Some properties of a~generalized solution of the second boundary-value problem for a~parabolic equation

A. K. Gushchin

Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 225-244

Résumé

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.