Some properties of a generalized solution of the second boundary-value problem for a parabolic equation
Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 225-244
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We establish some properties (bounds in $L_p(\Omega)$ for $p\geqslant1$, absolute continuity of the entropy, etc.) for a solution in a cylindrical domain $\Omega\times\{t>0\}$, where $\Omega$ is an arbitrary, unbounded in general, domain of $R_n$, of the second boundary-value problem for a linear uniformly-parabolic equation of second order: \begin{gather*} \frac{\partial u}{\partial t}=\sum_{i,j=1}^n\frac\partial{\partial x_i}\biggl(a_{ij}(t,x)\frac{\partial u(t,x)}{\partial x_j}\biggr), \\ \frac{\partial u}{\partial N}\bigg|_{x\in\partial\Omega}=0,\qquad u\big|_{t=0}=\varphi(x),\quad\varphi(x)\in L_2(\Omega). \end{gather*} Bibliography: 2 titles.
@article{SM_1975_26_2_a4,
author = {A. K. Gushchin},
title = {Some properties of a~generalized solution of the second boundary-value problem for a~parabolic equation},
journal = {Sbornik. Mathematics},
pages = {225--244},
year = {1975},
volume = {26},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1975_26_2_a4/}
}
A. K. Gushchin. Some properties of a generalized solution of the second boundary-value problem for a parabolic equation. Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 225-244. http://geodesic.mathdoc.fr/item/SM_1975_26_2_a4/
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[2] J. Nash, “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80 (1958), 931–954 ; Matematika, 4:1 (1960), 31–52 | DOI | MR | Zbl