Exact local estimates for the supports of solutions in problems for non-linear parabolic equations

U. G. Abdullaev

Sbornik. Mathematics, Tome 186 (1995) no. 8, pp. 1085-1106

Résumé

The phenomenon of instantaneous shrinking of the support in the Cauchy problem for non-linear parabolic equations with a positive initial function that is infinitesimal as $|x|\to\infty$ is considered. Exact local estimates for the boundary of the support of the solutions are proved. For example, the exact asymptotic formula $$ u_0\bigl(\eta^\pm(t)\bigr)\sim\bigl[(1-\beta)t\bigr]^{1/(1-\beta)}, \qquad t\to 0, $$ holds for the solution of the equation $u_t=(u^nu_x)_x-u^\beta$, $0<\beta<1$, $n\geqslant 1-\beta$, where $\eta^+(t)=\sup\bigl\{x:u(x,t)>0\bigr\}$ and $\eta^-(t)=\inf\bigl\{x:u(x,t)>0\bigr\}$.

Bibliographie

[1] Kalashnikov A. S., “Nekotorye voprosy kachestvennoi teorii nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii vtorogo poryadka”, UMN, 42:2 (254) (1987), 135–176 | MR | Zbl

[2] Galaktionov V. A., Kurdyumov S. P., Samarskii A. A., “Ob asimptoticheskikh “sobstvennykh funktsiyakh” zadachi Koshi dlya odnogo nelineinogo parabolicheskogo uravneniya”, Matem. sb., 126 (168) (1985), 435–472 | MR

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

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

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

[6] Bertch M. A., “A class of degenerate diffision equations with a singular term”, Nonl. Anal. Theor. Math. and Appl., 7 (1983), 117–127 | DOI | MR

[7] 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

[8] Abdullaev U. G., “Rezhimy s obostreniem v zadachakh dlya kvazilineinogo uravneniya teplomassoperenosa s konvektsiei”, ZhVM i MF, 31:3 (1991), 462–466 | MR | Zbl

[9] Abdullaev U. G., “O neogranichennykh resheniyakh nelineinogo uravneniya teploprovodnosti so stokom”, ZhVM i MF, 32:8 (1992), 1244–1257 | MR | Zbl

[10] Abdullaev U. G., “O lokalizatsii neogranichennykh reshenii nelineinogo uravneniya teploprovodnosti s perenosom”, DAN, 329:5 (1993), 535–537 | Zbl

[11] Abdullaev U. G., “On the space localization of unbounded perturbations in nonlinear heat conduction with transfer”, Applied Math. Letters, 7:6 (1994), 91–95 | DOI | MR | Zbl

[12] Abdullaev U. G., “O suschestvovanii neogranichennykh reshenii nelineinogo uravneniya teploprovodnosti so stokom”, ZhVM i MF, 33:2 (1993), 232–245 | MR | Zbl

[13] Evans L. C., Knerr B. F., “Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities”, Ill. J. Math., 23:1 (1979), 153–166 | MR | Zbl

[14] Kalashnikov A. S., “O zavisimosti svoistv reshenii parabolicheskikh uravnenii v neogranichennykh oblastyakh ot povedeniya koeffitsientov na beskonechnosti”, Matem. sb., 125:3 (1984), 398–409 | MR | Zbl

[15] Gilding B. H., Kersner R., “Instantaneous shrinking in nonlinear diffusion convection”, Proc. Amer. Math. Soc., 109 (1990), 385–394 | DOI | MR | Zbl

[16] Kalashnikov A. S., “O povedenii vblizi nachalnoi giperploskosti reshenii zadachi Koshi dlya parabolicheskikh sistem s nelineinoi dissipatsiei”, Tr. sem. im. I. G. Petrovskogo, 16, Izd-vo Mosk. un-ta, M., 1992, 106–113