportement asymptotique de la solution d'une équation non linéaire de la chaleur dans $R^N$”, Comp. Rendus Acad. Sci. Paris, série I, 295 (1982), 727–730 | MR | Zbl

[15] Elenin G. G., Kurdyumov S. P., Samarskii A. A., “Nestatsionarnye dissipativnye struktury v nelineinoi teploprovodnoi srede”, ZhVM i MF, 23:2 (1983), 380–390 | MR | Zbl

[16] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., “O neogranichennykh resheniyakh zadachi Koshi dlya parabolicheskogo uravneniya $u_t=\nabla(u^{\sigma}\nabla u)+u^{\beta}$”, DAN SSSR, 252:6 (1980), 1362–1364 | MR | Zbl

[17] Ad'yutov M. M., Klokov Yu. A., Mikhailov A. P., “Avtomodelnye teplovye struktury s sokraschayuscheisya polushirinoi”, Differents. uravneniya, 19:7 (1983), 1107–1114 | MR | Zbl

[18] Haraux A., Weissler F. B., “Non-uniqueness for a semilinear initial value problem”, Indiana Univ. Math. J., 31 (1982), 167–189 | DOI | MR | Zbl

[19] Glensdorf P., Prigozhii I., Termodinamicheskaya teoriya struktury, ustoichivosti i fluktuatsii, Mir, M., 1973 | MR

[20] Nikolis G., Prigozhii I., Samoorganizatsiya v neravnovesnykh sistemakh, Mir, M., 1977

[21] Khaken G., Sinergetika, Mir, M., 1980 | MR

[22] Vasilev V. A., Romanovskii Yu. M., Yakhno V. G., “Avtovolnovye protsessy v raspredelennykh kineticheskikh sistemakh”, Uspekhi fiz. nauk, 128:4 (1979), 625–666 | MR

[23] Nelineinye volny. Samoorganizatsiya, Nauka, M., 1983

[24] Kurdyumov S. P., Malinetskii G. G., Poveschenko Yu. A., Popov Yu. P., Samarskii A. A., “Dissipativnye struktury v triggernykh sredakh”, Differents. uravneniya, 17:10 (1981), 1875–1885 | MR

[25] Elenin G. G., Obrazovanie kvazistatsionarnykh beguschikh voln s periodicheskoi strukturoi iz neustoichivykh potokov barotropnogo gaza, dvizhuschegosya pod deistviem nelineinykh ob'emnykh sil, Preprint. No 126, IPM, M., 1977 | MR

[26] Nirenberg L., Lektsii po nelineinomu funktsionalnomu analizu, Mir, M., 1977 | MR | Zbl

[27] Strannye attraktory, Mir, M., 1981 | MR

[28] Gidrodinamicheskie neustoichivosti i perekhod k turbulentnosti., Per. s angl./Pod red. X. Suinni, Dzh. Gollaba, Mir, M., 1984 | MR

[29] Kurdyumov S. P. Kurkina. E. S. Potapov A. B., Samarskii A. A., “Arkhitektura mnogomernykh teplovykh struktur”, DAN SSSR, 274:5 (1984), 1071–1075 | MR

[30] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., “Lokalizatsiya tepla v nelineinykh sredakh”, Differents. uravneniya, 17:10 (1981), 1826–1841 | MR | Zbl

[31] Berestycki H., Lions P.-L., “Nonlinear scalar field equations”, Arch. for Ration. Mech. and Anal., 82 (1983), 313–375 | MR

[32] Martinson L. K., Pavlov K. B., “K voprosu o prostranstvennoi lokalizatsii teplovykh vozmuschenii v teorii nelineinoi teploprovodnosti”, ZhVM i MF, 12:4 (1972), 1048–1054 | MR

[33] Golaido S. I., Martinson L. K., Pavlov K. B., “Nestatsionarnye zadachi nelineinoi teploprovodnosti s ob'emnym pogloscheniem tepla”, ZhVM i MF, 13:5 (1973), 1351–1356 | Zbl

[34] Kalashnikov A. S., “O kharaktere rasprostraneniya vozmuschenii v zadachakh nelineinoi teploprovodnosti s pogloscheniem”, ZhVM i MF, 14:4 (1974), 891–905 | MR | Zbl

[35] Kalashnikov A. S., “O vliyanii pogloscheniya na rasprostranenie tepla v srede s teploprovodnostyu, zavisyaschei ot temperatury”, ZhVM i MF, 16:3 (1976), 689–696 | MR | Zbl

[36] Kershner R., O nekotorykh svoistvakh obobschennykh reshenii kvazilineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii, Dis. kand. fiz.-matem. nauk, MGU, M., 1976

[37] Kalashnikov A. S., “O differentsialnykh svoistvakh obobschennykh reshenii uravnenii tipa nestatsionarnoi filtratsii”, Vestn. MGU, Ser. matem., mekhan., 1971, no. 1, 62–68

[38] Kershner R., “O povedenii temperaturnykh frontov v sredakh s nelineinoi teploprovodnostyu pri nalichii pogloscheniya”, Vestn. MGU. Ser. matem., mekhan., 1978, no. 5, 44–51 | MR | Zbl

[39] Knerr B. F., “The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension”, Trans, of Amer. Math. Soc., 249 (1979), 409–424 | DOI | MR | Zbl

[40] Martinson L. K., “Rasprostranenie teplovoi volny v nelineinoi srede s pogloscheniem”, Prikl. mekhanika i tekhn. fiz., 1979, no. 4, 36–39

[41] Kershner R., “O povedenii temperaturnykh frontov v sredakh s nelineinoi teploprovodnostyu pri nalichii pogloscheniya. II”, Acta Math. Acad. Sci. Hungaricae, 35 (1980), 13–22 | DOI | MR | Zbl

[42] Kalashnikov A. S., “O kharaktere rasprostraneniya vozmuschenii v protsessakh, opisyvaemykh kvazilineinymi vyrozhdayuschimisya parabolicheskimi uravneniyami”, Tr. semin. im. I. G. Petrovskogo, 1, MGU, M., 1975, 135–144

[43] Kershner R., “O nekotorykh svoistvakh obobschennykh reshenii kvazilineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii”, Acta Math. Acad. Sci. Hungaricae, 32 (1978), 301–330 | DOI | Zbl

[44] Martinson L. K., Pavlov K. B., “Nestatsionarnye sdvigovye techeniya provodyaschei zhidkosti so stepennym reologicheskim zakonom”, Magnitnaya gidrodinamika, 1971, no. 2, 50–58

[45] Golaido S. I., Martinson L. K, Pavlov K. B., “Magnitogidrodinamicheskie sdvigovye techeniya zhidkosti po stepennym reologicheskim zakonom v usloviyakh poperechnogo snosa”, Magnitnaya gidrodinamika, 1974, no. 2, 58–62

[46] Granin I. S, Martinson L. K., “O dvizhenii granitsy nositelya obobschennogo resheniya v zadachakh magnitnoi reologii”, Magnitnaya gidrodinamika, 1978, no. 1, 13–16

[47] Granin I. S, Martinson L. K., “Dvizhenie fronta teplovoi volny v nelineinoi srede s pogloscheniem”, Inzh.-fiz. zhurnal, 34:4 (1980), 728–731

[48] Pokrovskii L. D., Taranenko S. N., “Ob usloviyakh prostranstvennoi lokalizatsii reshenii nelineinogo uravneniya teploprovodnosti”, ZhVM i MF, 22:3 (1982), 747–751 | MR | Zbl

[49] Volosov K. A., Fedotov I. A., “Asimptoticheskie predstavleniya resheniya kvazilineinogo parabolicheskogo uravneniya v okrestnosti fronta”, ZhVM i MF, 23:5 (1983), 1249–1253 | MR | Zbl

[50] Kersner R., Filtration with absorption: necessary and sufficient condition for the propagation, of perturbations to have finite velocity, Preprint Issl. Inst. Vychisl. Tekhn. i Avt. Vengerskoi AN, MN/8, 1982 | MR

[51] Kersner R., “Nonlinear heat conduction with absorption: space localization and extinction in finite time”, SIAM J. Appl. Math., 43 (1983), 1274–1285 | DOI | MR | Zbl

[52] Sacks P. E., “The initial and boundary value problem for a class of degenerate parabolic equations”, Comm. in Part. Differ. Equat., 8 (1983), 693–733 | DOI | MR | Zbl

[53] Bertch M., “A class of degenerate diffusion equations with a singular term”, Nonlinear Anal., Theory, Meth. and Appl., 7 (1983), 117–127 | DOI | MR

[54] Dibenedetto E., “Continuity of weak solutions to a general porous medium equation”, Indiana Univ. Math. J., 32 (1983), 83–118 | DOI | MR | Zbl

[55] Evans L. C, Knerr B. F., “Instantaneous shrinking of the support of nonegative solutions to certain nonlinear parabolic equations and variational inequalities”, Illinois J. of Math., 23 (1979), 153–166 | MR | Zbl

[56] Alikakos N. D., Rostamian R., “Stabilization of solutions of the equation $\partial u/\partial t= \Delta\Phi(u)-\beta(u)$”, Nonlinear Anal., Theory, Meth. and Appl., 6 (1982), 637–647 | DOI | MR | Zbl

[57] Bertch M., Nanby T., Peletier L. A., “Decay of solutions of a degenerate nonlinear diffusion equation”, Nonlinear Anal., Theory, Meth. and Appl., 6 (1982), 539–554 | DOI | MR

[58] Kersner R., “On the behavior when $t$ growths to infinity of generalized solutions of a degenerate quasilinear parabolic equation”, Acta Math. Acad. Sci. Hungaricae, 34 (1979), 157–163 | DOI | MR | Zbl

[59] Bertsch M., Kersner R., Peletier L. A., “Sur le comportement de la frontière libre dans une équation en théorie de la filtration”, Comp. Rendus Acad. Sci. Paris, série I, 295 (1982), 63–66 | MR | Zbl

[60] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, t. I, Nauka, M., 1973

[61] Volpert A. I., Khudyaev S. I., Analiz v klassakh razryvnykh funktsii i uravneniya matematicheskoi fiziki, Nauka, M., 1975 | MR

[62] Bartman A. B., “Raspad plazmy pri opticheskom proboe gazov”, Tezisy dokl. V Vsesoyuz. sovesch. po nered. vzaimod. optich. izluch. s veschestvom, Izd-vo GOI im. S. I. Vavilova, L., 1981, 264–265

[63] Aronson D. G., Benilan P., “Régularité des solutions de l'équation des milieux poreux dans $R^N$”, Comp. Rendus Acad. Sci. Paris, serie A, 288 (1979), 103–105 | MR | Zbl

[64] Kamin S., “Similar solutions and the asymptotics of filtration equations”, Arch. for Ration. Mech. and Anal., 60 (1976), 171–183 | MR | Zbl

[65] Friedman A., Kamin S., “The asymptotics behavior of gas in an $n$-dimensional porous medium”, Trans. of Amer. Math. Soc., 262 (1980), 551–563 | DOI | MR | Zbl

[66] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd–vo LGU, L., 1950

[67] Likht M. K., “O rasprostranenii vozmuschenii v zadachakh, svyazannykh s vyrozhdayuschimisya kvazilineinymi uravneniyami parabolicheskogo tipa”, Differents. uravneniya, 2:7 (1969), 953–957

[68] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., “Ob odnom, podkhode k sravneniyu reshenii parabolicheskikh uravnenii”, ZhVM i MF, 19:6 (1979), 1451–1461 | MR | Zbl

[69] Galaktionov V. A., “Dva metoda sravneniya reshenii parabolicheskikh uravnenii”, DAN SSSR, 251:4 (1980), 832–835 | MR | Zbl

[70] Galaktionov V. A., Kurdyumov S. P., Samarskii A. A., “O metode statsionarnykh sostoyanii dlya nelineinykh evolyutsionnykh parabolicheskikh zadach”, DAN SSSR, 278:6 (1984), 1296–1300 | MR | Zbl

[71] Galaktionov V. A., “Ob usloviyakh lokalizatsii neogranichennykh reshenii kvazilineinykh parabolicheskikh uravnenii”, DAN SSSR, 264:6 (1982), 1035–1040 | MR | Zbl

[72] Galaktionov V. A., “Ob usloviyakh nesuschestvovaniya v tselom i lokalizatsii reshenii, zadachi Koshi dlya odnogo klassa nelineinykh parabolicheskikh uravnenii”, ZhVM i MF, 23:6 (1983), 1341–1354 | MR

[73] Samarskii A. A., Zmitrenko N. V., Kurdyumov S. P., Mikhailov A. P., “Effekt metastabilnoi lokalizatsii tepla v srede s nelineinoi teploprovodnostyu”, DAN SSSR, 223:6 (1975), 1344–1347

[74] Zmitrenko N. V., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., Metastabilnaya lokalizatsiya tepla v srede s nelineinoi teploprovodnostyu i usloviya proyavleniya ee v eksperimente, Preprint No 103, IPM AN SSSR, M., 1977 | MR

[75] Brezis H., Friedman A., “Nonlinear parabolic equations involving measures as initial conditions”, J. Math, pures et appl., 62 (1983), 73–97 | MR | Zbl

[76] Kamin S., Peletier L. A., Large time behavior of solutions of the heat equation with absorption, Preprint Math. Inst. Univ. Leiden. Nr. 25, 1983 | MR

[77] Kamin S., Peletier L. A., Source-type solutions of degenerate diffusion equations with absorption, Preprint Math. Inst. Univ. Leiden. Nr. 6, 1984 | MR

[78] Gmira A., Veron L., “Large time behavior of the solution of a semilinear parabolic equation in $R^N$”, J. Diff. Equat., 53 (1984), 258–276 | DOI | MR | Zbl