Evolution of the support of a~solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order

A. E. Shishkov

Sbornik. Mathematics, Tome 186 (1995) no. 12, pp. 1843-1864

Résumé

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: $$ \int_{|x|\tau}u_0(x)^2\,dx\tau^{n+\frac{2mp}{p-2}} \qquad \forall\,\tau>\tau'>0, \quad c\infty. $$ 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 $t\to\infty$.