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This article is cited in 19 scientific papers (total in 19 papers)
Dynamics of the supports of energy solutions of mixed problems for quasi-linear parabolic equations of arbitrary order
A. E. Shishkov, A. G. Shchelkov Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
Abstract:
We study the geometry of the supports of solutions of the Cauchy–Dirichlet problem for a wide class of quasi-linear degenerate parabolic equations of any order, whose model representative is the equation of non-stationary filtration with non-linear absorption:
$$
\dfrac{\partial}{\partial t}\bigl(|u|^{q-1}u\bigr)-\sum_{i=1}^n\,\dfrac{\partial}{\partial x_i}\biggl(|D_x u|^{p-1}\dfrac{\partial u}{\partial x_i}\biggr)+b_0|u|^{\lambda-1}u=0,\qquad
b_0>0,\quad n\geqslant 1.
$$
In the cases when $0<\lambda<p\leqslant q$ and $0<\lambda<q<p$, which correspond to “fast” and “slow” diffusion, we find conditions on the behaviour of the initial function
$u_0(x)\in L_{q+1}(\Omega)$ in a neighbourhood of the boundary of its support that ensure the effect of finite and infinite inertia of the support of an arbitrary energy solution; these conditions are, in a certain sense, exact. We establish a condition for the reverse motion of the front of the support boundary.
Received: 15.03.1996
Citation:
A. E. Shishkov, A. G. Shchelkov, “Dynamics of the supports of energy solutions of mixed problems for quasi-linear parabolic equations of arbitrary order”, Izv. Math., 62:3 (1998), 601–626
Linking options:
https://www.mathnet.ru/eng/im200https://doi.org/10.1070/im1998v062n03ABEH000200 https://www.mathnet.ru/eng/im/v62/i3/p175
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Abstract page: | 434 | Russian version PDF: | 223 | English version PDF: | 23 | References: | 44 | First page: | 1 |
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