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This article is cited in 2 scientific papers (total in 2 papers)
Evolution of the support of a solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order
A. E. Shishkov Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
Abstract:
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 $t\to\infty$.
Received: 03.11.1994
Citation:
A. E. Shishkov, “Evolution of the support of a solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order”, Mat. Sb., 186:12 (1995), 151–172; Sb. Math., 186:12 (1995), 1843–1864
Linking options:
https://www.mathnet.ru/eng/sm96https://doi.org/10.1070/SM1995v186n12ABEH000096 https://www.mathnet.ru/eng/sm/v186/i12/p151
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Abstract page: | 361 | Russian version PDF: | 93 | English version PDF: | 12 | References: | 62 | First page: | 1 |
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