Geodesic
Evolution of the support of a solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order
Sbornik. Mathematics, Tome 186 (1995) no. 12, pp. 1843-1864 
A. E. Shishkov. Evolution of the support of a solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order. Sbornik. Mathematics, Tome 186 (1995) no. 12, pp. 1843-1864. http://geodesic.mathdoc.fr/item/SM_1995_186_12_a7/
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     title = {Evolution of the support of a~solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order},
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The Cauchy problem for a quasi-linear degenerate parabolic equation in divergence from with energy space $L_p\bigl(0,T;W_{p,\operatorname{loc}}^m(\mathbb R^n)\bigr)$, $m\geqslant 1$, $p>2$, $n\geqslant 1$ and with initial function $u_0(x)\in L_{2,\operatorname{loc}}(\mathbb R^n)$ is considered. The existence of a generalized solution $u(x,t)$ is proved for $u_0(x)$ growing at infinity at the rate: $$ \int_{|x|<\tau}u_0(x)^2\,dx<c\tau^{n+\frac{2mp}{p-2}} \qquad \forall\,\tau>\tau'>0, \quad c<\infty. $$ For more sever constraints on the growth of $u_0(x)$ several fairly wide uniqueness classes for the above-mentioned solution are discovered. The question of describing the geometry of the domain $\Omega(t)\equiv\mathbb R^n\setminus\operatorname{supp}_xu(x,t)$ for $\Omega_0\equiv\mathbb R^n\setminus\operatorname{supp}u_0(x)\ne\varnothing$ is considered. In case when the domain $\Omega_0$ is unbounded, estimates in terms of the global properties of the initial function $u_0(x)$ are established that characterize the geometry of $\Omega(t)$ as $t\to\infty$.

[1] Kalashnikov A. S., “O rasprostranenii vozmuschenii v protsessakh, opisyvaemykh kvazilineinymi vyrozhdayuschimisya parabolicheskimi uravneniyami”, Trudy sem. im. I. G. Petrovskogo, no. 1, 1975, 135–144 | Zbl

[2] Kershner R., “Povedenie temperaturnogo fronta v sredakh s nelineinoi teploprovodnostyu pri nalichii pogloscheniya”, Vestnik MGU. Ser. 1. Matem., mekh., 1978, no. 5, 44–51 | MR | Zbl

[3] Antontsev S. N., “O lokalizatsii reshenii nelineinykh vyrozhdayuschikhsya ellipticheskikh i parabolicheskikh uravnenii”, DAN SSSR, 260:6 (1981), 1289–1293 | MR | Zbl

[4] Brezis H., Friedman A., “Estimates on the support of solutions of parabolic variational inequalities”, Illinois J. Math., 20:1 (1976), 82–97 | MR | Zbl

[5] Peletier L. A., “A necessary and sufficient condition for the existence of an interface in flow through porous media”, Arch. Ration. Mech. Anal., 56 (1974), 183–190 | DOI | MR | Zbl

[6] Diaz J. I., Veron L., “Local vanishing properties of solutions of elliptic and parabolic quasilinear equations”, Trans. Amer. Math. Soc., 290:2 (1985), 787–814 | DOI | MR | Zbl

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

[8] Barenblatt G. I., “O nekotorykh neustanovivshikhsya dvizheniyakh zhidkosti i gaza v poristoi srede”, Prikl. matem. i mekh., 16:1 (1952), 67–78 | MR | Zbl

[9] Kalashnikov A. S., “O zadache Koshi v klassakh rastuschikh funktsii dlya kvazilineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii vtorogo poryadka”, Differents. uravn., 9:4 (1973), 682–691 | MR | Zbl

[10] Di Benedetto E., Herrero M. A., “On the Cauchy problem and initial traces for a degenerate parabolic equation”, Trans. Amer. Math. Soc., 314:1 (1989), 197–224 | DOI | MR

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

[12] Mazya V. G., Prostranstva S. L. Soboleva, Izd-vo LGU, L., 1985 | MR

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

[14] Shishkov A. E., “Ob otsenkakh skorosti rasprostraneniya vozmuschenii v kvazilineinykh divergentnykh vyrozhdayuschikhsya parabolicheskikh uravneniyakh vysokogo poryadka”, Ukr. matem. zhurn., 44:10 (1992), 1451–1456 | MR | Zbl

[15] Shishkov A. E., “Dinamika geometrii nositelya obobschennogo resheniya kvazilineinogo divergentnogo parabolicheskogo uravneniya vysokogo poryadka”, Differents. uravneniya, 29:3 (1993), 537–547 | MR | Zbl