Asymptotic behaviour of supports of solutions of

D. A. Sapronov   ; A. E. Shishkov

Sbornik. Mathematics, Tome 197 (2006) no. 5, pp. 753-790

Résumé

We study the phenomenon of the finiteness of the rate of propagation of the supports of generalized energy solutions of mixed problems for a broad class of doubly degenerate parabolic equations of high order; a model example here is the equation $$ (|u|^{q-1}u)_t+(-1)^m \sum_{|\alpha|=m} D_x^\alpha(|D_x^\alpha u|^{p-1} D_x^\alpha u)+(|u|^{\lambda-1}u)_{x_1}=0, $$ $m \geqslant 1$, $p>0$, $q>0$, $\lambda>0$. Bounds (that are sharp in a certain sense) for the early evolution of the supports of solutions (in particular, of the ‘right’ and the ‘left’ fronts of the solutions), which depend on local properties of the initial function and the parameters of the equation, are established. The behaviour of the supports for large times is also studied. Bibliography: 31 titles.

Bibliographie

[1] O. A. Oleinik, Kalashnikov A. S., Chzhou Yui-lin, “Zadacha Koshi i kraevye zadachi dlya uravnenii tipa nestatsionarnoi filtratsii”, Izv. AN SSSR. Ser. matem., 22:5 (1958), 667–704 | MR | Zbl

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

[3] E. Di Benedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993 | MR | Zbl

[4] A. S. Kalashnikov, “O kharaktere rasprostraneniya vozmuschenii v protsessakh, opisyvaemykh kvazilineinymi parabolicheskimi uravneniyami”, Trudy sem. im. I. G. Petrovskogo, no. 1, Izd-vo MGU, M., 1975, 135–144

[5] B. H. Gilding, L. A. Peletier, “The Cauchy problem for an equation in the theory of infiltration”, Arch. Ration. Mech. Anal., 61 (1976), 127–140 | DOI | MR | Zbl

[6] J. I. Diaz, R. Kersner, “On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium”, J. Differential Equations, 69 (1987), 368–403 | DOI | MR | Zbl

[7] B. H. Gilding, “Improved theory for a nonlinear degenerate parabolic equation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16:2 (1989), 165–224 | MR | Zbl

[8] Zh.-L. Lions, Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[9] Yu. A. Dubinskii, “Nelineinye ellipticheskie i parabolicheskie uravneniya”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 9, VINITI, M., 1976, 5–130

[10] O. Grange, F. Mignot, “Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires”, J. Funct. Anal., 11:1 (1972), 77–92 | DOI | MR | Zbl

[11] F. Bernis, “Existence results for doubly nonlinear higher order parabolic equations on unbounded domains”, Math. Ann., 279:3 (1988), 373–394 | DOI | MR | Zbl

[12] H. W. Alt, S. Luckhaus, “Quasilinear elliptic-parabolic differential equations”, Math. Z., 183 (1983), 311–341 | DOI | MR | Zbl

[13] Ph. Benilan, P. Wittbold, “On mild and weak solutions of elliptic-parabolic problems”, Adv. Differential Equations, 1:6 (1996), 1053–1073 | MR | Zbl

[14] L. Alvarez, J. I. Diaz, R. Kersner, “On the initial growth of the interfaces in nonlinear diffusion-convection processes”, Nonlinear diffusion equations and their equilibrium states I, eds. W.-M. Ni, L. A. Peletier, J. Serrin, Springer-Verlag, New York, 1987, 1–20 | MR

[15] L. Alvarez, J. I. Diaz, “Sufficiant and necessary initial mass conditions for the existence of a waiting time in nonlinear-convection processes”, J. Math. Anal. Appl., 155 (1991), 378–393 | DOI | MR

[16] B. H. Gilding, “Properties of solutions of an equation in the theory of infiltration”, Arch. Ration. Mech. Anal., 65 (1977), 203–225 | DOI | MR | Zbl

[17] R. E. Grundy, “Asymptotic solution of a model nonlinear convective diffusion eauation”, IMA J. Appl. Math., 31 (1983), 121–137 | DOI | MR

[18] Ph. Laurencot, F. Simondon, Sourse-type solutions to porous medium equations with convection, Preprint No. 96/17, Universite de Franche Comte | MR

[19] Ph. Laurencot, F. Simondon, Sourse-type solutions to porous medium equations with convection II, Preprint No. 96/33, Universite de Franche Comte | MR

[20] J. I. Diaz, R. Kersner, “Non existence d'une des frontières libres dans une équation dégénerée en théorie de la filtration”, C. R. Acad. Sci. Paris Ser. I Math., 296 (1983), 505–508 | MR | Zbl

[21] B. H. Gilding, “The occurrence of interfaces in nonlinear diffusion-advection processes”, Arch. Ration. Mech. Anal., 100:3 (1988), 243–263 | DOI | MR | Zbl

[22] B. H. Gilding, R. Kersner, “Instantaneous shrinking in nonlinear diffusion-convection”, Proc. Amer. Math. Soc., 109:2 (1990), 385–394 | DOI | MR | Zbl

[23] A. E. Shishkov, “Instantaneous shrinking phenomenon for solutions of higher-dimensional nonlinear diffusion-convection equations”, Methods Funct. Anal. Topology, 5:3 (1999), 54–76 | MR | Zbl

[24] M. Guedda, D. Hilhorst, C. Picard, “The one-dimensional porous medium equation with convection: Continuous differentiability of interfaces after the waiting time”, Appl. Math. Lett., 5:1 (1992), 59–62 | DOI | MR | Zbl

[25] S. Bonafede, G. R. Cirmi, A. F. Tedeev, “Finite speed of propagation for the porous media equation with lower order terms”, Discrete Contin. Dynam. Systems, 6:2 (2000), 305–314 | DOI | MR | Zbl

[26] B. Gilding, A. E. Shishkov, “Rasprostranenie nositelei reshenii uravnenii tipa mnogomernoi nelineinoi diffuzii s konvektsiei”, UMN, 53:4 (1998), 142

[27] D. A. Sapronov, A. E. Shishkov, “Rasprostranenie vozmuschenii v kvazilineinykh mnogomernykh parabolicheskikh uravneniyakh s konvektivnym chlenom”, Ukr. matem. zhurn., 53:7 (2001), 953–969 | MR | Zbl

[28] E. Zuazua, “Weakly nonlinear large time behavior in scalar convection-diffusion equations”, Differential Integral Equations, 6:6 (1993), 1481–1491 | MR | Zbl

[29] S. Claudi, F. R. Guarguaglini, “Large time behaviour for the heat equation with absorption and convection”, Adv. Appl. Math., 16:4 (1995), 377–401 | DOI | MR | Zbl

[30] F. Bernis, “Qualitative properties for some nonlinear higher order degenerate parabolic equations”, Houston J. Math., 14:3 (1988), 319–352 | MR | Zbl

[31] O. A. Oleinik, G. A. Iosifyan, “Analog printsipa Sen-Venana i edinstvennost reshenii kraevykh zadach v neogranichennykh oblastyakh dlya parabolicheskikh uravnenii”, UMN, 31:6 (1976), 142–166 | MR | Zbl